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Most recent proofs    These are the 100 (Unicode, GIF) or 1000 (Unicode, GIF) most recent proofs in the iset.mm database for the Intuitionistic Logic Explorer. The iset.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from the commit given on the MPE Most Recent Proofs page. The database from that commit is also available here: iset.mm.

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Last updated on 27-Apr-2025 at 6:44 AM ET.
Recent Additions to the Intuitionistic Logic Explorer
DateLabelDescription
Theorem
 
25-Apr-2025rspex 13570 Existence of the ring span. (Contributed by Jim Kingdon, 25-Apr-2025.)
(π‘Š ∈ 𝑉 β†’ (RSpanβ€˜π‘Š) ∈ V)
 
25-Apr-2025lspex 13493 Existence of the span of a set of vectors. (Contributed by Jim Kingdon, 25-Apr-2025.)
(π‘Š ∈ 𝑋 β†’ (LSpanβ€˜π‘Š) ∈ V)
 
25-Apr-2025eqgex 13091 The left coset equivalence relation exists. (Contributed by Jim Kingdon, 25-Apr-2025.)
((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ π‘Š) β†’ (𝐺 ~QG 𝑆) ∈ V)
 
25-Apr-2025qusex 12752 Existence of a quotient structure. (Contributed by Jim Kingdon, 25-Apr-2025.)
((𝑅 ∈ 𝑉 ∧ ∼ ∈ π‘Š) β†’ (𝑅 /s ∼ ) ∈ V)
 
23-Apr-20251dom1el 14904 If a set is dominated by one, then any two of its elements are equal. (Contributed by Jim Kingdon, 23-Apr-2025.)
((𝐴 β‰Ό 1o ∧ 𝐡 ∈ 𝐴 ∧ 𝐢 ∈ 𝐴) β†’ 𝐡 = 𝐢)
 
22-Apr-2025mulgex 12996 Existence of the group multiple operation. (Contributed by Jim Kingdon, 22-Apr-2025.)
(𝐺 ∈ 𝑉 β†’ (.gβ€˜πΊ) ∈ V)
 
18-Apr-2025lidlmex 13571 Existence of the set a left ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
𝐼 = (LIdealβ€˜π‘Š)    β‡’   (π‘ˆ ∈ 𝐼 β†’ π‘Š ∈ V)
 
18-Apr-2025lsslsp 13527 Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) Terms in the equation were swapped as proposed by NM on 15-Mar-2015. (Revised by AV, 18-Apr-2025.)
𝑋 = (π‘Š β†Ύs π‘ˆ)    &   π‘€ = (LSpanβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘‹)    &   πΏ = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝐿 ∧ 𝐺 βŠ† π‘ˆ) β†’ (π‘β€˜πΊ) = (π‘€β€˜πΊ))
 
16-Apr-2025sraex 13544 Existence of a subring algebra. (Contributed by Jim Kingdon, 16-Apr-2025.)
(πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))    &   (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑋)    β‡’   (πœ‘ β†’ 𝐴 ∈ V)
 
10-Apr-2025cndcap 14969 Real number trichotomy is equivalent to decidability of complex number apartness. (Contributed by Jim Kingdon, 10-Apr-2025.)
(βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ < 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 < π‘₯) ↔ βˆ€π‘§ ∈ β„‚ βˆ€π‘€ ∈ β„‚ DECID 𝑧 # 𝑀)
 
20-Mar-2025ccoslid 12693 Slot property of comp. (Contributed by Jim Kingdon, 20-Mar-2025.)
(comp = Slot (compβ€˜ndx) ∧ (compβ€˜ndx) ∈ β„•)
 
20-Mar-2025homslid 12691 Slot property of Hom. (Contributed by Jim Kingdon, 20-Mar-2025.)
(Hom = Slot (Hom β€˜ndx) ∧ (Hom β€˜ndx) ∈ β„•)
 
19-Mar-2025ptex 12719 Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.)
(𝐹 ∈ 𝑉 β†’ (∏tβ€˜πΉ) ∈ V)
 
18-Mar-2025prdsex 12724 Existence of the structure product. (Contributed by Jim Kingdon, 18-Mar-2025.)
((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ π‘Š) β†’ (𝑆Xs𝑅) ∈ V)
 
13-Mar-2025imasex 12732 Existence of the image structure. (Contributed by Jim Kingdon, 13-Mar-2025.)
((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ π‘Š) β†’ (𝐹 β€œs 𝑅) ∈ V)
 
11-Mar-2025imasival 12733 Value of an image structure. The is a lemma for the theorems imasbas 12734, imasplusg 12735, and imasmulr 12736 and should not be needed once they are proved. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Jim Kingdon, 11-Mar-2025.) (New usage is discouraged.)
(πœ‘ β†’ π‘ˆ = (𝐹 β€œs 𝑅))    &   (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘…))    &    + = (+gβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘…)    &   (πœ‘ β†’ ✚ = βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝 + π‘ž))⟩})    &   (πœ‘ β†’ βˆ™ = βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝 Γ— π‘ž))⟩})    &   (πœ‘ β†’ 𝐹:𝑉–onto→𝐡)    &   (πœ‘ β†’ 𝑅 ∈ 𝑍)    β‡’   (πœ‘ β†’ π‘ˆ = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), ✚ ⟩, ⟨(.rβ€˜ndx), βˆ™ ⟩})
 
8-Mar-2025subgex 13046 The class of subgroups of a group is a set. (Contributed by Jim Kingdon, 8-Mar-2025.)
(𝐺 ∈ Grp β†’ (SubGrpβ€˜πΊ) ∈ V)
 
28-Feb-2025ringressid 13249 A ring restricted to its base set is a ring. It will usually be the original ring exactly, of course, but to show that needs additional conditions such as those in strressid 12533. (Contributed by Jim Kingdon, 28-Feb-2025.)
𝐡 = (Baseβ€˜πΊ)    β‡’   (𝐺 ∈ Ring β†’ (𝐺 β†Ύs 𝐡) ∈ Ring)
 
28-Feb-2025grpressid 12938 A group restricted to its base set is a group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 12533. (Contributed by Jim Kingdon, 28-Feb-2025.)
𝐡 = (Baseβ€˜πΊ)    β‡’   (𝐺 ∈ Grp β†’ (𝐺 β†Ύs 𝐡) ∈ Grp)
 
26-Feb-2025strext 12567 Extending the upper range of a structure. This works because when we say that a structure has components in 𝐴...𝐢 we are not saying that every slot in that range is present, just that all the slots that are present are within that range. (Contributed by Jim Kingdon, 26-Feb-2025.)
(πœ‘ β†’ 𝐹 Struct ⟨𝐴, 𝐡⟩)    &   (πœ‘ β†’ 𝐢 ∈ (β„€β‰₯β€˜π΅))    β‡’   (πœ‘ β†’ 𝐹 Struct ⟨𝐴, 𝐢⟩)
 
23-Feb-2025ltlenmkv 14980 If < can be expressed as holding exactly when ≀ holds and the values are not equal, then the analytic Markov's Principle applies. (To get the regular Markov's Principle, combine with neapmkv 14978). (Contributed by Jim Kingdon, 23-Feb-2025.)
(βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ < 𝑦 ↔ (π‘₯ ≀ 𝑦 ∧ 𝑦 β‰  π‘₯)) β†’ βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ β‰  𝑦 β†’ π‘₯ # 𝑦))
 
23-Feb-2025neap0mkv 14979 The analytic Markov principle can be expressed either with two arbitrary real numbers, or one arbitrary number and zero. (Contributed by Jim Kingdon, 23-Feb-2025.)
(βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ β‰  𝑦 β†’ π‘₯ # 𝑦) ↔ βˆ€π‘₯ ∈ ℝ (π‘₯ β‰  0 β†’ π‘₯ # 0))
 
23-Feb-2025lringuplu 13348 If the sum of two elements of a local ring is invertible, then at least one of the summands must be invertible. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘…))    &   (πœ‘ β†’ + = (+gβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ LRing)    &   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 ∈ π‘ˆ ∨ π‘Œ ∈ π‘ˆ))
 
23-Feb-2025lringnz 13347 A local ring is a nonzero ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
1 = (1rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ LRing β†’ 1 β‰  0 )
 
23-Feb-2025lringring 13346 A local ring is a ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
(𝑅 ∈ LRing β†’ 𝑅 ∈ Ring)
 
23-Feb-2025lringnzr 13345 A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
(𝑅 ∈ LRing β†’ 𝑅 ∈ NzRing)
 
23-Feb-2025islring 13344 The predicate "is a local ring". (Contributed by SN, 23-Feb-2025.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    β‡’   (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯ + 𝑦) = 1 β†’ (π‘₯ ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ))))
 
23-Feb-2025df-lring 13343 A local ring is a nonzero ring where for any two elements summing to one, at least one is invertible. Any field is a local ring; the ring of integers is an example of a ring which is not a local ring. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
LRing = {π‘Ÿ ∈ NzRing ∣ βˆ€π‘₯ ∈ (Baseβ€˜π‘Ÿ)βˆ€π‘¦ ∈ (Baseβ€˜π‘Ÿ)((π‘₯(+gβ€˜π‘Ÿ)𝑦) = (1rβ€˜π‘Ÿ) β†’ (π‘₯ ∈ (Unitβ€˜π‘Ÿ) ∨ 𝑦 ∈ (Unitβ€˜π‘Ÿ)))}
 
23-Feb-202501eq0ring 13341 If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019.) (Proof shortened by SN, 23-Feb-2025.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 0 = 1 ) β†’ 𝐡 = { 0 })
 
23-Feb-2025nzrring 13338 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof shortened by SN, 23-Feb-2025.)
(𝑅 ∈ NzRing β†’ 𝑅 ∈ Ring)
 
21-Feb-2025dftap2 7253 Tight apartness with the apartness properties from df-pap 7250 expanded. (Contributed by Jim Kingdon, 21-Feb-2025.)
(𝑅 TAp 𝐴 ↔ (𝑅 βŠ† (𝐴 Γ— 𝐴) ∧ (βˆ€π‘₯ ∈ 𝐴 Β¬ π‘₯𝑅π‘₯ ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯𝑅𝑦 β†’ 𝑦𝑅π‘₯)) ∧ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 βˆ€π‘§ ∈ 𝐴 (π‘₯𝑅𝑦 β†’ (π‘₯𝑅𝑧 ∨ 𝑦𝑅𝑧)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (Β¬ π‘₯𝑅𝑦 β†’ π‘₯ = 𝑦))))
 
20-Feb-2025aprap 13387 The relation given by df-apr 13382 for a local ring is an apartness relation. (Contributed by Jim Kingdon, 20-Feb-2025.)
(𝑅 ∈ LRing β†’ (#rβ€˜π‘…) Ap (Baseβ€˜π‘…))
 
20-Feb-2025setscomd 12506 Different components can be set in any order. (Contributed by Jim Kingdon, 20-Feb-2025.)
(πœ‘ β†’ 𝐴 ∈ π‘Œ)    &   (πœ‘ β†’ 𝐡 ∈ 𝑍)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    &   (πœ‘ β†’ 𝐢 ∈ π‘Š)    &   (πœ‘ β†’ 𝐷 ∈ 𝑋)    β‡’   (πœ‘ β†’ ((𝑆 sSet ⟨𝐴, 𝐢⟩) sSet ⟨𝐡, 𝐷⟩) = ((𝑆 sSet ⟨𝐡, 𝐷⟩) sSet ⟨𝐴, 𝐢⟩))
 
17-Feb-2025aprcotr 13386 The apartness relation given by df-apr 13382 for a local ring is cotransitive. (Contributed by Jim Kingdon, 17-Feb-2025.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ # = (#rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ LRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 # π‘Œ β†’ (𝑋 # 𝑍 ∨ π‘Œ # 𝑍)))
 
17-Feb-2025aprsym 13385 The apartness relation given by df-apr 13382 for a ring is symmetric. (Contributed by Jim Kingdon, 17-Feb-2025.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ # = (#rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 # π‘Œ β†’ π‘Œ # 𝑋))
 
17-Feb-2025aprval 13383 Expand Definition df-apr 13382. (Contributed by Jim Kingdon, 17-Feb-2025.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ # = (#rβ€˜π‘…))    &   (πœ‘ β†’ βˆ’ = (-gβ€˜π‘…))    &   (πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 # π‘Œ ↔ (𝑋 βˆ’ π‘Œ) ∈ π‘ˆ))
 
16-Feb-2025aprirr 13384 The apartness relation given by df-apr 13382 for a nonzero ring is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2025.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ # = (#rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ (1rβ€˜π‘…) β‰  (0gβ€˜π‘…))    β‡’   (πœ‘ β†’ Β¬ 𝑋 # 𝑋)
 
16-Feb-2025aptap 8610 Complex apartness (as defined at df-ap 8542) is a tight apartness (as defined at df-tap 7252). (Contributed by Jim Kingdon, 16-Feb-2025.)
# TAp β„‚
 
15-Feb-2025tapeq2 7255 Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.)
(𝐴 = 𝐡 β†’ (𝑅 TAp 𝐴 ↔ 𝑅 TAp 𝐡))
 
14-Feb-2025exmidmotap 7263 The proposition that every class has at most one tight apartness is equivalent to excluded middle. (Contributed by Jim Kingdon, 14-Feb-2025.)
(EXMID ↔ βˆ€π‘₯βˆƒ*π‘Ÿ π‘Ÿ TAp π‘₯)
 
14-Feb-2025exmidapne 7262 Excluded middle implies there is only one tight apartness on any class, namely negated equality. (Contributed by Jim Kingdon, 14-Feb-2025.)
(EXMID β†’ (𝑅 TAp 𝐴 ↔ 𝑅 = {βŸ¨π‘’, π‘£βŸ© ∣ ((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑒 β‰  𝑣)}))
 
14-Feb-2025df-pap 7250 Apartness predicate. A relation 𝑅 is an apartness if it is irreflexive, symmetric, and cotransitive. (Contributed by Jim Kingdon, 14-Feb-2025.)
(𝑅 Ap 𝐴 ↔ ((𝑅 βŠ† (𝐴 Γ— 𝐴) ∧ βˆ€π‘₯ ∈ 𝐴 Β¬ π‘₯𝑅π‘₯) ∧ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯𝑅𝑦 β†’ 𝑦𝑅π‘₯) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 βˆ€π‘§ ∈ 𝐴 (π‘₯𝑅𝑦 β†’ (π‘₯𝑅𝑧 ∨ 𝑦𝑅𝑧)))))
 
13-Feb-2025df-apr 13382 The relation between elements whose difference is invertible, which for a local ring is an apartness relation by aprap 13387. (Contributed by Jim Kingdon, 13-Feb-2025.)
#r = (𝑀 ∈ V ↦ {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜π‘€) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) ∧ (π‘₯(-gβ€˜π‘€)𝑦) ∈ (Unitβ€˜π‘€))})
 
8-Feb-20252oneel 7258 βˆ… and 1o are two unequal elements of 2o. (Contributed by Jim Kingdon, 8-Feb-2025.)
βŸ¨βˆ…, 1o⟩ ∈ {βŸ¨π‘’, π‘£βŸ© ∣ ((𝑒 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑒 β‰  𝑣)}
 
8-Feb-2025tapeq1 7254 Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.)
(𝑅 = 𝑆 β†’ (𝑅 TAp 𝐴 ↔ 𝑆 TAp 𝐴))
 
6-Feb-20252omotap 7261 If there is at most one tight apartness on 2o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.)
(βˆƒ*π‘Ÿ π‘Ÿ TAp 2o β†’ EXMID)
 
6-Feb-20252omotaplemst 7260 Lemma for 2omotap 7261. (Contributed by Jim Kingdon, 6-Feb-2025.)
((βˆƒ*π‘Ÿ π‘Ÿ TAp 2o ∧ Β¬ Β¬ πœ‘) β†’ πœ‘)
 
6-Feb-20252omotaplemap 7259 Lemma for 2omotap 7261. (Contributed by Jim Kingdon, 6-Feb-2025.)
(Β¬ Β¬ πœ‘ β†’ {βŸ¨π‘’, π‘£βŸ© ∣ ((𝑒 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (πœ‘ ∧ 𝑒 β‰  𝑣))} TAp 2o)
 
6-Feb-20252onetap 7257 Negated equality is a tight apartness on 2o. (Contributed by Jim Kingdon, 6-Feb-2025.)
{βŸ¨π‘’, π‘£βŸ© ∣ ((𝑒 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑒 β‰  𝑣)} TAp 2o
 
5-Feb-2025netap 7256 Negated equality on a set with decidable equality is a tight apartness. (Contributed by Jim Kingdon, 5-Feb-2025.)
(βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 β†’ {βŸ¨π‘’, π‘£βŸ© ∣ ((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑒 β‰  𝑣)} TAp 𝐴)
 
5-Feb-2025df-tap 7252 Tight apartness predicate. A relation 𝑅 is a tight apartness if it is irreflexive, symmetric, cotransitive, and tight. (Contributed by Jim Kingdon, 5-Feb-2025.)
(𝑅 TAp 𝐴 ↔ (𝑅 Ap 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (Β¬ π‘₯𝑅𝑦 β†’ π‘₯ = 𝑦)))
 
1-Feb-2025mulgnn0cld 13014 Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 13009. (Contributed by SN, 1-Feb-2025.)
𝐡 = (Baseβ€˜πΊ)    &    Β· = (.gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Mnd)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑁 Β· 𝑋) ∈ 𝐡)
 
31-Jan-20250subg 13069 The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.) (Proof shortened by SN, 31-Jan-2025.)
0 = (0gβ€˜πΊ)    β‡’   (𝐺 ∈ Grp β†’ { 0 } ∈ (SubGrpβ€˜πΊ))
 
28-Jan-2025dvdsrex 13278 Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)
(𝑅 ∈ SRing β†’ (βˆ₯rβ€˜π‘…) ∈ V)
 
24-Jan-2025reldvdsrsrg 13272 The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2025.)
(𝑅 ∈ SRing β†’ Rel (βˆ₯rβ€˜π‘…))
 
18-Jan-2025rerecapb 8803 A real number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 18-Jan-2025.)
(𝐴 ∈ ℝ β†’ (𝐴 # 0 ↔ βˆƒπ‘₯ ∈ ℝ (𝐴 Β· π‘₯) = 1))
 
18-Jan-2025recapb 8631 A complex number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies), generalized from real to complex numbers. (Contributed by Jim Kingdon, 18-Jan-2025.)
(𝐴 ∈ β„‚ β†’ (𝐴 # 0 ↔ βˆƒπ‘₯ ∈ β„‚ (𝐴 Β· π‘₯) = 1))
 
17-Jan-2025ressval3d 12534 Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)
𝑅 = (𝑆 β†Ύs 𝐴)    &   π΅ = (Baseβ€˜π‘†)    &   πΈ = (Baseβ€˜ndx)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ Fun 𝑆)    &   (πœ‘ β†’ 𝐸 ∈ dom 𝑆)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐡)    β‡’   (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
 
17-Jan-2025strressid 12533 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 17-Jan-2025.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š Struct βŸ¨π‘€, π‘βŸ©)    &   (πœ‘ β†’ Fun π‘Š)    &   (πœ‘ β†’ (Baseβ€˜ndx) ∈ dom π‘Š)    β‡’   (πœ‘ β†’ (π‘Š β†Ύs 𝐡) = π‘Š)
 
16-Jan-2025ressex 12528 Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ (π‘Š β†Ύs 𝐴) ∈ V)
 
16-Jan-2025ressvalsets 12527 Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩))
 
10-Jan-2025opprex 13256 Existence of the opposite ring. If you know that 𝑅 is a ring, see opprring 13260. (Contributed by Jim Kingdon, 10-Jan-2025.)
𝑂 = (opprβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ 𝑂 ∈ V)
 
10-Jan-2025mgpex 13146 Existence of the multiplication group. If 𝑅 is known to be a semiring, see srgmgp 13162. (Contributed by Jim Kingdon, 10-Jan-2025.)
𝑀 = (mulGrpβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ 𝑀 ∈ V)
 
5-Jan-2025imbibi 252 The antecedent of one side of a biconditional can be moved out of the biconditional to become the antecedent of the remaining biconditional. (Contributed by BJ, 1-Jan-2025.) (Proof shortened by Wolf Lammen, 5-Jan-2025.)
(((πœ‘ β†’ πœ“) ↔ πœ’) β†’ (πœ‘ β†’ (πœ“ ↔ πœ’)))
 
1-Jan-2025snss 3729 The singleton of an element of a class is a subset of the class (inference form of snssg 3728). Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.) (Proof shortened by BJ, 1-Jan-2025.)
𝐴 ∈ V    β‡’   (𝐴 ∈ 𝐡 ↔ {𝐴} βŠ† 𝐡)
 
1-Jan-2025snssg 3728 The singleton formed on a set is included in a class if and only if the set is an element of that class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) (Proof shortened by BJ, 1-Jan-2025.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ 𝐡 ↔ {𝐴} βŠ† 𝐡))
 
1-Jan-2025snssb 3727 Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.)
({𝐴} βŠ† 𝐡 ↔ (𝐴 ∈ V β†’ 𝐴 ∈ 𝐡))
 
9-Dec-2024nninfwlpoim 7179 Decidable equality for β„•βˆž implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.)
(βˆ€π‘₯ ∈ β„•βˆž βˆ€π‘¦ ∈ β„•βˆž DECID π‘₯ = 𝑦 β†’ Ο‰ ∈ WOmni)
 
8-Dec-2024nninfwlpoimlemdc 7178 Lemma for nninfwlpoim 7179. (Contributed by Jim Kingdon, 8-Dec-2024.)
(πœ‘ β†’ 𝐹:Ο‰βŸΆ2o)    &   πΊ = (𝑖 ∈ Ο‰ ↦ if(βˆƒπ‘₯ ∈ suc 𝑖(πΉβ€˜π‘₯) = βˆ…, βˆ…, 1o))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„•βˆž βˆ€π‘¦ ∈ β„•βˆž DECID π‘₯ = 𝑦)    β‡’   (πœ‘ β†’ DECID βˆ€π‘› ∈ Ο‰ (πΉβ€˜π‘›) = 1o)
 
8-Dec-2024nninfwlpoimlemginf 7177 Lemma for nninfwlpoim 7179. (Contributed by Jim Kingdon, 8-Dec-2024.)
(πœ‘ β†’ 𝐹:Ο‰βŸΆ2o)    &   πΊ = (𝑖 ∈ Ο‰ ↦ if(βˆƒπ‘₯ ∈ suc 𝑖(πΉβ€˜π‘₯) = βˆ…, βˆ…, 1o))    β‡’   (πœ‘ β†’ (𝐺 = (𝑖 ∈ Ο‰ ↦ 1o) ↔ βˆ€π‘› ∈ Ο‰ (πΉβ€˜π‘›) = 1o))
 
8-Dec-2024nninfwlpoimlemg 7176 Lemma for nninfwlpoim 7179. (Contributed by Jim Kingdon, 8-Dec-2024.)
(πœ‘ β†’ 𝐹:Ο‰βŸΆ2o)    &   πΊ = (𝑖 ∈ Ο‰ ↦ if(βˆƒπ‘₯ ∈ suc 𝑖(πΉβ€˜π‘₯) = βˆ…, βˆ…, 1o))    β‡’   (πœ‘ β†’ 𝐺 ∈ β„•βˆž)
 
7-Dec-2024nninfwlpor 7175 The Weak Limited Principle of Omniscience (WLPO) implies that equality for β„•βˆž is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.)
(Ο‰ ∈ WOmni β†’ βˆ€π‘₯ ∈ β„•βˆž βˆ€π‘¦ ∈ β„•βˆž DECID π‘₯ = 𝑦)
 
7-Dec-2024nninfwlporlem 7174 Lemma for nninfwlpor 7175. The result. (Contributed by Jim Kingdon, 7-Dec-2024.)
(πœ‘ β†’ 𝑋:Ο‰βŸΆ2o)    &   (πœ‘ β†’ π‘Œ:Ο‰βŸΆ2o)    &   π· = (𝑖 ∈ Ο‰ ↦ if((π‘‹β€˜π‘–) = (π‘Œβ€˜π‘–), 1o, βˆ…))    &   (πœ‘ β†’ Ο‰ ∈ WOmni)    β‡’   (πœ‘ β†’ DECID 𝑋 = π‘Œ)
 
6-Dec-2024nninfwlporlemd 7173 Given two countably infinite sequences of zeroes and ones, they are equal if and only if a sequence formed by pointwise comparing them is all ones. (Contributed by Jim Kingdon, 6-Dec-2024.)
(πœ‘ β†’ 𝑋:Ο‰βŸΆ2o)    &   (πœ‘ β†’ π‘Œ:Ο‰βŸΆ2o)    &   π· = (𝑖 ∈ Ο‰ ↦ if((π‘‹β€˜π‘–) = (π‘Œβ€˜π‘–), 1o, βˆ…))    β‡’   (πœ‘ β†’ (𝑋 = π‘Œ ↔ 𝐷 = (𝑖 ∈ Ο‰ ↦ 1o)))
 
3-Dec-2024nninfwlpo 7180 Decidability of equality for β„•βˆž is equivalent to the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 3-Dec-2024.)
(βˆ€π‘₯ ∈ β„•βˆž βˆ€π‘¦ ∈ β„•βˆž DECID π‘₯ = 𝑦 ↔ Ο‰ ∈ WOmni)
 
3-Dec-2024nninfdcinf 7172 The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of β„•βˆž equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.)
(πœ‘ β†’ Ο‰ ∈ WOmni)    &   (πœ‘ β†’ 𝑁 ∈ β„•βˆž)    β‡’   (πœ‘ β†’ DECID 𝑁 = (𝑖 ∈ Ο‰ ↦ 1o))
 
28-Nov-2024basmexd 12525 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 28-Nov-2024.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΊ))    &   (πœ‘ β†’ 𝐴 ∈ 𝐡)    β‡’   (πœ‘ β†’ 𝐺 ∈ V)
 
22-Nov-2024eliotaeu 5207 An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.)
(𝐴 ∈ (β„©π‘₯πœ‘) β†’ βˆƒ!π‘₯πœ‘)
 
22-Nov-2024eliota 5206 An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.)
(𝐴 ∈ (β„©π‘₯πœ‘) ↔ βˆƒπ‘¦(𝐴 ∈ 𝑦 ∧ βˆ€π‘₯(πœ‘ ↔ π‘₯ = 𝑦)))
 
18-Nov-2024basmex 12524 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 18-Nov-2024.)
𝐡 = (Baseβ€˜πΊ)    β‡’   (𝐴 ∈ 𝐡 β†’ 𝐺 ∈ V)
 
12-Nov-2024sravscag 13541 The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)
(πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))    &   (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑋)    β‡’   (πœ‘ β†’ (.rβ€˜π‘Š) = ( ·𝑠 β€˜π΄))
 
12-Nov-2024srascag 13540 The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)
(πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))    &   (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑋)    β‡’   (πœ‘ β†’ (π‘Š β†Ύs 𝑆) = (Scalarβ€˜π΄))
 
12-Nov-2024slotsdifipndx 12636 The slot for the scalar is not the index of other slots. (Contributed by AV, 12-Nov-2024.)
(( ·𝑠 β€˜ndx) β‰  (Β·π‘–β€˜ndx) ∧ (Scalarβ€˜ndx) β‰  (Β·π‘–β€˜ndx))
 
11-Nov-2024bj-con1st 14664 Contraposition when the antecedent is a negated stable proposition. See con1dc 856. (Contributed by BJ, 11-Nov-2024.)
(STAB πœ‘ β†’ ((Β¬ πœ‘ β†’ πœ“) β†’ (Β¬ πœ“ β†’ πœ‘)))
 
11-Nov-2024slotsdifdsndx 12682 The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.)
((*π‘Ÿβ€˜ndx) β‰  (distβ€˜ndx) ∧ (leβ€˜ndx) β‰  (distβ€˜ndx))
 
11-Nov-2024slotsdifplendx 12671 The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.)
((*π‘Ÿβ€˜ndx) β‰  (leβ€˜ndx) ∧ (TopSetβ€˜ndx) β‰  (leβ€˜ndx))
 
11-Nov-2024tsetndxnstarvndx 12655 The slot for the topology is not the slot for the involution in an extensible structure. (Contributed by AV, 11-Nov-2024.)
(TopSetβ€˜ndx) β‰  (*π‘Ÿβ€˜ndx)
 
11-Nov-2024const 852 Contraposition when the antecedent is a negated stable proposition. See comment of condc 853. (Contributed by BJ, 18-Nov-2023.) (Proof shortened by BJ, 11-Nov-2024.)
(STAB πœ‘ β†’ ((Β¬ πœ‘ β†’ Β¬ πœ“) β†’ (πœ“ β†’ πœ‘)))
 
10-Nov-2024slotsdifunifndx 12689 The index of the slot for the uniform set is not the index of other slots. (Contributed by AV, 10-Nov-2024.)
(((+gβ€˜ndx) β‰  (UnifSetβ€˜ndx) ∧ (.rβ€˜ndx) β‰  (UnifSetβ€˜ndx) ∧ (*π‘Ÿβ€˜ndx) β‰  (UnifSetβ€˜ndx)) ∧ ((leβ€˜ndx) β‰  (UnifSetβ€˜ndx) ∧ (distβ€˜ndx) β‰  (UnifSetβ€˜ndx)))
 
7-Nov-2024ressbasd 12530 Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
(πœ‘ β†’ 𝑅 = (π‘Š β†Ύs 𝐴))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑋)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝐴 ∩ 𝐡) = (Baseβ€˜π‘…))
 
6-Nov-2024oppraddg 13259 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
𝑂 = (opprβ€˜π‘…)    &    + = (+gβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ + = (+gβ€˜π‘‚))
 
6-Nov-2024opprbasg 13258 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
𝑂 = (opprβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ 𝐡 = (Baseβ€˜π‘‚))
 
6-Nov-2024opprsllem 13257 Lemma for opprbasg 13258 and oppraddg 13259. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)
𝑂 = (opprβ€˜π‘…)    &   (𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   (πΈβ€˜ndx) β‰  (.rβ€˜ndx)    β‡’   (𝑅 ∈ 𝑉 β†’ (πΈβ€˜π‘…) = (πΈβ€˜π‘‚))
 
4-Nov-2024lgsfvalg 14567 Value of the function 𝐹 which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by Jim Kingdon, 4-Nov-2024.)
𝐹 = (𝑛 ∈ β„• ↦ if(𝑛 ∈ β„™, (if(𝑛 = 2, if(2 βˆ₯ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 βˆ’ 1) / 2)) + 1) mod 𝑛) βˆ’ 1))↑(𝑛 pCnt 𝑁)), 1))    β‡’   ((𝐴 ∈ β„€ ∧ 𝑁 ∈ β„• ∧ 𝑀 ∈ β„•) β†’ (πΉβ€˜π‘€) = if(𝑀 ∈ β„™, (if(𝑀 = 2, if(2 βˆ₯ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑀 βˆ’ 1) / 2)) + 1) mod 𝑀) βˆ’ 1))↑(𝑀 pCnt 𝑁)), 1))
 
3-Nov-2024zlmmulrg 13654 Ring operation of a β„€-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
π‘Š = (β„€Modβ€˜πΊ)    &    Β· = (.rβ€˜πΊ)    β‡’   (𝐺 ∈ 𝑉 β†’ Β· = (.rβ€˜π‘Š))
 
3-Nov-2024zlmplusgg 13653 Group operation of a β„€-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
π‘Š = (β„€Modβ€˜πΊ)    &    + = (+gβ€˜πΊ)    β‡’   (𝐺 ∈ 𝑉 β†’ + = (+gβ€˜π‘Š))
 
3-Nov-2024zlmbasg 13652 Base set of a β„€-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
π‘Š = (β„€Modβ€˜πΊ)    &   π΅ = (Baseβ€˜πΊ)    β‡’   (𝐺 ∈ 𝑉 β†’ 𝐡 = (Baseβ€˜π‘Š))
 
3-Nov-2024zlmlemg 13651 Lemma for zlmbasg 13652 and zlmplusgg 13653. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
π‘Š = (β„€Modβ€˜πΊ)    &   πΈ = Slot (πΈβ€˜ndx)    &   (πΈβ€˜ndx) ∈ β„•    &   (πΈβ€˜ndx) β‰  (Scalarβ€˜ndx)    &   (πΈβ€˜ndx) β‰  ( ·𝑠 β€˜ndx)    β‡’   (𝐺 ∈ 𝑉 β†’ (πΈβ€˜πΊ) = (πΈβ€˜π‘Š))
 
2-Nov-2024zlmsca 13655 Scalar ring of a β„€-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) (Proof shortened by AV, 2-Nov-2024.)
π‘Š = (β„€Modβ€˜πΊ)    β‡’   (𝐺 ∈ 𝑉 β†’ β„€ring = (Scalarβ€˜π‘Š))
 
1-Nov-2024plendxnvscandx 12670 The slot for the "less than or equal to" ordering is not the slot for the scalar product in an extensible structure. (Contributed by AV, 1-Nov-2024.)
(leβ€˜ndx) β‰  ( ·𝑠 β€˜ndx)
 
1-Nov-2024plendxnscandx 12669 The slot for the "less than or equal to" ordering is not the slot for the scalar in an extensible structure. (Contributed by AV, 1-Nov-2024.)
(leβ€˜ndx) β‰  (Scalarβ€˜ndx)
 
1-Nov-2024plendxnmulrndx 12668 The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 1-Nov-2024.)
(leβ€˜ndx) β‰  (.rβ€˜ndx)
 
1-Nov-2024qsqeqor 10634 The squares of two rational numbers are equal iff one number equals the other or its negative. (Contributed by Jim Kingdon, 1-Nov-2024.)
((𝐴 ∈ β„š ∧ 𝐡 ∈ β„š) β†’ ((𝐴↑2) = (𝐡↑2) ↔ (𝐴 = 𝐡 ∨ 𝐴 = -𝐡)))
 
31-Oct-2024dsndxnmulrndx 12679 The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(distβ€˜ndx) β‰  (.rβ€˜ndx)
 
31-Oct-2024tsetndxnmulrndx 12654 The slot for the topology is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(TopSetβ€˜ndx) β‰  (.rβ€˜ndx)
 
31-Oct-2024tsetndxnbasendx 12652 The slot for the topology is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 31-Oct-2024.)
(TopSetβ€˜ndx) β‰  (Baseβ€˜ndx)
 
31-Oct-2024basendxlttsetndx 12651 The index of the slot for the base set is less then the index of the slot for the topology in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(Baseβ€˜ndx) < (TopSetβ€˜ndx)
 
31-Oct-2024tsetndxnn 12650 The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 31-Oct-2024.)
(TopSetβ€˜ndx) ∈ β„•
 
30-Oct-2024plendxnbasendx 12666 The slot for the order is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 30-Oct-2024.)
(leβ€˜ndx) β‰  (Baseβ€˜ndx)
 
30-Oct-2024basendxltplendx 12665 The index value of the Base slot is less than the index value of the le slot. (Contributed by AV, 30-Oct-2024.)
(Baseβ€˜ndx) < (leβ€˜ndx)
 
30-Oct-2024plendxnn 12664 The index value of the order slot is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 30-Oct-2024.)
(leβ€˜ndx) ∈ β„•
 
29-Oct-2024sradsg 13546 Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))    &   (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑋)    β‡’   (πœ‘ β†’ (distβ€˜π‘Š) = (distβ€˜π΄))
 
29-Oct-2024sratsetg 13543 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))    &   (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑋)    β‡’   (πœ‘ β†’ (TopSetβ€˜π‘Š) = (TopSetβ€˜π΄))
 
29-Oct-2024sramulrg 13539 Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))    &   (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑋)    β‡’   (πœ‘ β†’ (.rβ€˜π‘Š) = (.rβ€˜π΄))
 
29-Oct-2024sraaddgg 13538 Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))    &   (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑋)    β‡’   (πœ‘ β†’ (+gβ€˜π‘Š) = (+gβ€˜π΄))
 
29-Oct-2024srabaseg 13537 Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))    &   (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑋)    β‡’   (πœ‘ β†’ (Baseβ€˜π‘Š) = (Baseβ€˜π΄))
 
29-Oct-2024sralemg 13536 Lemma for srabaseg 13537 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))    &   (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑋)    &   (𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   (Scalarβ€˜ndx) β‰  (πΈβ€˜ndx)    &   ( ·𝑠 β€˜ndx) β‰  (πΈβ€˜ndx)    &   (Β·π‘–β€˜ndx) β‰  (πΈβ€˜ndx)    β‡’   (πœ‘ β†’ (πΈβ€˜π‘Š) = (πΈβ€˜π΄))
 
29-Oct-2024dsndxntsetndx 12681 The slot for the distance function is not the slot for the topology in an extensible structure. (Contributed by AV, 29-Oct-2024.)
(distβ€˜ndx) β‰  (TopSetβ€˜ndx)
 
29-Oct-2024slotsdnscsi 12680 The slots Scalar, ·𝑠 and ·𝑖 are different from the slot dist. (Contributed by AV, 29-Oct-2024.)
((distβ€˜ndx) β‰  (Scalarβ€˜ndx) ∧ (distβ€˜ndx) β‰  ( ·𝑠 β€˜ndx) ∧ (distβ€˜ndx) β‰  (Β·π‘–β€˜ndx))
 
29-Oct-2024slotstnscsi 12656 The slots Scalar, ·𝑠 and ·𝑖 are different from the slot TopSet. (Contributed by AV, 29-Oct-2024.)
((TopSetβ€˜ndx) β‰  (Scalarβ€˜ndx) ∧ (TopSetβ€˜ndx) β‰  ( ·𝑠 β€˜ndx) ∧ (TopSetβ€˜ndx) β‰  (Β·π‘–β€˜ndx))
 
29-Oct-2024ipndxnmulrndx 12635 The slot for the inner product is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
(Β·π‘–β€˜ndx) β‰  (.rβ€˜ndx)
 
29-Oct-2024ipndxnplusgndx 12634 The slot for the inner product is not the slot for the group operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
(Β·π‘–β€˜ndx) β‰  (+gβ€˜ndx)
 
29-Oct-2024vscandxnmulrndx 12622 The slot for the scalar product is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
( ·𝑠 β€˜ndx) β‰  (.rβ€˜ndx)
 
29-Oct-2024scandxnmulrndx 12617 The slot for the scalar field is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
(Scalarβ€˜ndx) β‰  (.rβ€˜ndx)
 
29-Oct-2024fiubnn 10813 A finite set of natural numbers has an upper bound which is a a natural number. (Contributed by Jim Kingdon, 29-Oct-2024.)
((𝐴 βŠ† β„• ∧ 𝐴 ∈ Fin) β†’ βˆƒπ‘₯ ∈ β„• βˆ€π‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)
 
29-Oct-2024fiubz 10812 A finite set of integers has an upper bound which is an integer. (Contributed by Jim Kingdon, 29-Oct-2024.)
((𝐴 βŠ† β„€ ∧ 𝐴 ∈ Fin) β†’ βˆƒπ‘₯ ∈ β„€ βˆ€π‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)
 
29-Oct-2024fiubm 10811 Lemma for fiubz 10812 and fiubnn 10813. A general form of those theorems. (Contributed by Jim Kingdon, 29-Oct-2024.)
(πœ‘ β†’ 𝐴 βŠ† 𝐡)    &   (πœ‘ β†’ 𝐡 βŠ† β„š)    &   (πœ‘ β†’ 𝐢 ∈ 𝐡)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)
 
28-Oct-2024unifndxntsetndx 12688 The slot for the uniform set is not the slot for the topology in an extensible structure. (Contributed by AV, 28-Oct-2024.)
(UnifSetβ€˜ndx) β‰  (TopSetβ€˜ndx)
 
28-Oct-2024basendxltunifndx 12686 The index of the slot for the base set is less then the index of the slot for the uniform set in an extensible structure. (Contributed by AV, 28-Oct-2024.)
(Baseβ€˜ndx) < (UnifSetβ€˜ndx)
 
28-Oct-2024unifndxnn 12685 The index of the slot for the uniform set in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
(UnifSetβ€˜ndx) ∈ β„•
 
28-Oct-2024dsndxnbasendx 12677 The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 28-Oct-2024.)
(distβ€˜ndx) β‰  (Baseβ€˜ndx)
 
28-Oct-2024basendxltdsndx 12676 The index of the slot for the base set is less then the index of the slot for the distance in an extensible structure. (Contributed by AV, 28-Oct-2024.)
(Baseβ€˜ndx) < (distβ€˜ndx)
 
28-Oct-2024dsndxnn 12675 The index of the slot for the distance in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
(distβ€˜ndx) ∈ β„•
 
27-Oct-2024bj-nnst 14656 Double negation of stability of a formula. Intuitionistic logic refutes unstability (but does not prove stability) of any formula. This theorem can also be proved in classical refutability calculus (see https://us.metamath.org/mpeuni/bj-peircestab.html) but not in minimal calculus (see https://us.metamath.org/mpeuni/bj-stabpeirce.html). See nnnotnotr 14903 for the version not using the definition of stability. (Contributed by BJ, 9-Oct-2019.) Prove it in ( β†’ , Β¬ ) -intuitionistic calculus with definitions (uses of ax-ia1 106, ax-ia2 107, ax-ia3 108 are via sylibr 134, necessary for definition unpackaging), and in ( β†’ , ↔ , Β¬ )-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)
Β¬ Β¬ STAB πœ‘
 
27-Oct-2024bj-imnimnn 14651 If a formula is implied by both a formula and its negation, then it is not refutable. There is another proof using the inference associated with bj-nnclavius 14650 as its last step. (Contributed by BJ, 27-Oct-2024.)
(πœ‘ β†’ πœ“)    &   (Β¬ πœ‘ β†’ πœ“)    β‡’    Β¬ Β¬ πœ“
 
25-Oct-2024nnwosdc 12043 Well-ordering principle: any inhabited decidable set of positive integers has a least element (schema form). (Contributed by NM, 17-Aug-2001.) (Revised by Jim Kingdon, 25-Oct-2024.)
(π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   ((βˆƒπ‘₯ ∈ β„• πœ‘ ∧ βˆ€π‘₯ ∈ β„• DECID πœ‘) β†’ βˆƒπ‘₯ ∈ β„• (πœ‘ ∧ βˆ€π‘¦ ∈ β„• (πœ“ β†’ π‘₯ ≀ 𝑦)))
 
23-Oct-2024nnwodc 12040 Well-ordering principle: any inhabited decidable set of positive integers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34. (Contributed by NM, 17-Aug-2001.) (Revised by Jim Kingdon, 23-Oct-2024.)
((𝐴 βŠ† β„• ∧ βˆƒπ‘€ 𝑀 ∈ 𝐴 ∧ βˆ€π‘— ∈ β„• DECID 𝑗 ∈ 𝐴) β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 π‘₯ ≀ 𝑦)
 
22-Oct-2024uzwodc 12041 Well-ordering principle: any inhabited decidable subset of an upper set of integers has a least element. (Contributed by NM, 8-Oct-2005.) (Revised by Jim Kingdon, 22-Oct-2024.)
((𝑆 βŠ† (β„€β‰₯β€˜π‘€) ∧ βˆƒπ‘₯ π‘₯ ∈ 𝑆 ∧ βˆ€π‘₯ ∈ (β„€β‰₯β€˜π‘€)DECID π‘₯ ∈ 𝑆) β†’ βˆƒπ‘— ∈ 𝑆 βˆ€π‘˜ ∈ 𝑆 𝑗 ≀ π‘˜)
 
21-Oct-2024nnnotnotr 14903 Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 850, it can be proved with reference only to implication and negation (that is, without use of disjunction). (Contributed by Jim Kingdon, 21-Oct-2024.)
Β¬ Β¬ (Β¬ Β¬ πœ‘ β†’ πœ‘)
 
21-Oct-2024unifndxnbasendx 12687 The slot for the uniform set is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(UnifSetβ€˜ndx) β‰  (Baseβ€˜ndx)
 
21-Oct-2024ipndxnbasendx 12633 The slot for the inner product is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(Β·π‘–β€˜ndx) β‰  (Baseβ€˜ndx)
 
21-Oct-2024scandxnbasendx 12615 The slot for the scalar is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(Scalarβ€˜ndx) β‰  (Baseβ€˜ndx)
 
20-Oct-2024isprm5lem 12144 Lemma for isprm5 12145. The interesting direction (showing that one only needs to check prime divisors up to the square root of 𝑃). (Contributed by Jim Kingdon, 20-Oct-2024.)
(πœ‘ β†’ 𝑃 ∈ (β„€β‰₯β€˜2))    &   (πœ‘ β†’ βˆ€π‘§ ∈ β„™ ((𝑧↑2) ≀ 𝑃 β†’ Β¬ 𝑧 βˆ₯ 𝑃))    &   (πœ‘ β†’ 𝑋 ∈ (2...(𝑃 βˆ’ 1)))    β‡’   (πœ‘ β†’ Β¬ 𝑋 βˆ₯ 𝑃)
 
19-Oct-2024resseqnbasd 12535 The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.)
𝑅 = (π‘Š β†Ύs 𝐴)    &   πΆ = (πΈβ€˜π‘Š)    &   (𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   (πΈβ€˜ndx) β‰  (Baseβ€˜ndx)    &   (πœ‘ β†’ π‘Š ∈ 𝑋)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    β‡’   (πœ‘ β†’ 𝐢 = (πΈβ€˜π‘…))
 
18-Oct-2024rmodislmod 13452 The right module 𝑅 induces a left module 𝐿 by replacing the scalar multiplication with a reversed multiplication if the scalar ring is commutative. The hypothesis "rmodislmod.r" is a definition of a right module analogous to Definition df-lmod 13390 of a left module, see also islmod 13392. (Contributed by AV, 3-Dec-2021.) (Proof shortened by AV, 18-Oct-2024.)
𝑉 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘…)    &   πΉ = (Scalarβ€˜π‘…)    &   πΎ = (Baseβ€˜πΉ)    &    ⨣ = (+gβ€˜πΉ)    &    Γ— = (.rβ€˜πΉ)    &    1 = (1rβ€˜πΉ)    &   (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ βˆ€π‘ž ∈ 𝐾 βˆ€π‘Ÿ ∈ 𝐾 βˆ€π‘₯ ∈ 𝑉 βˆ€π‘€ ∈ 𝑉 (((𝑀 Β· π‘Ÿ) ∈ 𝑉 ∧ ((𝑀 + π‘₯) Β· π‘Ÿ) = ((𝑀 Β· π‘Ÿ) + (π‘₯ Β· π‘Ÿ)) ∧ (𝑀 Β· (π‘ž ⨣ π‘Ÿ)) = ((𝑀 Β· π‘ž) + (𝑀 Β· π‘Ÿ))) ∧ ((𝑀 Β· (π‘ž Γ— π‘Ÿ)) = ((𝑀 Β· π‘ž) Β· π‘Ÿ) ∧ (𝑀 Β· 1 ) = 𝑀)))    &    βˆ— = (𝑠 ∈ 𝐾, 𝑣 ∈ 𝑉 ↦ (𝑣 Β· 𝑠))    &   πΏ = (𝑅 sSet ⟨( ·𝑠 β€˜ndx), βˆ— ⟩)    β‡’   (𝐹 ∈ CRing β†’ 𝐿 ∈ LMod)
 
18-Oct-2024mgpress 13152 Subgroup commutes with the multiplicative group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.)
𝑆 = (𝑅 β†Ύs 𝐴)    &   π‘€ = (mulGrpβ€˜π‘…)    β‡’   ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ (𝑀 β†Ύs 𝐴) = (mulGrpβ€˜π‘†))
 
18-Oct-2024dsndxnplusgndx 12678 The slot for the distance function is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(distβ€˜ndx) β‰  (+gβ€˜ndx)
 
18-Oct-2024plendxnplusgndx 12667 The slot for the "less than or equal to" ordering is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(leβ€˜ndx) β‰  (+gβ€˜ndx)
 
18-Oct-2024tsetndxnplusgndx 12653 The slot for the topology is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(TopSetβ€˜ndx) β‰  (+gβ€˜ndx)
 
18-Oct-2024vscandxnscandx 12623 The slot for the scalar product is not the slot for the scalar field in an extensible structure. (Contributed by AV, 18-Oct-2024.)
( ·𝑠 β€˜ndx) β‰  (Scalarβ€˜ndx)
 
18-Oct-2024vscandxnplusgndx 12621 The slot for the scalar product is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
( ·𝑠 β€˜ndx) β‰  (+gβ€˜ndx)
 
18-Oct-2024vscandxnbasendx 12620 The slot for the scalar product is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
( ·𝑠 β€˜ndx) β‰  (Baseβ€˜ndx)
 
18-Oct-2024scandxnplusgndx 12616 The slot for the scalar field is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(Scalarβ€˜ndx) β‰  (+gβ€˜ndx)
 
18-Oct-2024starvndxnmulrndx 12605 The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(*π‘Ÿβ€˜ndx) β‰  (.rβ€˜ndx)
 
18-Oct-2024starvndxnplusgndx 12604 The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(*π‘Ÿβ€˜ndx) β‰  (+gβ€˜ndx)
 
18-Oct-2024starvndxnbasendx 12603 The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(*π‘Ÿβ€˜ndx) β‰  (Baseβ€˜ndx)
 
17-Oct-2024basendxltplusgndx 12575 The index of the slot for the base set is less then the index of the slot for the group operation in an extensible structure. (Contributed by AV, 17-Oct-2024.)
(Baseβ€˜ndx) < (+gβ€˜ndx)
 
17-Oct-2024plusgndxnn 12573 The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 17-Oct-2024.)
(+gβ€˜ndx) ∈ β„•
 
17-Oct-2024elnndc 9615 Membership of an integer in β„• is decidable. (Contributed by Jim Kingdon, 17-Oct-2024.)
(𝑁 ∈ β„€ β†’ DECID 𝑁 ∈ β„•)
 
14-Oct-20242zinfmin 11254 Two ways to express the minimum of two integers. Because order of integers is decidable, we have more flexibility than for real numbers. (Contributed by Jim Kingdon, 14-Oct-2024.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ inf({𝐴, 𝐡}, ℝ, < ) = if(𝐴 ≀ 𝐡, 𝐴, 𝐡))
 
14-Oct-2024mingeb 11253 Equivalence of ≀ and being equal to the minimum of two reals. (Contributed by Jim Kingdon, 14-Oct-2024.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 ≀ 𝐡 ↔ inf({𝐴, 𝐡}, ℝ, < ) = 𝐴))
 
13-Oct-2024pcxnn0cl 12313 Extended nonnegative integer closure of the general prime count function. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝑃 ∈ β„™ ∧ 𝑁 ∈ β„€) β†’ (𝑃 pCnt 𝑁) ∈ β„•0*)
 
13-Oct-2024xnn0letri 9806 Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ β„•0* ∧ 𝐡 ∈ β„•0*) β†’ (𝐴 ≀ 𝐡 ∨ 𝐡 ≀ 𝐴))
 
13-Oct-2024xnn0dcle 9805 Decidability of ≀ for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ β„•0* ∧ 𝐡 ∈ β„•0*) β†’ DECID 𝐴 ≀ 𝐡)
 
9-Oct-2024nn0leexp2 10693 Ordering law for exponentiation. (Contributed by Jim Kingdon, 9-Oct-2024.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) ∧ 1 < 𝐴) β†’ (𝑀 ≀ 𝑁 ↔ (𝐴↑𝑀) ≀ (𝐴↑𝑁)))
 
8-Oct-2024pclemdc 12291 Lemma for the prime power pre-function's properties. (Contributed by Jim Kingdon, 8-Oct-2024.)
𝐴 = {𝑛 ∈ β„•0 ∣ (𝑃↑𝑛) βˆ₯ 𝑁}    β‡’   ((𝑃 ∈ (β„€β‰₯β€˜2) ∧ (𝑁 ∈ β„€ ∧ 𝑁 β‰  0)) β†’ βˆ€π‘₯ ∈ β„€ DECID π‘₯ ∈ 𝐴)
 
8-Oct-2024elnn0dc 9614 Membership of an integer in β„•0 is decidable. (Contributed by Jim Kingdon, 8-Oct-2024.)
(𝑁 ∈ β„€ β†’ DECID 𝑁 ∈ β„•0)
 
7-Oct-2024pclemub 12290 Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Jim Kingdon, 7-Oct-2024.)
𝐴 = {𝑛 ∈ β„•0 ∣ (𝑃↑𝑛) βˆ₯ 𝑁}    β‡’   ((𝑃 ∈ (β„€β‰₯β€˜2) ∧ (𝑁 ∈ β„€ ∧ 𝑁 β‰  0)) β†’ βˆƒπ‘₯ ∈ β„€ βˆ€π‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)
 
7-Oct-2024pclem0 12289 Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Jim Kingdon, 7-Oct-2024.)
𝐴 = {𝑛 ∈ β„•0 ∣ (𝑃↑𝑛) βˆ₯ 𝑁}    β‡’   ((𝑃 ∈ (β„€β‰₯β€˜2) ∧ (𝑁 ∈ β„€ ∧ 𝑁 β‰  0)) β†’ 0 ∈ 𝐴)
 
7-Oct-2024nn0ltexp2 10692 Special case of ltexp2 14521 which we use here because we haven't yet defined df-rpcxp 14441 which is used in the current proof of ltexp2 14521. (Contributed by Jim Kingdon, 7-Oct-2024.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) ∧ 1 < 𝐴) β†’ (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁)))
 
6-Oct-2024suprzcl2dc 11959 The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 7935.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 6-Oct-2024.)
(πœ‘ β†’ 𝐴 βŠ† β„€)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„€ DECID π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ β„€ βˆ€π‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)    &   (πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    β‡’   (πœ‘ β†’ sup(𝐴, ℝ, < ) ∈ 𝐴)
 
5-Oct-2024zsupssdc 11958 An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 7935.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
(πœ‘ β†’ 𝐴 βŠ† β„€)    &   (πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„€ DECID π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ β„€ βˆ€π‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ 𝐡 (𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 < 𝑧)))
 
5-Oct-2024suprzubdc 11956 The supremum of a bounded-above decidable set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
(πœ‘ β†’ 𝐴 βŠ† β„€)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„€ DECID π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ β„€ βˆ€π‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    β‡’   (πœ‘ β†’ 𝐡 ≀ sup(𝐴, ℝ, < ))
 
1-Oct-2024infex2g 7036 Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
(𝐴 ∈ 𝐢 β†’ inf(𝐡, 𝐴, 𝑅) ∈ V)
 
30-Sep-2024unbendc 12458 An unbounded decidable set of positive integers is infinite. (Contributed by NM, 5-May-2005.) (Revised by Jim Kingdon, 30-Sep-2024.)
((𝐴 βŠ† β„• ∧ βˆ€π‘₯ ∈ β„• DECID π‘₯ ∈ 𝐴 ∧ βˆ€π‘š ∈ β„• βˆƒπ‘› ∈ 𝐴 π‘š < 𝑛) β†’ 𝐴 β‰ˆ β„•)
 
30-Sep-2024prmdc 12133 Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.)
(𝑁 ∈ β„• β†’ DECID 𝑁 ∈ β„™)
 
30-Sep-2024dcfi 6983 Decidability of a family of propositions indexed by a finite set. (Contributed by Jim Kingdon, 30-Sep-2024.)
((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 DECID πœ‘) β†’ DECID βˆ€π‘₯ ∈ 𝐴 πœ‘)
 
29-Sep-2024ssnnct 12451 A decidable subset of β„• is countable. (Contributed by Jim Kingdon, 29-Sep-2024.)
((𝐴 βŠ† β„• ∧ βˆ€π‘₯ ∈ β„• DECID π‘₯ ∈ 𝐴) β†’ βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o))
 
29-Sep-2024ssnnctlemct 12450 Lemma for ssnnct 12451. The result. (Contributed by Jim Kingdon, 29-Sep-2024.)
𝐺 = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 1)    β‡’   ((𝐴 βŠ† β„• ∧ βˆ€π‘₯ ∈ β„• DECID π‘₯ ∈ 𝐴) β†’ βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o))
 
28-Sep-2024nninfdcex 11957 A decidable set of natural numbers has an infimum. (Contributed by Jim Kingdon, 28-Sep-2024.)
(πœ‘ β†’ 𝐴 βŠ† β„•)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„• DECID π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘¦ 𝑦 ∈ 𝐴)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ 𝑦 < π‘₯ ∧ βˆ€π‘¦ ∈ ℝ (π‘₯ < 𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 𝑧 < 𝑦)))
 
27-Sep-2024infregelbex 9601 Any lower bound of a set of real numbers with an infimum is less than or equal to the infimum. (Contributed by Jim Kingdon, 27-Sep-2024.)
(πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ 𝑦 < π‘₯ ∧ βˆ€π‘¦ ∈ ℝ (π‘₯ < 𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 𝑧 < 𝑦)))    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (𝐡 ≀ inf(𝐴, ℝ, < ) ↔ βˆ€π‘§ ∈ 𝐴 𝐡 ≀ 𝑧))
 
26-Sep-2024nninfdclemp1 12454 Lemma for nninfdc 12457. Each element of the sequence 𝐹 is greater than the previous element. (Contributed by Jim Kingdon, 26-Sep-2024.)
(πœ‘ β†’ 𝐴 βŠ† β„•)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„• DECID π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘š ∈ β„• βˆƒπ‘› ∈ 𝐴 π‘š < 𝑛)    &   (πœ‘ β†’ (𝐽 ∈ 𝐴 ∧ 1 < 𝐽))    &   πΉ = seq1((𝑦 ∈ β„•, 𝑧 ∈ β„• ↦ inf((𝐴 ∩ (β„€β‰₯β€˜(𝑦 + 1))), ℝ, < )), (𝑖 ∈ β„• ↦ 𝐽))    &   (πœ‘ β†’ π‘ˆ ∈ β„•)    β‡’   (πœ‘ β†’ (πΉβ€˜π‘ˆ) < (πΉβ€˜(π‘ˆ + 1)))
 
26-Sep-2024nnminle 12039 The infimum of a decidable subset of the natural numbers is less than an element of the set. The infimum is also a minimum as shown at nnmindc 12038. (Contributed by Jim Kingdon, 26-Sep-2024.)
((𝐴 βŠ† β„• ∧ βˆ€π‘₯ ∈ β„• DECID π‘₯ ∈ 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ inf(𝐴, ℝ, < ) ≀ 𝐡)
 
25-Sep-2024nninfdclemcl 12452 Lemma for nninfdc 12457. (Contributed by Jim Kingdon, 25-Sep-2024.)
(πœ‘ β†’ 𝐴 βŠ† β„•)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„• DECID π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘š ∈ β„• βˆƒπ‘› ∈ 𝐴 π‘š < 𝑛)    &   (πœ‘ β†’ 𝑃 ∈ 𝐴)    &   (πœ‘ β†’ 𝑄 ∈ 𝐴)    β‡’   (πœ‘ β†’ (𝑃(𝑦 ∈ β„•, 𝑧 ∈ β„• ↦ inf((𝐴 ∩ (β„€β‰₯β€˜(𝑦 + 1))), ℝ, < ))𝑄) ∈ 𝐴)
 
24-Sep-2024nninfdclemlt 12455 Lemma for nninfdc 12457. The function from nninfdclemf 12453 is strictly monotonic. (Contributed by Jim Kingdon, 24-Sep-2024.)
(πœ‘ β†’ 𝐴 βŠ† β„•)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„• DECID π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘š ∈ β„• βˆƒπ‘› ∈ 𝐴 π‘š < 𝑛)    &   (πœ‘ β†’ (𝐽 ∈ 𝐴 ∧ 1 < 𝐽))    &   πΉ = seq1((𝑦 ∈ β„•, 𝑧 ∈ β„• ↦ inf((𝐴 ∩ (β„€β‰₯β€˜(𝑦 + 1))), ℝ, < )), (𝑖 ∈ β„• ↦ 𝐽))    &   (πœ‘ β†’ π‘ˆ ∈ β„•)    &   (πœ‘ β†’ 𝑉 ∈ β„•)    &   (πœ‘ β†’ π‘ˆ < 𝑉)    β‡’   (πœ‘ β†’ (πΉβ€˜π‘ˆ) < (πΉβ€˜π‘‰))
 
23-Sep-2024nninfdc 12457 An unbounded decidable set of positive integers is infinite. (Contributed by Jim Kingdon, 23-Sep-2024.)
((𝐴 βŠ† β„• ∧ βˆ€π‘₯ ∈ β„• DECID π‘₯ ∈ 𝐴 ∧ βˆ€π‘š ∈ β„• βˆƒπ‘› ∈ 𝐴 π‘š < 𝑛) β†’ Ο‰ β‰Ό 𝐴)
 
23-Sep-2024nninfdclemf1 12456 Lemma for nninfdc 12457. The function from nninfdclemf 12453 is one-to-one. (Contributed by Jim Kingdon, 23-Sep-2024.)
(πœ‘ β†’ 𝐴 βŠ† β„•)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„• DECID π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘š ∈ β„• βˆƒπ‘› ∈ 𝐴 π‘š < 𝑛)    &   (πœ‘ β†’ (𝐽 ∈ 𝐴 ∧ 1 < 𝐽))    &   πΉ = seq1((𝑦 ∈ β„•, 𝑧 ∈ β„• ↦ inf((𝐴 ∩ (β„€β‰₯β€˜(𝑦 + 1))), ℝ, < )), (𝑖 ∈ β„• ↦ 𝐽))    β‡’   (πœ‘ β†’ 𝐹:ℕ–1-1→𝐴)
 
23-Sep-2024nninfdclemf 12453 Lemma for nninfdc 12457. A function from the natural numbers into 𝐴. (Contributed by Jim Kingdon, 23-Sep-2024.)
(πœ‘ β†’ 𝐴 βŠ† β„•)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„• DECID π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘š ∈ β„• βˆƒπ‘› ∈ 𝐴 π‘š < 𝑛)    &   (πœ‘ β†’ (𝐽 ∈ 𝐴 ∧ 1 < 𝐽))    &   πΉ = seq1((𝑦 ∈ β„•, 𝑧 ∈ β„• ↦ inf((𝐴 ∩ (β„€β‰₯β€˜(𝑦 + 1))), ℝ, < )), (𝑖 ∈ β„• ↦ 𝐽))    β‡’   (πœ‘ β†’ 𝐹:β„•βŸΆπ΄)
 
23-Sep-2024nnmindc 12038 An inhabited decidable subset of the natural numbers has a minimum. (Contributed by Jim Kingdon, 23-Sep-2024.)
((𝐴 βŠ† β„• ∧ βˆ€π‘₯ ∈ β„• DECID π‘₯ ∈ 𝐴 ∧ βˆƒπ‘¦ 𝑦 ∈ 𝐴) β†’ inf(𝐴, ℝ, < ) ∈ 𝐴)
 
19-Sep-2024ssomct 12449 A decidable subset of Ο‰ is countable. (Contributed by Jim Kingdon, 19-Sep-2024.)
((𝐴 βŠ† Ο‰ ∧ βˆ€π‘₯ ∈ Ο‰ DECID π‘₯ ∈ 𝐴) β†’ βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o))
 
14-Sep-2024nnpredlt 4625 The predecessor (see nnpredcl 4624) of a nonzero natural number is less than (see df-iord 4368) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
((𝐴 ∈ Ο‰ ∧ 𝐴 β‰  βˆ…) β†’ βˆͺ 𝐴 ∈ 𝐴)
 
13-Sep-2024nninfisollemeq 7133 Lemma for nninfisol 7134. The case where 𝑁 is a successor and 𝑁 and 𝑋 are equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
(πœ‘ β†’ 𝑋 ∈ β„•βˆž)    &   (πœ‘ β†’ (π‘‹β€˜π‘) = βˆ…)    &   (πœ‘ β†’ 𝑁 ∈ Ο‰)    &   (πœ‘ β†’ 𝑁 β‰  βˆ…)    &   (πœ‘ β†’ (π‘‹β€˜βˆͺ 𝑁) = 1o)    β‡’   (πœ‘ β†’ DECID (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑁, 1o, βˆ…)) = 𝑋)
 
13-Sep-2024nninfisollemne 7132 Lemma for nninfisol 7134. A case where 𝑁 is a successor and 𝑁 and 𝑋 are not equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
(πœ‘ β†’ 𝑋 ∈ β„•βˆž)    &   (πœ‘ β†’ (π‘‹β€˜π‘) = βˆ…)    &   (πœ‘ β†’ 𝑁 ∈ Ο‰)    &   (πœ‘ β†’ 𝑁 β‰  βˆ…)    &   (πœ‘ β†’ (π‘‹β€˜βˆͺ 𝑁) = βˆ…)    β‡’   (πœ‘ β†’ DECID (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑁, 1o, βˆ…)) = 𝑋)
 
13-Sep-2024nninfisollem0 7131 Lemma for nninfisol 7134. The case where 𝑁 is zero. (Contributed by Jim Kingdon, 13-Sep-2024.)
(πœ‘ β†’ 𝑋 ∈ β„•βˆž)    &   (πœ‘ β†’ (π‘‹β€˜π‘) = βˆ…)    &   (πœ‘ β†’ 𝑁 ∈ Ο‰)    &   (πœ‘ β†’ 𝑁 = βˆ…)    β‡’   (πœ‘ β†’ DECID (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑁, 1o, βˆ…)) = 𝑋)
 
12-Sep-2024nninfisol 7134 Finite elements of β„•βˆž are isolated. That is, given a natural number and any element of β„•βˆž, it is decidable whether the natural number (when converted to an element of β„•βˆž) is equal to the given element of β„•βˆž. Stated in an online post by Martin Escardo. One way to understand this theorem is that you do not need to look at an unbounded number of elements of the sequence 𝑋 to decide whether it is equal to 𝑁 (in fact, you only need to look at two elements and 𝑁 tells you where to look). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.)
((𝑁 ∈ Ο‰ ∧ 𝑋 ∈ β„•βˆž) β†’ DECID (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑁, 1o, βˆ…)) = 𝑋)
 
7-Sep-2024eulerthlemfi 12231 Lemma for eulerth 12236. The set 𝑆 is finite. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.)
(πœ‘ β†’ (𝑁 ∈ β„• ∧ 𝐴 ∈ β„€ ∧ (𝐴 gcd 𝑁) = 1))    &   π‘† = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    β‡’   (πœ‘ β†’ 𝑆 ∈ Fin)
 
7-Sep-2024modqexp 10650 Exponentiation property of the modulo operation, see theorem 5.2(c) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.)
(πœ‘ β†’ 𝐴 ∈ β„€)    &   (πœ‘ β†’ 𝐡 ∈ β„€)    &   (πœ‘ β†’ 𝐢 ∈ β„•0)    &   (πœ‘ β†’ 𝐷 ∈ β„š)    &   (πœ‘ β†’ 0 < 𝐷)    &   (πœ‘ β†’ (𝐴 mod 𝐷) = (𝐡 mod 𝐷))    β‡’   (πœ‘ β†’ ((𝐴↑𝐢) mod 𝐷) = ((𝐡↑𝐢) mod 𝐷))
 
5-Sep-2024eulerthlemh 12234 Lemma for eulerth 12236. A permutation of (1...(Ο•β€˜π‘)). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 5-Sep-2024.)
(πœ‘ β†’ (𝑁 ∈ β„• ∧ 𝐴 ∈ β„€ ∧ (𝐴 gcd 𝑁) = 1))    &   π‘† = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   (πœ‘ β†’ 𝐹:(1...(Ο•β€˜π‘))–1-1-onto→𝑆)    &   π» = (◑𝐹 ∘ (𝑦 ∈ (1...(Ο•β€˜π‘)) ↦ ((𝐴 Β· (πΉβ€˜π‘¦)) mod 𝑁)))    β‡’   (πœ‘ β†’ 𝐻:(1...(Ο•β€˜π‘))–1-1-ontoβ†’(1...(Ο•β€˜π‘)))
 
2-Sep-2024eulerthlemth 12235 Lemma for eulerth 12236. The result. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
(πœ‘ β†’ (𝑁 ∈ β„• ∧ 𝐴 ∈ β„€ ∧ (𝐴 gcd 𝑁) = 1))    &   π‘† = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   (πœ‘ β†’ 𝐹:(1...(Ο•β€˜π‘))–1-1-onto→𝑆)    β‡’   (πœ‘ β†’ ((𝐴↑(Ο•β€˜π‘)) mod 𝑁) = (1 mod 𝑁))
 
2-Sep-2024eulerthlema 12233 Lemma for eulerth 12236. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
(πœ‘ β†’ (𝑁 ∈ β„• ∧ 𝐴 ∈ β„€ ∧ (𝐴 gcd 𝑁) = 1))    &   π‘† = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   (πœ‘ β†’ 𝐹:(1...(Ο•β€˜π‘))–1-1-onto→𝑆)    β‡’   (πœ‘ β†’ (((𝐴↑(Ο•β€˜π‘)) Β· ∏π‘₯ ∈ (1...(Ο•β€˜π‘))(πΉβ€˜π‘₯)) mod 𝑁) = (∏π‘₯ ∈ (1...(Ο•β€˜π‘))((𝐴 Β· (πΉβ€˜π‘₯)) mod 𝑁) mod 𝑁))
 
2-Sep-2024eulerthlemrprm 12232 Lemma for eulerth 12236. 𝑁 and ∏π‘₯ ∈ (1...(Ο•β€˜π‘))(πΉβ€˜π‘₯) are relatively prime. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
(πœ‘ β†’ (𝑁 ∈ β„• ∧ 𝐴 ∈ β„€ ∧ (𝐴 gcd 𝑁) = 1))    &   π‘† = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   (πœ‘ β†’ 𝐹:(1...(Ο•β€˜π‘))–1-1-onto→𝑆)    β‡’   (πœ‘ β†’ (𝑁 gcd ∏π‘₯ ∈ (1...(Ο•β€˜π‘))(πΉβ€˜π‘₯)) = 1)
 
30-Aug-2024fprodap0f 11647 A finite product of terms apart from zero is apart from zero. A version of fprodap0 11632 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by Jim Kingdon, 30-Aug-2024.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 # 0)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 # 0)
 
28-Aug-2024fprodrec 11640 The finite product of reciprocals is the reciprocal of the product. (Contributed by Jim Kingdon, 28-Aug-2024.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 # 0)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 (1 / 𝐡) = (1 / βˆπ‘˜ ∈ 𝐴 𝐡))
 
26-Aug-2024exmidontri2or 7245 Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
(EXMID ↔ βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On (π‘₯ βŠ† 𝑦 ∨ 𝑦 βŠ† π‘₯))
 
26-Aug-2024exmidontri 7241 Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
(EXMID ↔ βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On (π‘₯ ∈ 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 ∈ π‘₯))
 
26-Aug-2024ontri2orexmidim 4573 Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4572. (Contributed by Jim Kingdon, 26-Aug-2024.)
(βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On (π‘₯ βŠ† 𝑦 ∨ 𝑦 βŠ† π‘₯) β†’ DECID πœ‘)
 
26-Aug-2024ontriexmidim 4523 Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4522. (Contributed by Jim Kingdon, 26-Aug-2024.)
(βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On (π‘₯ ∈ 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 ∈ π‘₯) β†’ DECID πœ‘)
 
25-Aug-2024onntri2or 7248 Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
(Β¬ Β¬ EXMID ↔ βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On Β¬ Β¬ (π‘₯ βŠ† 𝑦 ∨ 𝑦 βŠ† π‘₯))
 
25-Aug-2024onntri3or 7247 Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
(Β¬ Β¬ EXMID ↔ βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On Β¬ Β¬ (π‘₯ ∈ 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 ∈ π‘₯))
 
25-Aug-2024csbcow 3070 Composition law for chained substitutions into a class. Version of csbco 3069 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-Nov-2005.) (Revised by Gino Giotto, 25-Aug-2024.)
⦋𝐴 / π‘¦β¦Œβ¦‹π‘¦ / π‘₯⦌𝐡 = ⦋𝐴 / π‘₯⦌𝐡
 
25-Aug-2024cbvreuvw 2711 Version of cbvreuv 2707 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
(π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆƒ!π‘₯ ∈ 𝐴 πœ‘ ↔ βˆƒ!𝑦 ∈ 𝐴 πœ“)
 
25-Aug-2024cbvrexvw 2710 Version of cbvrexv 2706 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
(π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆƒπ‘₯ ∈ 𝐴 πœ‘ ↔ βˆƒπ‘¦ ∈ 𝐴 πœ“)
 
25-Aug-2024cbvralvw 2709 Version of cbvralv 2705 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
(π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆ€π‘₯ ∈ 𝐴 πœ‘ ↔ βˆ€π‘¦ ∈ 𝐴 πœ“)
 
25-Aug-2024cbvabw 2300 Version of cbvab 2301 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
β„²π‘¦πœ‘    &   β„²π‘₯πœ“    &   (π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   {π‘₯ ∣ πœ‘} = {𝑦 ∣ πœ“}
 
25-Aug-2024nfsbv 1947 If 𝑧 is not free in πœ‘, it is not free in [𝑦 / π‘₯]πœ‘ when 𝑧 is distinct from π‘₯ and 𝑦. Version of nfsb 1946 requiring more disjoint variables. (Contributed by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on π‘₯, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
β„²π‘§πœ‘    β‡’   β„²π‘§[𝑦 / π‘₯]πœ‘
 
25-Aug-2024cbvexvw 1920 Change bound variable. See cbvexv 1918 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1448. (Revised by Gino Giotto, 25-Aug-2024.)
(π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆƒπ‘₯πœ‘ ↔ βˆƒπ‘¦πœ“)
 
25-Aug-2024cbvalvw 1919 Change bound variable. See cbvalv 1917 for a version with fewer disjoint variable conditions. (Contributed by NM, 9-Apr-2017.) Avoid ax-7 1448. (Revised by Gino Giotto, 25-Aug-2024.)
(π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆ€π‘₯πœ‘ ↔ βˆ€π‘¦πœ“)
 
25-Aug-2024nfal 1576 If π‘₯ is not free in πœ‘, it is not free in βˆ€π‘¦πœ‘. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-4 1510. (Revised by Gino Giotto, 25-Aug-2024.)
β„²π‘₯πœ‘    β‡’   β„²π‘₯βˆ€π‘¦πœ‘
 
24-Aug-2024gcdcomd 11978 The gcd operator is commutative, deduction version. (Contributed by SN, 24-Aug-2024.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    β‡’   (πœ‘ β†’ (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀))
 
21-Aug-2024dvds2addd 11839 Deduction form of dvds2add 11835. (Contributed by SN, 21-Aug-2024.)
(πœ‘ β†’ 𝐾 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝐾 βˆ₯ 𝑀)    &   (πœ‘ β†’ 𝐾 βˆ₯ 𝑁)    β‡’   (πœ‘ β†’ 𝐾 βˆ₯ (𝑀 + 𝑁))
 
17-Aug-2024fprodcl2lem 11616 Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.) (Revised by Jim Kingdon, 17-Aug-2024.)
(πœ‘ β†’ 𝑆 βŠ† β„‚)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ Β· 𝑦) ∈ 𝑆)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ 𝐴 β‰  βˆ…)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 ∈ 𝑆)
 
16-Aug-2024if0ab 14718 Expression of a conditional class as a class abstraction when the False alternative is the empty class: in that case, the conditional class is the extension, in the True alternative, of the condition.

Remark: a consequence which could be formalized is the inclusion if(πœ‘, 𝐴, βˆ…) βŠ† 𝐴 and therefore, using elpwg 3585, (𝐴 ∈ 𝑉 β†’ if(πœ‘, 𝐴, βˆ…) ∈ 𝒫 𝐴), from which fmelpw1o 14719 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)

if(πœ‘, 𝐴, βˆ…) = {π‘₯ ∈ 𝐴 ∣ πœ‘}
 
16-Aug-2024fprodunsn 11615 Multiply in an additional term in a finite product. See also fprodsplitsn 11644 which is the same but with a β„²π‘˜πœ‘ hypothesis in place of the distinct variable condition between πœ‘ and π‘˜. (Contributed by Jim Kingdon, 16-Aug-2024.)
β„²π‘˜π·    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝐡 ∈ 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐷 ∈ β„‚)    &   (π‘˜ = 𝐡 β†’ 𝐢 = 𝐷)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ (𝐴 βˆͺ {𝐡})𝐢 = (βˆπ‘˜ ∈ 𝐴 𝐢 Β· 𝐷))
 
15-Aug-2024bj-charfundcALT 14722 Alternate proof of bj-charfundc 14721. It was expected to be much shorter since it uses bj-charfun 14720 for the main part of the proof and the rest is basic computations, but these turn out to be lengthy, maybe because of the limited library of available lemmas. (Contributed by BJ, 15-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ 𝐴, 1o, βˆ…)))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 DECID π‘₯ ∈ 𝐴)    β‡’   (πœ‘ β†’ (𝐹:π‘‹βŸΆ2o ∧ (βˆ€π‘₯ ∈ (𝑋 ∩ 𝐴)(πΉβ€˜π‘₯) = 1o ∧ βˆ€π‘₯ ∈ (𝑋 βˆ– 𝐴)(πΉβ€˜π‘₯) = βˆ…)))
 
15-Aug-2024bj-charfun 14720 Properties of the characteristic function on the class 𝑋 of the class 𝐴. (Contributed by BJ, 15-Aug-2024.)
(πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ 𝐴, 1o, βˆ…)))    β‡’   (πœ‘ β†’ ((𝐹:π‘‹βŸΆπ’« 1o ∧ (𝐹 β†Ύ ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))):((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))⟢2o) ∧ (βˆ€π‘₯ ∈ (𝑋 ∩ 𝐴)(πΉβ€˜π‘₯) = 1o ∧ βˆ€π‘₯ ∈ (𝑋 βˆ– 𝐴)(πΉβ€˜π‘₯) = βˆ…)))
 
15-Aug-2024fmelpw1o 14719 With a formula πœ‘ one can associate an element of 𝒫 1o, which can therefore be thought of as the set of "truth values" (but recall that there are no other genuine truth values than ⊀ and βŠ₯, by nndc 851, which translate to 1o and βˆ… respectively by iftrue 3541 and iffalse 3544, giving pwtrufal 14909).

As proved in if0ab 14718, the associated element of 𝒫 1o is the extension, in 𝒫 1o, of the formula πœ‘. (Contributed by BJ, 15-Aug-2024.)

if(πœ‘, 1o, βˆ…) ∈ 𝒫 1o
 
15-Aug-2024cnstab 8605 Equality of complex numbers is stable. Stability here means Β¬ Β¬ 𝐴 = 𝐡 β†’ 𝐴 = 𝐡 as defined at df-stab 831. This theorem for real numbers is Proposition 5.2 of [BauerHanson], p. 27. (Contributed by Jim Kingdon, 1-Aug-2023.) (Proof shortened by BJ, 15-Aug-2024.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ STAB 𝐴 = 𝐡)
 
15-Aug-2024subap0d 8604 Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.) (Proof shortened by BJ, 15-Aug-2024.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐴 # 𝐡)    β‡’   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) # 0)
 
15-Aug-2024ifexd 4486 Existence of a conditional class (deduction form). (Contributed by BJ, 15-Aug-2024.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ π‘Š)    β‡’   (πœ‘ β†’ if(πœ“, 𝐴, 𝐡) ∈ V)
 
15-Aug-2024ifelpwun 4485 Existence of a conditional class, quantitative version (inference form). (Contributed by BJ, 15-Aug-2024.)
𝐴 ∈ V    &   π΅ ∈ V    β‡’   if(πœ‘, 𝐴, 𝐡) ∈ 𝒫 (𝐴 βˆͺ 𝐡)
 
15-Aug-2024ifelpwund 4484 Existence of a conditional class, quantitative version (deduction form). (Contributed by BJ, 15-Aug-2024.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ π‘Š)    β‡’   (πœ‘ β†’ if(πœ“, 𝐴, 𝐡) ∈ 𝒫 (𝐴 βˆͺ 𝐡))
 
15-Aug-2024ifelpwung 4483 Existence of a conditional class, quantitative version (closed form). (Contributed by BJ, 15-Aug-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ if(πœ‘, 𝐴, 𝐡) ∈ 𝒫 (𝐴 βˆͺ 𝐡))
 
15-Aug-2024ifidss 3551 A conditional class whose two alternatives are equal is included in that alternative. With excluded middle, we can prove it is equal to it. (Contributed by BJ, 15-Aug-2024.)
if(πœ‘, 𝐴, 𝐴) βŠ† 𝐴
 
15-Aug-2024ifssun 3550 A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.)
if(πœ‘, 𝐴, 𝐡) βŠ† (𝐴 βˆͺ 𝐡)
 
12-Aug-2024exmidontriimlem2 7224 Lemma for exmidontriim 7227. (Contributed by Jim Kingdon, 12-Aug-2024.)
(πœ‘ β†’ 𝐡 ∈ On)    &   (πœ‘ β†’ EXMID)    &   (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐡 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴))    β‡’   (πœ‘ β†’ (𝐴 ∈ 𝐡 ∨ βˆ€π‘¦ ∈ 𝐡 𝑦 ∈ 𝐴))
 
12-Aug-2024exmidontriimlem1 7223 Lemma for exmidontriim 7227. A variation of r19.30dc 2624. (Contributed by Jim Kingdon, 12-Aug-2024.)
((βˆ€π‘₯ ∈ 𝐴 (πœ‘ ∨ πœ“ ∨ πœ’) ∧ EXMID) β†’ (βˆƒπ‘₯ ∈ 𝐴 πœ‘ ∨ βˆƒπ‘₯ ∈ 𝐴 πœ“ ∨ βˆ€π‘₯ ∈ 𝐴 πœ’))
 
11-Aug-2024nndc 851 Double negation of decidability of a formula. Intuitionistic logic refutes the negation of decidability (but does not prove decidability) of any formula.

This should not trick the reader into thinking that Β¬ Β¬ EXMID is provable in intuitionistic logic. Indeed, if we could quantify over formula metavariables, then generalizing nnexmid 850 over πœ‘ would give "βˆ€πœ‘Β¬ Β¬ DECID πœ‘", but EXMID is "βˆ€πœ‘DECID πœ‘", so proving Β¬ Β¬ EXMID would amount to proving "Β¬ Β¬ βˆ€πœ‘DECID πœ‘", which is not implied by the above theorem. Indeed, the converse of nnal 1649 does not hold. Since our system does not allow quantification over formula metavariables, we can reproduce this argument by representing formulas as subsets of 𝒫 1o, like we do in our definition of EXMID (df-exmid 4197): then, we can prove βˆ€π‘₯ ∈ 𝒫 1oΒ¬ Β¬ DECID π‘₯ = 1o but we cannot prove Β¬ Β¬ βˆ€π‘₯ ∈ 𝒫 1oDECID π‘₯ = 1o because the converse of nnral 2467 does not hold.

Actually, Β¬ Β¬ EXMID is not provable in intuitionistic logic since intuitionistic logic has models satisfying Β¬ EXMID and noncontradiction holds (pm3.24 693). (Contributed by BJ, 9-Oct-2019.) Add explanation on non-provability of Β¬ Β¬ EXMID. (Revised by BJ, 11-Aug-2024.)

Β¬ Β¬ DECID πœ‘
 
10-Aug-2024exmidontriim 7227 Excluded middle implies ordinal trichotomy. Lemma 10.4.1 of [HoTT], p. (varies). The proof follows the proof from the HoTT book fairly closely. (Contributed by Jim Kingdon, 10-Aug-2024.)
(EXMID β†’ βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On (π‘₯ ∈ 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 ∈ π‘₯))
 
10-Aug-2024exmidontriimlem4 7226 Lemma for exmidontriim 7227. The induction step for the induction on 𝐴. (Contributed by Jim Kingdon, 10-Aug-2024.)
(πœ‘ β†’ 𝐴 ∈ On)    &   (πœ‘ β†’ 𝐡 ∈ On)    &   (πœ‘ β†’ EXMID)    &   (πœ‘ β†’ βˆ€π‘§ ∈ 𝐴 βˆ€π‘¦ ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))    β‡’   (πœ‘ β†’ (𝐴 ∈ 𝐡 ∨ 𝐴 = 𝐡 ∨ 𝐡 ∈ 𝐴))
 
10-Aug-2024exmidontriimlem3 7225 Lemma for exmidontriim 7227. What we get to do based on induction on both 𝐴 and 𝐡. (Contributed by Jim Kingdon, 10-Aug-2024.)
(πœ‘ β†’ 𝐴 ∈ On)    &   (πœ‘ β†’ 𝐡 ∈ On)    &   (πœ‘ β†’ EXMID)    &   (πœ‘ β†’ βˆ€π‘§ ∈ 𝐴 βˆ€π‘¦ ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))    &   (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐡 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴))    β‡’   (πœ‘ β†’ (𝐴 ∈ 𝐡 ∨ 𝐴 = 𝐡 ∨ 𝐡 ∈ 𝐴))
 
10-Aug-2024nnnninf2 7128 Canonical embedding of suc Ο‰ into β„•βˆž. (Contributed by BJ, 10-Aug-2024.)
(𝑁 ∈ suc Ο‰ β†’ (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑁, 1o, βˆ…)) ∈ β„•βˆž)
 
10-Aug-2024infnninf 7125 The point at infinity in β„•βˆž is the constant sequence equal to 1o. Note that with our encoding of functions, that constant function can also be expressed as (Ο‰ Γ— {1o}), as fconstmpt 4675 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
(𝑖 ∈ Ο‰ ↦ 1o) ∈ β„•βˆž
 
9-Aug-2024ss1o0el1o 6915 Reformulation of ss1o0el1 4199 using 1o instead of {βˆ…}. (Contributed by BJ, 9-Aug-2024.)
(𝐴 βŠ† 1o β†’ (βˆ… ∈ 𝐴 ↔ 𝐴 = 1o))
 
9-Aug-2024pw1dc0el 6914 Another equivalent of excluded middle, which is a mere reformulation of the definition. (Contributed by BJ, 9-Aug-2024.)
(EXMID ↔ βˆ€π‘₯ ∈ 𝒫 1oDECID βˆ… ∈ π‘₯)
 
9-Aug-2024ss1o0el1 4199 A subclass of {βˆ…} contains the empty set if and only if it equals {βˆ…}. (Contributed by BJ and Jim Kingdon, 9-Aug-2024.)
(𝐴 βŠ† {βˆ…} β†’ (βˆ… ∈ 𝐴 ↔ 𝐴 = {βˆ…}))
 
8-Aug-2024pw1dc1 6916 If, in the set of truth values (the powerset of 1o), equality to 1o is decidable, then excluded middle holds (and conversely). (Contributed by BJ and Jim Kingdon, 8-Aug-2024.)
(EXMID ↔ βˆ€π‘₯ ∈ 𝒫 1oDECID π‘₯ = 1o)
 
7-Aug-2024pw1fin 6913 Excluded middle is equivalent to the power set of 1o being finite. (Contributed by SN and Jim Kingdon, 7-Aug-2024.)
(EXMID ↔ 𝒫 1o ∈ Fin)
 
7-Aug-2024elomssom 4606 A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4607. (Revised by BJ, 7-Aug-2024.)
(𝐴 ∈ Ο‰ β†’ 𝐴 βŠ† Ο‰)
 
6-Aug-2024bj-charfunbi 14724 In an ambient set 𝑋, if membership in 𝐴 is stable, then it is decidable if and only if 𝐴 has a characteristic function.

This characterization can be applied to singletons when the set 𝑋 has stable equality, which is the case as soon as it has a tight apartness relation. (Contributed by BJ, 6-Aug-2024.)

(πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 STAB π‘₯ ∈ 𝐴)    β‡’   (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝑋 DECID π‘₯ ∈ 𝐴 ↔ βˆƒπ‘“ ∈ (2o β†‘π‘š 𝑋)(βˆ€π‘₯ ∈ (𝑋 ∩ 𝐴)(π‘“β€˜π‘₯) = 1o ∧ βˆ€π‘₯ ∈ (𝑋 βˆ– 𝐴)(π‘“β€˜π‘₯) = βˆ…)))
 
6-Aug-2024bj-charfunr 14723 If a class 𝐴 has a "weak" characteristic function on a class 𝑋, then negated membership in 𝐴 is decidable (in other words, membership in 𝐴 is testable) in 𝑋.

The hypothesis imposes that 𝑋 be a set. As usual, it could be formulated as (πœ‘ β†’ (𝐹:π‘‹βŸΆΟ‰ ∧ ...)) to deal with general classes, but that extra generality would not make the theorem much more useful.

The theorem would still hold if the codomain of 𝑓 were any class with testable equality to the point where (𝑋 βˆ– 𝐴) is sent. (Contributed by BJ, 6-Aug-2024.)

(πœ‘ β†’ βˆƒπ‘“ ∈ (Ο‰ β†‘π‘š 𝑋)(βˆ€π‘₯ ∈ (𝑋 ∩ 𝐴)(π‘“β€˜π‘₯) β‰  βˆ… ∧ βˆ€π‘₯ ∈ (𝑋 βˆ– 𝐴)(π‘“β€˜π‘₯) = βˆ…))    β‡’   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 DECID Β¬ π‘₯ ∈ 𝐴)
 
6-Aug-2024bj-charfundc 14721 Properties of the characteristic function on the class 𝑋 of the class 𝐴, provided membership in 𝐴 is decidable in 𝑋. (Contributed by BJ, 6-Aug-2024.)
(πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ 𝐴, 1o, βˆ…)))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 DECID π‘₯ ∈ 𝐴)    β‡’   (πœ‘ β†’ (𝐹:π‘‹βŸΆ2o ∧ (βˆ€π‘₯ ∈ (𝑋 ∩ 𝐴)(πΉβ€˜π‘₯) = 1o ∧ βˆ€π‘₯ ∈ (𝑋 βˆ– 𝐴)(πΉβ€˜π‘₯) = βˆ…)))
 
6-Aug-2024prodssdc 11600 Change the index set to a subset in an upper integer product. (Contributed by Scott Fenton, 11-Dec-2017.) (Revised by Jim Kingdon, 6-Aug-2024.)
(πœ‘ β†’ 𝐴 βŠ† 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ βˆƒπ‘› ∈ (β„€β‰₯β€˜π‘€)βˆƒπ‘¦(𝑦 # 0 ∧ seq𝑛( Β· , (π‘˜ ∈ (β„€β‰₯β€˜π‘€) ↦ if(π‘˜ ∈ 𝐡, 𝐢, 1))) ⇝ 𝑦))    &   (πœ‘ β†’ βˆ€π‘— ∈ (β„€β‰₯β€˜π‘€)DECID 𝑗 ∈ 𝐴)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 = 1)    &   (πœ‘ β†’ 𝐡 βŠ† (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ βˆ€π‘— ∈ (β„€β‰₯β€˜π‘€)DECID 𝑗 ∈ 𝐡)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢)
 
5-Aug-2024fnmptd 14717 The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.)
(πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐴 ↦ 𝐡))    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ 𝑉)    β‡’   (πœ‘ β†’ 𝐹 Fn 𝐴)
 
5-Aug-2024funmptd 14716 The maps-to notation defines a function (deduction form).

Note: one should similarly prove a deduction form of funopab4 5255, then prove funmptd 14716 from it, and then prove funmpt 5256 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.)

(πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐴 ↦ 𝐡))    β‡’   (πœ‘ β†’ Fun 𝐹)
 
5-Aug-2024bj-dcfal 14668 The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID βŠ₯
 
5-Aug-2024bj-dctru 14666 The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID ⊀
 
5-Aug-2024bj-stfal 14655 The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)
STAB βŠ₯
 
5-Aug-2024bj-sttru 14653 The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)
STAB ⊀
 
5-Aug-2024prod1dc 11597 Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.) (Revised by Jim Kingdon, 5-Aug-2024.)
(((𝑀 ∈ β„€ ∧ 𝐴 βŠ† (β„€β‰₯β€˜π‘€) ∧ βˆ€π‘— ∈ (β„€β‰₯β€˜π‘€)DECID 𝑗 ∈ 𝐴) ∨ 𝐴 ∈ Fin) β†’ βˆπ‘˜ ∈ 𝐴 1 = 1)
 
5-Aug-20242ssom 6528 The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
2o βŠ† Ο‰
 
2-Aug-2024onntri52 7246 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(Β¬ Β¬ EXMID β†’ Β¬ Β¬ βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On (π‘₯ βŠ† 𝑦 ∨ 𝑦 βŠ† π‘₯))
 
2-Aug-2024onntri24 7244 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(Β¬ Β¬ βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On (π‘₯ βŠ† 𝑦 ∨ 𝑦 βŠ† π‘₯) β†’ βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On Β¬ Β¬ (π‘₯ βŠ† 𝑦 ∨ 𝑦 βŠ† π‘₯))
 
2-Aug-2024onntri45 7243 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On Β¬ Β¬ (π‘₯ βŠ† 𝑦 ∨ 𝑦 βŠ† π‘₯) β†’ Β¬ Β¬ EXMID)
 
2-Aug-2024onntri51 7242 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(Β¬ Β¬ EXMID β†’ Β¬ Β¬ βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On (π‘₯ ∈ 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 ∈ π‘₯))
 
2-Aug-2024onntri13 7240 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(Β¬ Β¬ βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On (π‘₯ ∈ 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 ∈ π‘₯) β†’ βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On Β¬ Β¬ (π‘₯ ∈ 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 ∈ π‘₯))
 
2-Aug-2024onntri35 7239 Double negated ordinal trichotomy.

There are five equivalent statements: (1) Β¬ Β¬ βˆ€π‘₯ ∈ Onβˆ€π‘¦ ∈ On(π‘₯ ∈ 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 ∈ π‘₯), (2) Β¬ Β¬ βˆ€π‘₯ ∈ Onβˆ€π‘¦ ∈ On(π‘₯ βŠ† 𝑦 ∨ 𝑦 βŠ† π‘₯), (3) βˆ€π‘₯ ∈ Onβˆ€π‘¦ ∈ OnΒ¬ Β¬ (π‘₯ ∈ 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 ∈ π‘₯), (4) βˆ€π‘₯ ∈ Onβˆ€π‘¦ ∈ OnΒ¬ Β¬ (π‘₯ βŠ† 𝑦 ∨ 𝑦 βŠ† π‘₯), and (5) Β¬ Β¬ EXMID. That these are all equivalent is expressed by (1) implies (3) (onntri13 7240), (3) implies (5) (onntri35 7239), (5) implies (1) (onntri51 7242), (2) implies (4) (onntri24 7244), (4) implies (5) (onntri45 7243), and (5) implies (2) (onntri52 7246).

Another way of stating this is that EXMID is equivalent to trichotomy, either the π‘₯ ∈ 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 ∈ π‘₯ or the π‘₯ βŠ† 𝑦 ∨ 𝑦 βŠ† π‘₯ form, as shown in exmidontri 7241 and exmidontri2or 7245, respectively. Thus Β¬ Β¬ EXMID is equivalent to (1) or (2). In addition, Β¬ Β¬ EXMID is equivalent to (3) by onntri3or 7247 and (4) by onntri2or 7248.

(Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)

(βˆ€π‘₯ ∈ On βˆ€π‘¦ ∈ On Β¬ Β¬ (π‘₯ ∈ 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 ∈ π‘₯) β†’ Β¬ Β¬ EXMID)
 
1-Aug-2024nnral 2467 The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1649. (Contributed by Jim Kingdon, 1-Aug-2024.)
(Β¬ Β¬ βˆ€π‘₯ ∈ 𝐴 πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 Β¬ Β¬ πœ‘)
 
31-Jul-20243nsssucpw1 7238 Negated excluded middle implies that 3o is not a subset of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
(Β¬ EXMID β†’ Β¬ 3o βŠ† suc 𝒫 1o)
 
31-Jul-2024sucpw1nss3 7237 Negated excluded middle implies that the successor of the power set of 1o is not a subset of 3o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
(Β¬ EXMID β†’ Β¬ suc 𝒫 1o βŠ† 3o)
 
30-Jul-20243nelsucpw1 7236 Three is not an element of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
Β¬ 3o ∈ suc 𝒫 1o
 
30-Jul-2024sucpw1nel3 7235 The successor of the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
Β¬ suc 𝒫 1o ∈ 3o
 
30-Jul-2024sucpw1ne3 7234 Negated excluded middle implies that the successor of the power set of 1o is not three . (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
(Β¬ EXMID β†’ suc 𝒫 1o β‰  3o)
 
30-Jul-2024pw1nel3 7233 Negated excluded middle implies that the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
(Β¬ EXMID β†’ Β¬ 𝒫 1o ∈ 3o)
 
30-Jul-2024pw1ne3 7232 The power set of 1o is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
𝒫 1o β‰  3o
 
30-Jul-2024pw1ne1 7231 The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
𝒫 1o β‰  1o
 
30-Jul-2024pw1ne0 7230 The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
𝒫 1o β‰  βˆ…
 
29-Jul-2024grpcld 12897 Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Grp)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ 𝐡)
 
29-Jul-2024pw1on 7228 The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
𝒫 1o ∈ On
 
28-Jul-2024exmidpweq 6912 Excluded middle is equivalent to the power set of 1o being 2o. (Contributed by Jim Kingdon, 28-Jul-2024.)
(EXMID ↔ 𝒫 1o = 2o)
 
27-Jul-2024dcapnconstALT 14972 Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 14971 by means of dceqnconst 14970. (Contributed by Jim Kingdon, 27-Jul-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
(βˆ€π‘₯ ∈ ℝ DECID π‘₯ # 0 β†’ βˆƒπ‘“(𝑓:β„βŸΆβ„€ ∧ (π‘“β€˜0) = 0 ∧ βˆ€π‘₯ ∈ ℝ+ (π‘“β€˜π‘₯) β‰  0))
 
27-Jul-2024reap0 14968 Real number trichotomy is equivalent to decidability of apartness from zero. (Contributed by Jim Kingdon, 27-Jul-2024.)
(βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ < 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 < π‘₯) ↔ βˆ€π‘§ ∈ ℝ DECID 𝑧 # 0)
 
26-Jul-2024nconstwlpolemgt0 14974 Lemma for nconstwlpo 14976. If one of the terms of series is positive, so is the sum. (Contributed by Jim Kingdon, 26-Jul-2024.)
(πœ‘ β†’ 𝐺:β„•βŸΆ{0, 1})    &   π΄ = Σ𝑖 ∈ β„• ((1 / (2↑𝑖)) Β· (πΊβ€˜π‘–))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ β„• (πΊβ€˜π‘₯) = 1)    β‡’   (πœ‘ β†’ 0 < 𝐴)
 
26-Jul-2024nconstwlpolem0 14973 Lemma for nconstwlpo 14976. If all the terms of the series are zero, so is their sum. (Contributed by Jim Kingdon, 26-Jul-2024.)
(πœ‘ β†’ 𝐺:β„•βŸΆ{0, 1})    &   π΄ = Σ𝑖 ∈ β„• ((1 / (2↑𝑖)) Β· (πΊβ€˜π‘–))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„• (πΊβ€˜π‘₯) = 0)    β‡’   (πœ‘ β†’ 𝐴 = 0)
 
24-Jul-2024tridceq 14966 Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 14953 and redcwlpo 14965). Thus, this is an analytic analogue to lpowlpo 7169. (Contributed by Jim Kingdon, 24-Jul-2024.)
(βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ < 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 < π‘₯) β†’ βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ DECID π‘₯ = 𝑦)
 
24-Jul-2024iswomni0 14961 Weak omniscience stated in terms of equality with 0. Like iswomninn 14960 but with zero in place of one. (Contributed by Jim Kingdon, 24-Jul-2024.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ WOmni ↔ βˆ€π‘“ ∈ ({0, 1} β†‘π‘š 𝐴)DECID βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 0))
 
24-Jul-2024lpowlpo 7169 LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7168. There is an analogue in terms of analytic omniscience principles at tridceq 14966. (Contributed by Jim Kingdon, 24-Jul-2024.)
(Ο‰ ∈ Omni β†’ Ο‰ ∈ WOmni)
 
23-Jul-2024nconstwlpolem 14975 Lemma for nconstwlpo 14976. (Contributed by Jim Kingdon, 23-Jul-2024.)
(πœ‘ β†’ 𝐹:β„βŸΆβ„€)    &   (πœ‘ β†’ (πΉβ€˜0) = 0)    &   ((πœ‘ ∧ π‘₯ ∈ ℝ+) β†’ (πΉβ€˜π‘₯) β‰  0)    &   (πœ‘ β†’ 𝐺:β„•βŸΆ{0, 1})    &   π΄ = Σ𝑖 ∈ β„• ((1 / (2↑𝑖)) Β· (πΊβ€˜π‘–))    β‡’   (πœ‘ β†’ (βˆ€π‘¦ ∈ β„• (πΊβ€˜π‘¦) = 0 ∨ Β¬ βˆ€π‘¦ ∈ β„• (πΊβ€˜π‘¦) = 0))
 
23-Jul-2024dceqnconst 14970 Decidability of real number equality implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See redcwlpo 14965 for more discussion of decidability of real number equality. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) (Revised by Jim Kingdon, 23-Jul-2024.)
(βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 β†’ βˆƒπ‘“(𝑓:β„βŸΆβ„€ ∧ (π‘“β€˜0) = 0 ∧ βˆ€π‘₯ ∈ ℝ+ (π‘“β€˜π‘₯) β‰  0))
 
23-Jul-2024redc0 14967 Two ways to express decidability of real number equality. (Contributed by Jim Kingdon, 23-Jul-2024.)
(βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ DECID π‘₯ = 𝑦 ↔ βˆ€π‘§ ∈ ℝ DECID 𝑧 = 0)
 
23-Jul-2024canth 5832 No set 𝐴 is equinumerous to its power set (Cantor's theorem), i.e., no function can map 𝐴 onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. (Use nex 1500 if you want the form Β¬ βˆƒπ‘“π‘“:𝐴–onto→𝒫 𝐴.) (Contributed by NM, 7-Aug-1994.) (Revised by Noah R Kingdon, 23-Jul-2024.)
𝐴 ∈ V    β‡’    Β¬ 𝐹:𝐴–onto→𝒫 𝐴
 
22-Jul-2024nconstwlpo 14976 Existence of a certain non-constant function from reals to integers implies Ο‰ ∈ WOmni (the Weak Limited Principle of Omniscience or WLPO). Based on Exercise 11.6(ii) of [HoTT], p. (varies). (Contributed by BJ and Jim Kingdon, 22-Jul-2024.)
(πœ‘ β†’ 𝐹:β„βŸΆβ„€)    &   (πœ‘ β†’ (πΉβ€˜0) = 0)    &   ((πœ‘ ∧ π‘₯ ∈ ℝ+) β†’ (πΉβ€˜π‘₯) β‰  0)    β‡’   (πœ‘ β†’ Ο‰ ∈ WOmni)
 
15-Jul-2024fprodseq 11594 The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.) (Revised by Jim Kingdon, 15-Jul-2024.)
(π‘˜ = (πΉβ€˜π‘›) β†’ 𝐡 = 𝐢)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝐹:(1...𝑀)–1-1-onto→𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ 𝑛 ∈ (1...𝑀)) β†’ (πΊβ€˜π‘›) = 𝐢)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = (seq1( Β· , (𝑛 ∈ β„• ↦ if(𝑛 ≀ 𝑀, (πΊβ€˜π‘›), 1)))β€˜π‘€))
 
14-Jul-2024rexbid2 2482 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ∧ πœ“) ↔ (π‘₯ ∈ 𝐡 ∧ πœ’)))    β‡’   (πœ‘ β†’ (βˆƒπ‘₯ ∈ 𝐴 πœ“ ↔ βˆƒπ‘₯ ∈ 𝐡 πœ’))
 
14-Jul-2024ralbid2 2481 Formula-building rule for restricted universal quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ ((π‘₯ ∈ 𝐴 β†’ πœ“) ↔ (π‘₯ ∈ 𝐡 β†’ πœ’)))    β‡’   (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝐴 πœ“ ↔ βˆ€π‘₯ ∈ 𝐡 πœ’))
 
12-Jul-20242irrexpqap 14557 There exist real numbers π‘Ž and 𝑏 which are irrational (in the sense of being apart from any rational number) such that (π‘Žβ†‘π‘) is rational. Statement in the Metamath book, section 1.1.5, footnote 27 on page 17, and the "constructive proof" for theorem 1.2 of [Bauer], p. 483. This is a constructive proof because it is based on two explicitly named irrational numbers (βˆšβ€˜2) and (2 logb 9), see sqrt2irrap 12183, 2logb9irrap 14556 and sqrt2cxp2logb9e3 14554. Therefore, this proof is acceptable/usable in intuitionistic logic. (Contributed by Jim Kingdon, 12-Jul-2024.)
βˆƒπ‘Ž ∈ ℝ βˆƒπ‘ ∈ ℝ (βˆ€π‘ ∈ β„š π‘Ž # 𝑝 ∧ βˆ€π‘ž ∈ β„š 𝑏 # π‘ž ∧ (π‘Žβ†‘π‘π‘) ∈ β„š)
 
12-Jul-20242logb9irrap 14556 Example for logbgcd1irrap 14549. The logarithm of nine to base two is irrational (in the sense of being apart from any rational number). (Contributed by Jim Kingdon, 12-Jul-2024.)
(𝑄 ∈ β„š β†’ (2 logb 9) # 𝑄)
 
12-Jul-2024erlecpbl 12758 Translate the relation compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
(πœ‘ β†’ ∼ Er 𝑉)    &   (πœ‘ β†’ 𝑉 ∈ π‘Š)    &   πΉ = (π‘₯ ∈ 𝑉 ↦ [π‘₯] ∼ )    &   (πœ‘ β†’ ((𝐴 ∼ 𝐢 ∧ 𝐡 ∼ 𝐷) β†’ (𝐴𝑁𝐡 ↔ 𝐢𝑁𝐷)))    β‡’   ((πœ‘ ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐢 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) β†’ (((πΉβ€˜π΄) = (πΉβ€˜πΆ) ∧ (πΉβ€˜π΅) = (πΉβ€˜π·)) β†’ (𝐴𝑁𝐡 ↔ 𝐢𝑁𝐷)))
 
12-Jul-2024ercpbl 12757 Translate the function compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
(πœ‘ β†’ ∼ Er 𝑉)    &   (πœ‘ β†’ 𝑉 ∈ π‘Š)    &   πΉ = (π‘₯ ∈ 𝑉 ↦ [π‘₯] ∼ )    &   ((πœ‘ ∧ (π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) β†’ (π‘Ž + 𝑏) ∈ 𝑉)    &   (πœ‘ β†’ ((𝐴 ∼ 𝐢 ∧ 𝐡 ∼ 𝐷) β†’ (𝐴 + 𝐡) ∼ (𝐢 + 𝐷)))    β‡’   ((πœ‘ ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐢 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) β†’ (((πΉβ€˜π΄) = (πΉβ€˜πΆ) ∧ (πΉβ€˜π΅) = (πΉβ€˜π·)) β†’ (πΉβ€˜(𝐴 + 𝐡)) = (πΉβ€˜(𝐢 + 𝐷))))
 
12-Jul-2024ercpbllemg 12756 Lemma for ercpbl 12757. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by AV, 12-Jul-2024.)
(πœ‘ β†’ ∼ Er 𝑉)    &   (πœ‘ β†’ 𝑉 ∈ π‘Š)    &   πΉ = (π‘₯ ∈ 𝑉 ↦ [π‘₯] ∼ )    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    β‡’   (πœ‘ β†’ ((πΉβ€˜π΄) = (πΉβ€˜π΅) ↔ 𝐴 ∼ 𝐡))
 
12-Jul-2024divsfvalg 12755 Value of the function in qusval 12750. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
(πœ‘ β†’ ∼ Er 𝑉)    &   (πœ‘ β†’ 𝑉 ∈ π‘Š)    &   πΉ = (π‘₯ ∈ 𝑉 ↦ [π‘₯] ∼ )    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    β‡’   (πœ‘ β†’ (πΉβ€˜π΄) = [𝐴] ∼ )
 
11-Jul-2024logbgcd1irraplemexp 14547 Lemma for logbgcd1irrap 14549. Apartness of 𝑋↑𝑁 and 𝐡↑𝑀. (Contributed by Jim Kingdon, 11-Jul-2024.)
(πœ‘ β†’ 𝑋 ∈ (β„€β‰₯β€˜2))    &   (πœ‘ β†’ 𝐡 ∈ (β„€β‰₯β€˜2))    &   (πœ‘ β†’ (𝑋 gcd 𝐡) = 1)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (𝑋↑𝑁) # (𝐡↑𝑀))
 
11-Jul-2024reapef 14360 Apartness and the exponential function for reals. (Contributed by Jim Kingdon, 11-Jul-2024.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 # 𝐡 ↔ (expβ€˜π΄) # (expβ€˜π΅)))
 
10-Jul-2024apcxp2 14519 Apartness and real exponentiation. (Contributed by Jim Kingdon, 10-Jul-2024.)
(((𝐴 ∈ ℝ+ ∧ 𝐴 # 1) ∧ (𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ)) β†’ (𝐡 # 𝐢 ↔ (𝐴↑𝑐𝐡) # (𝐴↑𝑐𝐢)))
 
9-Jul-2024logbgcd1irraplemap 14548 Lemma for logbgcd1irrap 14549. The result, with the rational number expressed as numerator and denominator. (Contributed by Jim Kingdon, 9-Jul-2024.)
(πœ‘ β†’ 𝑋 ∈ (β„€β‰₯β€˜2))    &   (πœ‘ β†’ 𝐡 ∈ (β„€β‰₯β€˜2))    &   (πœ‘ β†’ (𝑋 gcd 𝐡) = 1)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (𝐡 logb 𝑋) # (𝑀 / 𝑁))
 
9-Jul-2024apexp1 10701 Exponentiation and apartness. (Contributed by Jim Kingdon, 9-Jul-2024.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝑁 ∈ β„•) β†’ ((𝐴↑𝑁) # (𝐡↑𝑁) β†’ 𝐴 # 𝐡))
 
5-Jul-2024logrpap0 14459 The logarithm is apart from 0 if its argument is apart from 1. (Contributed by Jim Kingdon, 5-Jul-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐴 # 1) β†’ (logβ€˜π΄) # 0)
 
3-Jul-2024rplogbval 14524 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by Jim Kingdon, 3-Jul-2024.)
((𝐡 ∈ ℝ+ ∧ 𝐡 # 1 ∧ 𝑋 ∈ ℝ+) β†’ (𝐡 logb 𝑋) = ((logβ€˜π‘‹) / (logβ€˜π΅)))
 
3-Jul-2024logrpap0d 14460 Deduction form of logrpap0 14459. (Contributed by Jim Kingdon, 3-Jul-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐴 # 1)    β‡’   (πœ‘ β†’ (logβ€˜π΄) # 0)
 
3-Jul-2024logrpap0b 14458 The logarithm is apart from 0 if and only if its argument is apart from 1. (Contributed by Jim Kingdon, 3-Jul-2024.)
(𝐴 ∈ ℝ+ β†’ (𝐴 # 1 ↔ (logβ€˜π΄) # 0))
 
28-Jun-20242o01f 14908 Mapping zero and one between Ο‰ and β„•0 style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
𝐺 = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    β‡’   (𝐺 β†Ύ 2o):2o⟢{0, 1}
 
28-Jun-2024012of 14907 Mapping zero and one between β„•0 and Ο‰ style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
𝐺 = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    β‡’   (◑𝐺 β†Ύ {0, 1}):{0, 1}⟢2o
 
27-Jun-2024iooreen 14945 An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.)
(0(,)1) β‰ˆ ℝ
 
27-Jun-2024iooref1o 14944 A one-to-one mapping from the real numbers onto the open unit interval. (Contributed by Jim Kingdon, 27-Jun-2024.)
𝐹 = (π‘₯ ∈ ℝ ↦ (1 / (1 + (expβ€˜π‘₯))))    β‡’   πΉ:ℝ–1-1-ontoβ†’(0(,)1)
 
25-Jun-2024neapmkvlem 14977 Lemma for neapmkv 14978. The result, with a few hypotheses broken out for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.)
(πœ‘ β†’ 𝐹:β„•βŸΆ{0, 1})    &   π΄ = Σ𝑖 ∈ β„• ((1 / (2↑𝑖)) Β· (πΉβ€˜π‘–))    &   ((πœ‘ ∧ 𝐴 β‰  1) β†’ 𝐴 # 1)    β‡’   (πœ‘ β†’ (Β¬ βˆ€π‘₯ ∈ β„• (πΉβ€˜π‘₯) = 1 β†’ βˆƒπ‘₯ ∈ β„• (πΉβ€˜π‘₯) = 0))
 
25-Jun-2024ismkvnn 14963 The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 25-Jun-2024.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Markov ↔ βˆ€π‘“ ∈ ({0, 1} β†‘π‘š 𝐴)(Β¬ βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1 β†’ βˆƒπ‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 0)))
 
25-Jun-2024ismkvnnlem 14962 Lemma for ismkvnn 14963. The result, with a hypothesis to give a name to an expression for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.)
𝐺 = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    β‡’   (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Markov ↔ βˆ€π‘“ ∈ ({0, 1} β†‘π‘š 𝐴)(Β¬ βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1 β†’ βˆƒπ‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 0)))
 
25-Jun-2024enmkvlem 7162 Lemma for enmkv 7163. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
(𝐴 β‰ˆ 𝐡 β†’ (𝐴 ∈ Markov β†’ 𝐡 ∈ Markov))
 
24-Jun-2024neapmkv 14978 If negated equality for real numbers implies apartness, Markov's Principle follows. Exercise 11.10 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 24-Jun-2024.)
(βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ β‰  𝑦 β†’ π‘₯ # 𝑦) β†’ Ο‰ ∈ Markov)
 
24-Jun-2024dcapnconst 14971 Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See trilpo 14953 for more discussion of decidability of real number apartness.

This is a weaker form of dceqnconst 14970 and in fact this theorem can be proved using dceqnconst 14970 as shown at dcapnconstALT 14972. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.)

(βˆ€π‘₯ ∈ ℝ DECID π‘₯ # 0 β†’ βˆƒπ‘“(𝑓:β„βŸΆβ„€ ∧ (π‘“β€˜0) = 0 ∧ βˆ€π‘₯ ∈ ℝ+ (π‘“β€˜π‘₯) β‰  0))
 
24-Jun-2024enmkv 7163 Being Markov is invariant with respect to equinumerosity. For example, this means that we can express the Markov's Principle as either Ο‰ ∈ Markov or β„•0 ∈ Markov. The former is a better match to conventional notation in the sense that df2o3 6434 says that 2o = {βˆ…, 1o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 24-Jun-2024.)
(𝐴 β‰ˆ 𝐡 β†’ (𝐴 ∈ Markov ↔ 𝐡 ∈ Markov))
 
21-Jun-2024redcwlpolemeq1 14964 Lemma for redcwlpo 14965. A biconditionalized version of trilpolemeq1 14950. (Contributed by Jim Kingdon, 21-Jun-2024.)
(πœ‘ β†’ 𝐹:β„•βŸΆ{0, 1})    &   π΄ = Σ𝑖 ∈ β„• ((1 / (2↑𝑖)) Β· (πΉβ€˜π‘–))    β‡’   (πœ‘ β†’ (𝐴 = 1 ↔ βˆ€π‘₯ ∈ β„• (πΉβ€˜π‘₯) = 1))
 
20-Jun-2024redcwlpo 14965 Decidability of real number equality implies the Weak Limited Principle of Omniscience (WLPO). We expect that we'd need some form of countable choice to prove the converse.

Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 14964). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones.

Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO".

WLPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qdceq 10250 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)

(βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ DECID π‘₯ = 𝑦 β†’ Ο‰ ∈ WOmni)
 
20-Jun-2024iswomninn 14960 Weak omniscience stated in terms of natural numbers. Similar to iswomnimap 7167 but it will sometimes be more convenient to use 0 and 1 rather than βˆ… and 1o. (Contributed by Jim Kingdon, 20-Jun-2024.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ WOmni ↔ βˆ€π‘“ ∈ ({0, 1} β†‘π‘š 𝐴)DECID βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1))
 
20-Jun-2024iswomninnlem 14959 Lemma for iswomnimap 7167. The result, with a hypothesis for convenience. (Contributed by Jim Kingdon, 20-Jun-2024.)
𝐺 = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    β‡’   (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ WOmni ↔ βˆ€π‘“ ∈ ({0, 1} β†‘π‘š 𝐴)DECID βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1))
 
20-Jun-2024enwomni 7171 Weak omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Weak Limited Principle of Omniscience as either Ο‰ ∈ WOmni or β„•0 ∈ WOmni. The former is a better match to conventional notation in the sense that df2o3 6434 says that 2o = {βˆ…, 1o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 20-Jun-2024.)
(𝐴 β‰ˆ 𝐡 β†’ (𝐴 ∈ WOmni ↔ 𝐡 ∈ WOmni))
 
20-Jun-2024enwomnilem 7170 Lemma for enwomni 7171. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
(𝐴 β‰ˆ 𝐡 β†’ (𝐴 ∈ WOmni β†’ 𝐡 ∈ WOmni))
 
19-Jun-2024rpabscxpbnd 14520 Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Jim Kingdon, 19-Jun-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 0 < (β„œβ€˜π΅))    &   (πœ‘ β†’ 𝑀 ∈ ℝ)    &   (πœ‘ β†’ (absβ€˜π΄) ≀ 𝑀)    β‡’   (πœ‘ β†’ (absβ€˜(𝐴↑𝑐𝐡)) ≀ ((𝑀↑𝑐(β„œβ€˜π΅)) Β· (expβ€˜((absβ€˜π΅) Β· Ο€))))
 
16-Jun-2024rpcxpsqrt 14503 The exponential function with exponent 1 / 2 exactly matches the square root function, and thus serves as a suitable generalization to other 𝑛-th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 16-Jun-2024.)
(𝐴 ∈ ℝ+ β†’ (𝐴↑𝑐(1 / 2)) = (βˆšβ€˜π΄))
 
13-Jun-2024rpcxpadd 14487 Sum of exponents law for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 13-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ (𝐴↑𝑐(𝐡 + 𝐢)) = ((𝐴↑𝑐𝐡) Β· (𝐴↑𝑐𝐢)))
 
12-Jun-2024cxpap0 14486 Complex exponentiation is apart from zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐡 ∈ β„‚) β†’ (𝐴↑𝑐𝐡) # 0)
 
12-Jun-2024rpcncxpcl 14484 Closure of the complex power function. (Contributed by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐡 ∈ β„‚) β†’ (𝐴↑𝑐𝐡) ∈ β„‚)
 
12-Jun-2024rpcxp0 14480 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
(𝐴 ∈ ℝ+ β†’ (𝐴↑𝑐0) = 1)
 
12-Jun-2024cxpexpnn 14478 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ β„• ∧ 𝐡 ∈ β„€) β†’ (𝐴↑𝑐𝐡) = (𝐴↑𝐡))
 
12-Jun-2024cxpexprp 14477 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐡 ∈ β„€) β†’ (𝐴↑𝑐𝐡) = (𝐴↑𝐡))
 
12-Jun-2024rpcxpef 14476 Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐡 ∈ β„‚) β†’ (𝐴↑𝑐𝐡) = (expβ€˜(𝐡 Β· (logβ€˜π΄))))
 
12-Jun-2024df-rpcxp 14441 Define the power function on complex numbers. Because df-relog 14440 is only defined on positive reals, this definition only allows for a base which is a positive real. (Contributed by Jim Kingdon, 12-Jun-2024.)
↑𝑐 = (π‘₯ ∈ ℝ+, 𝑦 ∈ β„‚ ↦ (expβ€˜(𝑦 Β· (logβ€˜π‘₯))))
 
10-Jun-2024trirec0xor 14955 Version of trirec0 14954 with exclusive-or.

The definition of a discrete field is sometimes stated in terms of exclusive-or but as proved here, this is equivalent to inclusive-or because the two disjuncts cannot be simultaneously true. (Contributed by Jim Kingdon, 10-Jun-2024.)

(βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ < 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 < π‘₯) ↔ βˆ€π‘₯ ∈ ℝ (βˆƒπ‘§ ∈ ℝ (π‘₯ Β· 𝑧) = 1 ⊻ π‘₯ = 0))
 
10-Jun-2024trirec0 14954 Every real number having a reciprocal or equaling zero is equivalent to real number trichotomy.

This is the key part of the definition of what is known as a discrete field, so "the real numbers are a discrete field" can be taken as an equivalent way to state real trichotomy (see further discussion at trilpo 14953). (Contributed by Jim Kingdon, 10-Jun-2024.)

(βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ < 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 < π‘₯) ↔ βˆ€π‘₯ ∈ ℝ (βˆƒπ‘§ ∈ ℝ (π‘₯ Β· 𝑧) = 1 ∨ π‘₯ = 0))
 
9-Jun-2024omniwomnimkv 7168 A set is omniscient if and only if it is weakly omniscient and Markov. The case 𝐴 = Ο‰ says that LPO ↔ WLPO ∧ MP which is a remark following Definition 2.5 of [Pierik], p. 9. (Contributed by Jim Kingdon, 9-Jun-2024.)
(𝐴 ∈ Omni ↔ (𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov))
 
9-Jun-2024iswomnimap 7167 The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ WOmni ↔ βˆ€π‘“ ∈ (2o β†‘π‘š 𝐴)DECID βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1o))
 
9-Jun-2024iswomni 7166 The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ WOmni ↔ βˆ€π‘“(𝑓:𝐴⟢2o β†’ DECID βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1o)))
 
9-Jun-2024df-womni 7165 A weakly omniscient set is one where we can decide whether a predicate (here represented by a function 𝑓) holds (is equal to 1o) for all elements or not. Generalization of definition 2.4 of [Pierik], p. 9.

In particular, Ο‰ ∈ WOmni is known as the Weak Limited Principle of Omniscience (WLPO).

The term WLPO is common in the literature; there appears to be no widespread term for what we are calling a weakly omniscient set. (Contributed by Jim Kingdon, 9-Jun-2024.)

WOmni = {𝑦 ∣ βˆ€π‘“(𝑓:π‘¦βŸΆ2o β†’ DECID βˆ€π‘₯ ∈ 𝑦 (π‘“β€˜π‘₯) = 1o)}
 
1-Jun-2024cmnmndd 13122 A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
(πœ‘ β†’ 𝐺 ∈ CMnd)    β‡’   (πœ‘ β†’ 𝐺 ∈ Mnd)
 
1-Jun-2024grpmndd 12896 A group is a monoid. (Contributed by SN, 1-Jun-2024.)
(πœ‘ β†’ 𝐺 ∈ Grp)    β‡’   (πœ‘ β†’ 𝐺 ∈ Mnd)
 
29-May-2024pw1nct 14914 A condition which ensures that the powerset of a singleton is not countable. The antecedent here can be referred to as the uniformity principle. Based on Mastodon posts by Andrej Bauer and Rahul Chhabra. (Contributed by Jim Kingdon, 29-May-2024.)
(βˆ€π‘Ÿ(π‘Ÿ βŠ† (𝒫 1o Γ— Ο‰) β†’ (βˆ€π‘ ∈ 𝒫 1oβˆƒπ‘› ∈ Ο‰ π‘π‘Ÿπ‘› β†’ βˆƒπ‘š ∈ Ο‰ βˆ€π‘ž ∈ 𝒫 1oπ‘žπ‘Ÿπ‘š)) β†’ Β¬ βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝒫 1o βŠ” 1o))
 
28-May-2024sssneq 14913 Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.)
(𝐴 βŠ† {𝐡} β†’ βˆ€π‘¦ ∈ 𝐴 βˆ€π‘§ ∈ 𝐴 𝑦 = 𝑧)
 
26-May-2024elpwi2 4160 Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof shortened by Wolf Lammen, 26-May-2024.)
𝐡 ∈ 𝑉    &   π΄ βŠ† 𝐡    β‡’   π΄ ∈ 𝒫 𝐡
 
24-May-2024dvmptcjx 14347 Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 24-May-2024.)
((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ β„‚)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ (ℝ D (π‘₯ ∈ 𝑋 ↦ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ 𝐡))    &   (πœ‘ β†’ 𝑋 βŠ† ℝ)    β‡’   (πœ‘ β†’ (ℝ D (π‘₯ ∈ 𝑋 ↦ (βˆ—β€˜π΄))) = (π‘₯ ∈ 𝑋 ↦ (βˆ—β€˜π΅)))
 
23-May-2024cbvralfw 2695 Rule used to change bound variables, using implicit substitution. Version of cbvralf 2697 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1507 and ax-bndl 1509 in the proof. (Contributed by NM, 7-Mar-2004.) (Revised by Gino Giotto, 23-May-2024.)
β„²π‘₯𝐴    &   β„²π‘¦π΄    &   β„²π‘¦πœ‘    &   β„²π‘₯πœ“    &   (π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆ€π‘₯ ∈ 𝐴 πœ‘ ↔ βˆ€π‘¦ ∈ 𝐴 πœ“)
 
22-May-2024efltlemlt 14356 Lemma for eflt 14357. The converse of efltim 11709 plus the epsilon-delta setup. (Contributed by Jim Kingdon, 22-May-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (expβ€˜π΄) < (expβ€˜π΅))    &   (πœ‘ β†’ 𝐷 ∈ ℝ+)    &   (πœ‘ β†’ ((absβ€˜(𝐴 βˆ’ 𝐡)) < 𝐷 β†’ (absβ€˜((expβ€˜π΄) βˆ’ (expβ€˜π΅))) < ((expβ€˜π΅) βˆ’ (expβ€˜π΄))))    β‡’   (πœ‘ β†’ 𝐴 < 𝐡)
 
21-May-2024eflt 14357 The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 21-May-2024.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 < 𝐡 ↔ (expβ€˜π΄) < (expβ€˜π΅)))
 
19-May-2024apdifflemr 14957 Lemma for apdiff 14958. (Contributed by Jim Kingdon, 19-May-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝑆 ∈ β„š)    &   (πœ‘ β†’ (absβ€˜(𝐴 βˆ’ -1)) # (absβ€˜(𝐴 βˆ’ 1)))    &   ((πœ‘ ∧ 𝑆 β‰  0) β†’ (absβ€˜(𝐴 βˆ’ 0)) # (absβ€˜(𝐴 βˆ’ (2 Β· 𝑆))))    β‡’   (πœ‘ β†’ 𝐴 # 𝑆)
 
18-May-2024apdifflemf 14956 Lemma for apdiff 14958. Being apart from the point halfway between 𝑄 and 𝑅 suffices for 𝐴 to be a different distance from 𝑄 and from 𝑅. (Contributed by Jim Kingdon, 18-May-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝑄 ∈ β„š)    &   (πœ‘ β†’ 𝑅 ∈ β„š)    &   (πœ‘ β†’ 𝑄 < 𝑅)    &   (πœ‘ β†’ ((𝑄 + 𝑅) / 2) # 𝐴)    β‡’   (πœ‘ β†’ (absβ€˜(𝐴 βˆ’ 𝑄)) # (absβ€˜(𝐴 βˆ’ 𝑅)))
 
17-May-2024apdiff 14958 The irrationals (reals apart from any rational) are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 17-May-2024.)
(𝐴 ∈ ℝ β†’ (βˆ€π‘ž ∈ β„š 𝐴 # π‘ž ↔ βˆ€π‘ž ∈ β„š βˆ€π‘Ÿ ∈ β„š (π‘ž β‰  π‘Ÿ β†’ (absβ€˜(𝐴 βˆ’ π‘ž)) # (absβ€˜(𝐴 βˆ’ π‘Ÿ)))))
 
16-May-2024lmodgrpd 13398 A left module is a group. (Contributed by SN, 16-May-2024.)
(πœ‘ β†’ π‘Š ∈ LMod)    β‡’   (πœ‘ β†’ π‘Š ∈ Grp)
 
16-May-2024crnggrpd 13204 A commutative ring is a group. (Contributed by SN, 16-May-2024.)
(πœ‘ β†’ 𝑅 ∈ CRing)    β‡’   (πœ‘ β†’ 𝑅 ∈ Grp)
 
16-May-2024crngringd 13203 A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
(πœ‘ β†’ 𝑅 ∈ CRing)    β‡’   (πœ‘ β†’ 𝑅 ∈ Ring)
 
16-May-2024ringgrpd 13199 A ring is a group. (Contributed by SN, 16-May-2024.)
(πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ 𝑅 ∈ Grp)
 
15-May-2024reeff1oleme 14354 Lemma for reeff1o 14355. (Contributed by Jim Kingdon, 15-May-2024.)
(π‘ˆ ∈ (0(,)e) β†’ βˆƒπ‘₯ ∈ ℝ (expβ€˜π‘₯) = π‘ˆ)
 
14-May-2024df-relog 14440 Define the natural logarithm function. Defining the logarithm on complex numbers is similar to square root - there are ways to define it but they tend to make use of excluded middle. Therefore, we merely define logarithms on positive reals. See http://en.wikipedia.org/wiki/Natural_logarithm and https://en.wikipedia.org/wiki/Complex_logarithm. (Contributed by Jim Kingdon, 14-May-2024.)
log = β—‘(exp β†Ύ ℝ)
 
12-May-2024dvdstrd 11840 The divides relation is transitive, a deduction version of dvdstr 11838. (Contributed by metakunt, 12-May-2024.)
(πœ‘ β†’ 𝐾 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝐾 βˆ₯ 𝑀)    &   (πœ‘ β†’ 𝑀 βˆ₯ 𝑁)    β‡’   (πœ‘ β†’ 𝐾 βˆ₯ 𝑁)
 
7-May-2024ioocosf1o 14436 The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.) (Revised by Jim Kingdon, 7-May-2024.)
(cos β†Ύ (0(,)Ο€)):(0(,)Ο€)–1-1-ontoβ†’(-1(,)1)
 
7-May-2024cos0pilt1 14434 Cosine is between minus one and one on the open interval between zero and Ο€. (Contributed by Jim Kingdon, 7-May-2024.)
(𝐴 ∈ (0(,)Ο€) β†’ (cosβ€˜π΄) ∈ (-1(,)1))
 
6-May-2024cos11 14435 Cosine is one-to-one over the closed interval from 0 to Ο€. (Contributed by Paul Chapman, 16-Mar-2008.) (Revised by Jim Kingdon, 6-May-2024.)
((𝐴 ∈ (0[,]Ο€) ∧ 𝐡 ∈ (0[,]Ο€)) β†’ (𝐴 = 𝐡 ↔ (cosβ€˜π΄) = (cosβ€˜π΅)))
 
5-May-2024omiunct 12448 The union of a countably infinite collection of countable sets is countable. Theorem 8.1.28 of [AczelRathjen], p. 78. Compare with ctiunct 12444 which has a stronger hypothesis but does not require countable choice. (Contributed by Jim Kingdon, 5-May-2024.)
(πœ‘ β†’ CCHOICE)    &   ((πœ‘ ∧ π‘₯ ∈ Ο‰) β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐡 βŠ” 1o))    β‡’   (πœ‘ β†’ βˆƒβ„Ž β„Ž:ω–ontoβ†’(βˆͺ π‘₯ ∈ Ο‰ 𝐡 βŠ” 1o))
 
5-May-2024ctiunctal 12445 Variation of ctiunct 12444 which allows π‘₯ to be present in πœ‘. (Contributed by Jim Kingdon, 5-May-2024.)
(πœ‘ β†’ 𝐹:ω–ontoβ†’(𝐴 βŠ” 1o))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 𝐺:ω–ontoβ†’(𝐡 βŠ” 1o))    β‡’   (πœ‘ β†’ βˆƒβ„Ž β„Ž:ω–ontoβ†’(βˆͺ π‘₯ ∈ 𝐴 𝐡 βŠ” 1o))
 
3-May-2024cc4n 7273 Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7272, the hypotheses only require an A(n) for each value of 𝑛, not a single set 𝐴 which suffices for every 𝑛 ∈ Ο‰. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
(πœ‘ β†’ CCHOICE)    &   (πœ‘ β†’ βˆ€π‘› ∈ 𝑁 {π‘₯ ∈ 𝐴 ∣ πœ“} ∈ 𝑉)    &   (πœ‘ β†’ 𝑁 β‰ˆ Ο‰)    &   (π‘₯ = (π‘“β€˜π‘›) β†’ (πœ“ ↔ πœ’))    &   (πœ‘ β†’ βˆ€π‘› ∈ 𝑁 βˆƒπ‘₯ ∈ 𝐴 πœ“)    β‡’   (πœ‘ β†’ βˆƒπ‘“(𝑓 Fn 𝑁 ∧ βˆ€π‘› ∈ 𝑁 πœ’))
 
3-May-2024cc4f 7271 Countable choice by showing the existence of a function 𝑓 which can choose a value at each index 𝑛 such that πœ’ holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
(πœ‘ β†’ CCHOICE)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   β„²π‘›π΄    &   (πœ‘ β†’ 𝑁 β‰ˆ Ο‰)    &   (π‘₯ = (π‘“β€˜π‘›) β†’ (πœ“ ↔ πœ’))    &   (πœ‘ β†’ βˆ€π‘› ∈ 𝑁 βˆƒπ‘₯ ∈ 𝐴 πœ“)    β‡’   (πœ‘ β†’ βˆƒπ‘“(𝑓:π‘βŸΆπ΄ ∧ βˆ€π‘› ∈ 𝑁 πœ’))
 
1-May-2024cc4 7272 Countable choice by showing the existence of a function 𝑓 which can choose a value at each index 𝑛 such that πœ’ holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 1-May-2024.)
(πœ‘ β†’ CCHOICE)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝑁 β‰ˆ Ο‰)    &   (π‘₯ = (π‘“β€˜π‘›) β†’ (πœ“ ↔ πœ’))    &   (πœ‘ β†’ βˆ€π‘› ∈ 𝑁 βˆƒπ‘₯ ∈ 𝐴 πœ“)    β‡’   (πœ‘ β†’ βˆƒπ‘“(𝑓:π‘βŸΆπ΄ ∧ βˆ€π‘› ∈ 𝑁 πœ’))
 
29-Apr-2024cc3 7270 Countable choice using a sequence F(n) . (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Jim Kingdon, 29-Apr-2024.)
(πœ‘ β†’ CCHOICE)    &   (πœ‘ β†’ βˆ€π‘› ∈ 𝑁 𝐹 ∈ V)    &   (πœ‘ β†’ βˆ€π‘› ∈ 𝑁 βˆƒπ‘€ 𝑀 ∈ 𝐹)    &   (πœ‘ β†’ 𝑁 β‰ˆ Ο‰)    β‡’   (πœ‘ β†’ βˆƒπ‘“(𝑓 Fn 𝑁 ∧ βˆ€π‘› ∈ 𝑁 (π‘“β€˜π‘›) ∈ 𝐹))
 
27-Apr-2024cc2 7269 Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
(πœ‘ β†’ CCHOICE)    &   (πœ‘ β†’ 𝐹 Fn Ο‰)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ Ο‰ βˆƒπ‘€ 𝑀 ∈ (πΉβ€˜π‘₯))    β‡’   (πœ‘ β†’ βˆƒπ‘”(𝑔 Fn Ο‰ ∧ βˆ€π‘› ∈ Ο‰ (π‘”β€˜π‘›) ∈ (πΉβ€˜π‘›)))
 
27-Apr-2024cc2lem 7268 Lemma for cc2 7269. (Contributed by Jim Kingdon, 27-Apr-2024.)
(πœ‘ β†’ CCHOICE)    &   (πœ‘ β†’ 𝐹 Fn Ο‰)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ Ο‰ βˆƒπ‘€ 𝑀 ∈ (πΉβ€˜π‘₯))    &   π΄ = (𝑛 ∈ Ο‰ ↦ ({𝑛} Γ— (πΉβ€˜π‘›)))    &   πΊ = (𝑛 ∈ Ο‰ ↦ (2nd β€˜(π‘“β€˜(π΄β€˜π‘›))))    β‡’   (πœ‘ β†’ βˆƒπ‘”(𝑔 Fn Ο‰ ∧ βˆ€π‘› ∈ Ο‰ (π‘”β€˜π‘›) ∈ (πΉβ€˜π‘›)))
 
27-Apr-2024cc1 7267 Countable choice in terms of a choice function on a countably infinite set of inhabited sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
(CCHOICE β†’ βˆ€π‘₯((π‘₯ β‰ˆ Ο‰ ∧ βˆ€π‘§ ∈ π‘₯ βˆƒπ‘€ 𝑀 ∈ 𝑧) β†’ βˆƒπ‘“βˆ€π‘§ ∈ π‘₯ (π‘“β€˜π‘§) ∈ 𝑧))
 
24-Apr-2024lsppropd 13530 If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   (πœ‘ β†’ 𝐡 βŠ† π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ π‘Š ∧ 𝑦 ∈ π‘Š)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) ∈ π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))    &   (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜πΎ)))    &   (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜πΏ)))    &   (πœ‘ β†’ 𝐾 ∈ 𝑋)    &   (πœ‘ β†’ 𝐿 ∈ π‘Œ)    β‡’   (πœ‘ β†’ (LSpanβ€˜πΎ) = (LSpanβ€˜πΏ))
 
19-Apr-2024omctfn 12447 Using countable choice to find a sequence of enumerations for a collection of countable sets. Lemma 8.1.27 of [AczelRathjen], p. 77. (Contributed by Jim Kingdon, 19-Apr-2024.)
(πœ‘ β†’ CCHOICE)    &   ((πœ‘ ∧ π‘₯ ∈ Ο‰) β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐡 βŠ” 1o))    β‡’   (πœ‘ β†’ βˆƒπ‘“(𝑓 Fn Ο‰ ∧ βˆ€π‘₯ ∈ Ο‰ (π‘“β€˜π‘₯):ω–ontoβ†’(𝐡 βŠ” 1o)))
 
13-Apr-2024prodmodclem2 11588 Lemma for prodmodc 11589. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 13-Apr-2024.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 1))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   πΊ = (𝑗 ∈ β„• ↦ if(𝑗 ≀ (β™―β€˜π΄), ⦋(π‘“β€˜π‘—) / π‘˜β¦Œπ΅, 1))    β‡’   ((πœ‘ ∧ βˆƒπ‘š ∈ β„€ ((𝐴 βŠ† (β„€β‰₯β€˜π‘š) ∧ βˆ€π‘— ∈ (β„€β‰₯β€˜π‘š)DECID 𝑗 ∈ 𝐴) ∧ (βˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)βˆƒπ‘¦(𝑦 # 0 ∧ seq𝑛( Β· , 𝐹) ⇝ 𝑦) ∧ seqπ‘š( Β· , 𝐹) ⇝ π‘₯))) β†’ (βˆƒπ‘š ∈ β„• βˆƒπ‘“(𝑓:(1...π‘š)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( Β· , 𝐺)β€˜π‘š)) β†’ π‘₯ = 𝑧))
 
11-Apr-2024prodmodclem2a 11587 Lemma for prodmodc 11589. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 1))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   πΊ = (𝑗 ∈ β„• ↦ if(𝑗 ≀ (β™―β€˜π΄), ⦋(π‘“β€˜π‘—) / π‘˜β¦Œπ΅, 1))    &   π» = (𝑗 ∈ β„• ↦ if(𝑗 ≀ (β™―β€˜π΄), ⦋(πΎβ€˜π‘—) / π‘˜β¦Œπ΅, 1))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ DECID π‘˜ ∈ 𝐴)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐴 βŠ† (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ 𝑓:(1...𝑁)–1-1-onto→𝐴)    &   (πœ‘ β†’ 𝐾 Isom < , < ((1...(β™―β€˜π΄)), 𝐴))    β‡’   (πœ‘ β†’ seq𝑀( Β· , 𝐹) ⇝ (seq1( Β· , 𝐺)β€˜π‘))
 
11-Apr-2024prodmodclem3 11586 Lemma for prodmodc 11589. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 1))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   πΊ = (𝑗 ∈ β„• ↦ if(𝑗 ≀ (β™―β€˜π΄), ⦋(π‘“β€˜π‘—) / π‘˜β¦Œπ΅, 1))    &   π» = (𝑗 ∈ β„• ↦ if(𝑗 ≀ (β™―β€˜π΄), ⦋(πΎβ€˜π‘—) / π‘˜β¦Œπ΅, 1))    &   (πœ‘ β†’ (𝑀 ∈ β„• ∧ 𝑁 ∈ β„•))    &   (πœ‘ β†’ 𝑓:(1...𝑀)–1-1-onto→𝐴)    &   (πœ‘ β†’ 𝐾:(1...𝑁)–1-1-onto→𝐴)    β‡’   (πœ‘ β†’ (seq1( Β· , 𝐺)β€˜π‘€) = (seq1( Β· , 𝐻)β€˜π‘))
 
10-Apr-2024jcnd 652 Deduction joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Apr-2024.)
(πœ‘ β†’ πœ“)    &   (πœ‘ β†’ Β¬ πœ’)    β‡’   (πœ‘ β†’ Β¬ (πœ“ β†’ πœ’))
 
4-Apr-2024prodrbdclem 11582 Lemma for prodrbdc 11585. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 4-Apr-2024.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 1))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ DECID π‘˜ ∈ 𝐴)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    β‡’   ((πœ‘ ∧ 𝐴 βŠ† (β„€β‰₯β€˜π‘)) β†’ (seq𝑀( Β· , 𝐹) β†Ύ (β„€β‰₯β€˜π‘)) = seq𝑁( Β· , 𝐹))
 
24-Mar-2024prodfdivap 11558 The quotient of two products. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΊβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΊβ€˜π‘˜) # 0)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (π»β€˜π‘˜) = ((πΉβ€˜π‘˜) / (πΊβ€˜π‘˜)))    β‡’   (πœ‘ β†’ (seq𝑀( Β· , 𝐻)β€˜π‘) = ((seq𝑀( Β· , 𝐹)β€˜π‘) / (seq𝑀( Β· , 𝐺)β€˜π‘)))
 
24-Mar-2024prodfrecap 11557 The reciprocal of a finite product. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) # 0)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΊβ€˜π‘˜) = (1 / (πΉβ€˜π‘˜)))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΊβ€˜π‘˜) ∈ β„‚)    β‡’   (πœ‘ β†’ (seq𝑀( Β· , 𝐺)β€˜π‘) = (1 / (seq𝑀( Β· , 𝐹)β€˜π‘)))
 
23-Mar-2024prodfap0 11556 The product of finitely many terms apart from zero is apart from zero. (Contributed by Scott Fenton, 14-Jan-2018.) (Revised by Jim Kingdon, 23-Mar-2024.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) # 0)    β‡’   (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) # 0)
 
22-Mar-2024prod3fmul 11552 The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017.) (Revised by Jim Kingdon, 22-Mar-2024.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΊβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (π»β€˜π‘˜) = ((πΉβ€˜π‘˜) Β· (πΊβ€˜π‘˜)))    β‡’   (πœ‘ β†’ (seq𝑀( Β· , 𝐻)β€˜π‘) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq𝑀( Β· , 𝐺)β€˜π‘)))
 
21-Mar-2024df-proddc 11562 Define the product of a series with an index set of integers 𝐴. This definition takes most of the aspects of df-sumdc 11365 and adapts them for multiplication instead of addition. However, we insist that in the infinite case, there is a nonzero tail of the sequence. This ensures that the convergence criteria match those of infinite sums. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 21-Mar-2024.)
βˆπ‘˜ ∈ 𝐴 𝐡 = (β„©π‘₯(βˆƒπ‘š ∈ β„€ ((𝐴 βŠ† (β„€β‰₯β€˜π‘š) ∧ βˆ€π‘— ∈ (β„€β‰₯β€˜π‘š)DECID 𝑗 ∈ 𝐴) ∧ (βˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)βˆƒπ‘¦(𝑦 # 0 ∧ seq𝑛( Β· , (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 1))) ⇝ 𝑦) ∧ seqπ‘š( Β· , (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 1))) ⇝ π‘₯)) ∨ βˆƒπ‘š ∈ β„• βˆƒπ‘“(𝑓:(1...π‘š)–1-1-onto→𝐴 ∧ π‘₯ = (seq1( Β· , (𝑛 ∈ β„• ↦ if(𝑛 ≀ π‘š, ⦋(π‘“β€˜π‘›) / π‘˜β¦Œπ΅, 1)))β€˜π‘š))))
 
19-Mar-2024cos02pilt1 14433 Cosine is less than one between zero and 2 Β· Ο€. (Contributed by Jim Kingdon, 19-Mar-2024.)
(𝐴 ∈ (0(,)(2 Β· Ο€)) β†’ (cosβ€˜π΄) < 1)
 
19-Mar-2024cosq34lt1 14432 Cosine is less than one in the third and fourth quadrants. (Contributed by Jim Kingdon, 19-Mar-2024.)
(𝐴 ∈ (Ο€[,)(2 Β· Ο€)) β†’ (cosβ€˜π΄) < 1)
 
14-Mar-2024coseq0q4123 14416 Location of the zeroes of cosine in (-(Ο€ / 2)(,)(3 Β· (Ο€ / 2))). (Contributed by Jim Kingdon, 14-Mar-2024.)
(𝐴 ∈ (-(Ο€ / 2)(,)(3 Β· (Ο€ / 2))) β†’ ((cosβ€˜π΄) = 0 ↔ 𝐴 = (Ο€ / 2)))
 
14-Mar-2024cosq23lt0 14415 The cosine of a number in the second and third quadrants is negative. (Contributed by Jim Kingdon, 14-Mar-2024.)
(𝐴 ∈ ((Ο€ / 2)(,)(3 Β· (Ο€ / 2))) β†’ (cosβ€˜π΄) < 0)
 
9-Mar-2024pilem3 14365 Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.)
(Ο€ ∈ (2(,)4) ∧ (sinβ€˜Ο€) = 0)
 
9-Mar-2024exmidonfin 7196 If a finite ordinal is a natural number, excluded middle follows. That excluded middle implies that a finite ordinal is a natural number is proved in the Metamath Proof Explorer. That a natural number is a finite ordinal is shown at nnfi 6875 and nnon 4611. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
(Ο‰ = (On ∩ Fin) β†’ EXMID)
 
9-Mar-2024exmidonfinlem 7195 Lemma for exmidonfin 7196. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
𝐴 = {{π‘₯ ∈ {βˆ…} ∣ πœ‘}, {π‘₯ ∈ {βˆ…} ∣ Β¬ πœ‘}}    β‡’   (Ο‰ = (On ∩ Fin) β†’ DECID πœ‘)
 
8-Mar-2024sin0pilem2 14364 Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.)
βˆƒπ‘ž ∈ (2(,)4)((sinβ€˜π‘ž) = 0 ∧ βˆ€π‘₯ ∈ (0(,)π‘ž)0 < (sinβ€˜π‘₯))
 
8-Mar-2024sin0pilem1 14363 Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.)
βˆƒπ‘ ∈ (1(,)2)((cosβ€˜π‘) = 0 ∧ βˆ€π‘₯ ∈ (𝑝(,)(2 Β· 𝑝))0 < (sinβ€˜π‘₯))
 
7-Mar-2024cosz12 14362 Cosine has a zero between 1 and 2. (Contributed by Mario Carneiro and Jim Kingdon, 7-Mar-2024.)
βˆƒπ‘ ∈ (1(,)2)(cosβ€˜π‘) = 0
 
6-Mar-2024cos12dec 11778 Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.)
((𝐴 ∈ (1[,]2) ∧ 𝐡 ∈ (1[,]2) ∧ 𝐴 < 𝐡) β†’ (cosβ€˜π΅) < (cosβ€˜π΄))
 
2-Mar-2024scaffvalg 13407 The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
𝐡 = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    βˆ™ = ( Β·sf β€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    β‡’   (π‘Š ∈ 𝑉 β†’ βˆ™ = (π‘₯ ∈ 𝐾, 𝑦 ∈ 𝐡 ↦ (π‘₯ Β· 𝑦)))
 
2-Mar-2024dvrfvald 13313 Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ Β· = (.rβ€˜π‘…))    &   (πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘…))    &   (πœ‘ β†’ 𝐼 = (invrβ€˜π‘…))    &   (πœ‘ β†’ / = (/rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ SRing)    β‡’   (πœ‘ β†’ / = (π‘₯ ∈ 𝐡, 𝑦 ∈ π‘ˆ ↦ (π‘₯ Β· (πΌβ€˜π‘¦))))
 
2-Mar-2024plusffvalg 12788 The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &    ⨣ = (+π‘“β€˜πΊ)    β‡’   (𝐺 ∈ 𝑉 β†’ ⨣ = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (π‘₯ + 𝑦)))
 
25-Feb-2024insubm 12879 The intersection of two submonoids is a submonoid. (Contributed by AV, 25-Feb-2024.)
((𝐴 ∈ (SubMndβ€˜π‘€) ∧ 𝐡 ∈ (SubMndβ€˜π‘€)) β†’ (𝐴 ∩ 𝐡) ∈ (SubMndβ€˜π‘€))
 
25-Feb-2024mul2lt0pn 9767 The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 0)    &   (πœ‘ β†’ 0 < 𝐡)    β‡’   (πœ‘ β†’ (𝐡 Β· 𝐴) < 0)
 
25-Feb-2024mul2lt0np 9766 The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 0)    &   (πœ‘ β†’ 0 < 𝐡)    β‡’   (πœ‘ β†’ (𝐴 Β· 𝐡) < 0)
 
25-Feb-2024lt0ap0 8608 A number which is less than zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐴 < 0) β†’ 𝐴 # 0)
 
25-Feb-2024negap0d 8591 The negative of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐴 # 0)    β‡’   (πœ‘ β†’ -𝐴 # 0)
 
24-Feb-2024lt0ap0d 8609 A real number less than zero is apart from zero. Deduction form. (Contributed by Jim Kingdon, 24-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 0)    β‡’   (πœ‘ β†’ 𝐴 # 0)
 
20-Feb-2024ivthdec 14283 The intermediate value theorem, decreasing case, for a strictly monotonic function. (Contributed by Jim Kingdon, 20-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ π‘ˆ ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ (𝐴[,]𝐡) βŠ† 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐷–cnβ†’β„‚))    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (πΉβ€˜π‘₯) ∈ ℝ)    &   (πœ‘ β†’ ((πΉβ€˜π΅) < π‘ˆ ∧ π‘ˆ < (πΉβ€˜π΄)))    &   (((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) ∧ (𝑦 ∈ (𝐴[,]𝐡) ∧ π‘₯ < 𝑦)) β†’ (πΉβ€˜π‘¦) < (πΉβ€˜π‘₯))    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ (𝐴(,)𝐡)(πΉβ€˜π‘) = π‘ˆ)
 
20-Feb-2024ivthinclemex 14281 Lemma for ivthinc 14282. Existence of a number between the lower cut and the upper cut. (Contributed by Jim Kingdon, 20-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ π‘ˆ ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ (𝐴[,]𝐡) βŠ† 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐷–cnβ†’β„‚))    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (πΉβ€˜π‘₯) ∈ ℝ)    &   (πœ‘ β†’ ((πΉβ€˜π΄) < π‘ˆ ∧ π‘ˆ < (πΉβ€˜π΅)))    &   (((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) ∧ (𝑦 ∈ (𝐴[,]𝐡) ∧ π‘₯ < 𝑦)) β†’ (πΉβ€˜π‘₯) < (πΉβ€˜π‘¦))    &   πΏ = {𝑀 ∈ (𝐴[,]𝐡) ∣ (πΉβ€˜π‘€) < π‘ˆ}    &   π‘… = {𝑀 ∈ (𝐴[,]𝐡) ∣ π‘ˆ < (πΉβ€˜π‘€)}    β‡’   (πœ‘ β†’ βˆƒ!𝑧 ∈ (𝐴(,)𝐡)(βˆ€π‘ž ∈ 𝐿 π‘ž < 𝑧 ∧ βˆ€π‘Ÿ ∈ 𝑅 𝑧 < π‘Ÿ))
 
19-Feb-2024ivthinclemuopn 14277 Lemma for ivthinc 14282. The upper cut is open. (Contributed by Jim Kingdon, 19-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ π‘ˆ ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ (𝐴[,]𝐡) βŠ† 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐷–cnβ†’β„‚))    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (πΉβ€˜π‘₯) ∈ ℝ)    &   (πœ‘ β†’ ((πΉβ€˜π΄) < π‘ˆ ∧ π‘ˆ < (πΉβ€˜π΅)))    &   (((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) ∧ (𝑦 ∈ (𝐴[,]𝐡) ∧ π‘₯ < 𝑦)) β†’ (πΉβ€˜π‘₯) < (πΉβ€˜π‘¦))    &   πΏ = {𝑀 ∈ (𝐴[,]𝐡) ∣ (πΉβ€˜π‘€) < π‘ˆ}    &   π‘… = {𝑀 ∈ (𝐴[,]𝐡) ∣ π‘ˆ < (πΉβ€˜π‘€)}    &   (πœ‘ β†’ 𝑆 ∈ 𝑅)    β‡’   (πœ‘ β†’ βˆƒπ‘ž ∈ 𝑅 π‘ž < 𝑆)
 
19-Feb-2024dedekindicc 14272 A Dedekind cut identifies a unique real number. Similar to df-inp 7468 except that the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 19-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐿 βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ π‘ˆ βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ βˆƒπ‘ž ∈ (𝐴[,]𝐡)π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ (𝐴[,]𝐡)π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)(π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    &   (πœ‘ β†’ 𝐴 < 𝐡)    β‡’   (πœ‘ β†’ βˆƒ!π‘₯ ∈ (𝐴(,)𝐡)(βˆ€π‘ž ∈ 𝐿 π‘ž < π‘₯ ∧ βˆ€π‘Ÿ ∈ π‘ˆ π‘₯ < π‘Ÿ))
 
19-Feb-2024grpsubfvalg 12925 Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &   πΌ = (invgβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    β‡’   (𝐺 ∈ 𝑉 β†’ βˆ’ = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (π‘₯ + (πΌβ€˜π‘¦))))
 
18-Feb-2024ivthinclemloc 14280 Lemma for ivthinc 14282. Locatedness. (Contributed by Jim Kingdon, 18-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ π‘ˆ ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ (𝐴[,]𝐡) βŠ† 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐷–cnβ†’β„‚))    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (πΉβ€˜π‘₯) ∈ ℝ)    &   (πœ‘ β†’ ((πΉβ€˜π΄) < π‘ˆ ∧ π‘ˆ < (πΉβ€˜π΅)))    &   (((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) ∧ (𝑦 ∈ (𝐴[,]𝐡) ∧ π‘₯ < 𝑦)) β†’ (πΉβ€˜π‘₯) < (πΉβ€˜π‘¦))    &   πΏ = {𝑀 ∈ (𝐴[,]𝐡) ∣ (πΉβ€˜π‘€) < π‘ˆ}    &   π‘… = {𝑀 ∈ (𝐴[,]𝐡) ∣ π‘ˆ < (πΉβ€˜π‘€)}    β‡’   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ 𝑅)))
 
18-Feb-2024ivthinclemdisj 14279 Lemma for ivthinc 14282. The lower and upper cuts are disjoint. (Contributed by Jim Kingdon, 18-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ π‘ˆ ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ (𝐴[,]𝐡) βŠ† 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐷–cnβ†’β„‚))    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (πΉβ€˜π‘₯) ∈ ℝ)    &   (πœ‘ β†’ ((πΉβ€˜π΄) < π‘ˆ ∧ π‘ˆ < (πΉβ€˜π΅)))    &   (((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) ∧ (𝑦 ∈ (𝐴[,]𝐡) ∧ π‘₯ < 𝑦)) β†’ (πΉβ€˜π‘₯) < (πΉβ€˜π‘¦))    &   πΏ = {𝑀 ∈ (𝐴[,]𝐡) ∣ (πΉβ€˜π‘€) < π‘ˆ}    &   π‘… = {𝑀 ∈ (𝐴[,]𝐡) ∣ π‘ˆ < (πΉβ€˜π‘€)}    β‡’   (πœ‘ β†’ (𝐿 ∩ 𝑅) = βˆ…)
 
18-Feb-2024ivthinclemur 14278 Lemma for ivthinc 14282. The upper cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ π‘ˆ ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ (𝐴[,]𝐡) βŠ† 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐷–cnβ†’β„‚))    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (πΉβ€˜π‘₯) ∈ ℝ)    &   (πœ‘ β†’ ((πΉβ€˜π΄) < π‘ˆ ∧ π‘ˆ < (πΉβ€˜π΅)))    &   (((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) ∧ (𝑦 ∈ (𝐴[,]𝐡) ∧ π‘₯ < 𝑦)) β†’ (πΉβ€˜π‘₯) < (πΉβ€˜π‘¦))    &   πΏ = {𝑀 ∈ (𝐴[,]𝐡) ∣ (πΉβ€˜π‘€) < π‘ˆ}    &   π‘… = {𝑀 ∈ (𝐴[,]𝐡) ∣ π‘ˆ < (πΉβ€˜π‘€)}    β‡’   (πœ‘ β†’ βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘Ÿ ∈ 𝑅 ↔ βˆƒπ‘ž ∈ 𝑅 π‘ž < π‘Ÿ))
 
18-Feb-2024ivthinclemlr 14276 Lemma for ivthinc 14282. The lower cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ π‘ˆ ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ (𝐴[,]𝐡) βŠ† 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐷–cnβ†’β„‚))    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (πΉβ€˜π‘₯) ∈ ℝ)    &   (πœ‘ β†’ ((πΉβ€˜π΄) < π‘ˆ ∧ π‘ˆ < (πΉβ€˜π΅)))    &   (((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) ∧ (𝑦 ∈ (𝐴[,]𝐡) ∧ π‘₯ < 𝑦)) β†’ (πΉβ€˜π‘₯) < (πΉβ€˜π‘¦))    &   πΏ = {𝑀 ∈ (𝐴[,]𝐡) ∣ (πΉβ€˜π‘€) < π‘ˆ}    &   π‘… = {𝑀 ∈ (𝐴[,]𝐡) ∣ π‘ˆ < (πΉβ€˜π‘€)}    β‡’   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)(π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))
 
18-Feb-2024ivthinclemum 14274 Lemma for ivthinc 14282. The upper cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ π‘ˆ ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ (𝐴[,]𝐡) βŠ† 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐷–cnβ†’β„‚))    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (πΉβ€˜π‘₯) ∈ ℝ)    &   (πœ‘ β†’ ((πΉβ€˜π΄) < π‘ˆ ∧ π‘ˆ < (πΉβ€˜π΅)))    &   (((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) ∧ (𝑦 ∈ (𝐴[,]𝐡) ∧ π‘₯ < 𝑦)) β†’ (πΉβ€˜π‘₯) < (πΉβ€˜π‘¦))    &   πΏ = {𝑀 ∈ (𝐴[,]𝐡) ∣ (πΉβ€˜π‘€) < π‘ˆ}    &   π‘… = {𝑀 ∈ (𝐴[,]𝐡) ∣ π‘ˆ < (πΉβ€˜π‘€)}    β‡’   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ (𝐴[,]𝐡)π‘Ÿ ∈ 𝑅)
 
18-Feb-2024ivthinclemlm 14273 Lemma for ivthinc 14282. The lower cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ π‘ˆ ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ (𝐴[,]𝐡) βŠ† 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐷–cnβ†’β„‚))    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (πΉβ€˜π‘₯) ∈ ℝ)    &   (πœ‘ β†’ ((πΉβ€˜π΄) < π‘ˆ ∧ π‘ˆ < (πΉβ€˜π΅)))    &   (((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) ∧ (𝑦 ∈ (𝐴[,]𝐡) ∧ π‘₯ < 𝑦)) β†’ (πΉβ€˜π‘₯) < (πΉβ€˜π‘¦))    &   πΏ = {𝑀 ∈ (𝐴[,]𝐡) ∣ (πΉβ€˜π‘€) < π‘ˆ}    &   π‘… = {𝑀 ∈ (𝐴[,]𝐡) ∣ π‘ˆ < (πΉβ€˜π‘€)}    β‡’   (πœ‘ β†’ βˆƒπ‘ž ∈ (𝐴[,]𝐡)π‘ž ∈ 𝐿)
 
17-Feb-20240subm 12878 The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)
0 = (0gβ€˜πΊ)    β‡’   (𝐺 ∈ Mnd β†’ { 0 } ∈ (SubMndβ€˜πΊ))
 
17-Feb-2024mndissubm 12873 If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the the other monoid is contained in the base set of the monoid, then the (base set of the) monoid is a submonoid of the other monoid. (Contributed by AV, 17-Feb-2024.)
𝐡 = (Baseβ€˜πΊ)    &   π‘† = (Baseβ€˜π»)    &    0 = (0gβ€˜πΊ)    β‡’   ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) β†’ ((𝑆 βŠ† 𝐡 ∧ 0 ∈ 𝑆 ∧ (+gβ€˜π») = ((+gβ€˜πΊ) β†Ύ (𝑆 Γ— 𝑆))) β†’ 𝑆 ∈ (SubMndβ€˜πΊ)))
 
17-Feb-2024mgmsscl 12787 If the base set of a magma is contained in the base set of another magma, and the group operation of the magma is the restriction of the group operation of the other magma to its base set, then the base set of the magma is closed under the group operation of the other magma. (Contributed by AV, 17-Feb-2024.)
𝐡 = (Baseβ€˜πΊ)    &   π‘† = (Baseβ€˜π»)    β‡’   (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 βŠ† 𝐡 ∧ (+gβ€˜π») = ((+gβ€˜πΊ) β†Ύ (𝑆 Γ— 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆)) β†’ (𝑋(+gβ€˜πΊ)π‘Œ) ∈ 𝑆)
 
15-Feb-2024dedekindicclemeu 14270 Lemma for dedekindicc 14272. Part of proving uniqueness. (Contributed by Jim Kingdon, 15-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐿 βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ π‘ˆ βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ βˆƒπ‘ž ∈ (𝐴[,]𝐡)π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ (𝐴[,]𝐡)π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)(π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴[,]𝐡))    &   (πœ‘ β†’ (βˆ€π‘ž ∈ 𝐿 π‘ž < 𝐢 ∧ βˆ€π‘Ÿ ∈ π‘ˆ 𝐢 < π‘Ÿ))    &   (πœ‘ β†’ 𝐷 ∈ (𝐴[,]𝐡))    &   (πœ‘ β†’ (βˆ€π‘ž ∈ 𝐿 π‘ž < 𝐷 ∧ βˆ€π‘Ÿ ∈ π‘ˆ 𝐷 < π‘Ÿ))    &   (πœ‘ β†’ 𝐢 < 𝐷)    β‡’   (πœ‘ β†’ βŠ₯)
 
15-Feb-2024dedekindicclemlu 14269 Lemma for dedekindicc 14272. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 15-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐿 βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ π‘ˆ βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ βˆƒπ‘ž ∈ (𝐴[,]𝐡)π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ (𝐴[,]𝐡)π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)(π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    &   (πœ‘ β†’ 𝐴 < 𝐡)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ (𝐴[,]𝐡)(βˆ€π‘ž ∈ 𝐿 π‘ž < π‘₯ ∧ βˆ€π‘Ÿ ∈ π‘ˆ π‘₯ < π‘Ÿ))
 
15-Feb-2024dedekindicclemlub 14268 Lemma for dedekindicc 14272. The set L has a least upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐿 βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ π‘ˆ βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ βˆƒπ‘ž ∈ (𝐴[,]𝐡)π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ (𝐴[,]𝐡)π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)(π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    &   (πœ‘ β†’ 𝐴 < 𝐡)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ (𝐴[,]𝐡)(βˆ€π‘¦ ∈ 𝐿 Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ (𝐴[,]𝐡)(𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ 𝐿 𝑦 < 𝑧)))
 
15-Feb-2024dedekindicclemloc 14267 Lemma for dedekindicc 14272. The set L is located. (Contributed by Jim Kingdon, 15-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐿 βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ π‘ˆ βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ βˆƒπ‘ž ∈ (𝐴[,]𝐡)π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ (𝐴[,]𝐡)π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)(π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    β‡’   (πœ‘ β†’ βˆ€π‘₯ ∈ (𝐴[,]𝐡)βˆ€π‘¦ ∈ (𝐴[,]𝐡)(π‘₯ < 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐿 π‘₯ < 𝑧 ∨ βˆ€π‘§ ∈ 𝐿 𝑧 < 𝑦)))
 
15-Feb-2024dedekindicclemub 14266 Lemma for dedekindicc 14272. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐿 βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ π‘ˆ βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ βˆƒπ‘ž ∈ (𝐴[,]𝐡)π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ (𝐴[,]𝐡)π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)(π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ (𝐴[,]𝐡)βˆ€π‘¦ ∈ 𝐿 𝑦 < π‘₯)
 
15-Feb-2024dedekindicclemuub 14265 Lemma for dedekindicc 14272. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 15-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐿 βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ π‘ˆ βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ βˆƒπ‘ž ∈ (𝐴[,]𝐡)π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ (𝐴[,]𝐡)π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)(π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    &   (πœ‘ β†’ 𝐢 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ βˆ€π‘§ ∈ 𝐿 𝑧 < 𝐢)
 
14-Feb-2024suplociccex 14264 An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8033 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
(πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 < 𝐢)    &   (πœ‘ β†’ 𝐴 βŠ† (𝐡[,]𝐢))    &   (πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ (𝐡[,]𝐢)βˆ€π‘¦ ∈ (𝐡[,]𝐢)(π‘₯ < 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯ < 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧 < 𝑦)))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ (𝐡[,]𝐢)(βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ (𝐡[,]𝐢)(𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 < 𝑧)))
 
14-Feb-2024suplociccreex 14263 An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8033 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
(πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 < 𝐢)    &   (πœ‘ β†’ 𝐴 βŠ† (𝐡[,]𝐢))    &   (πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ (𝐡[,]𝐢)βˆ€π‘¦ ∈ (𝐡[,]𝐢)(π‘₯ < 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯ < 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧 < 𝑦)))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 < 𝑧)))
 
6-Feb-2024ivthinclemlopn 14275 Lemma for ivthinc 14282. The lower cut is open. (Contributed by Jim Kingdon, 6-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ π‘ˆ ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ (𝐴[,]𝐡) βŠ† 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐷–cnβ†’β„‚))    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (πΉβ€˜π‘₯) ∈ ℝ)    &   (πœ‘ β†’ ((πΉβ€˜π΄) < π‘ˆ ∧ π‘ˆ < (πΉβ€˜π΅)))    &   (((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) ∧ (𝑦 ∈ (𝐴[,]𝐡) ∧ π‘₯ < 𝑦)) β†’ (πΉβ€˜π‘₯) < (πΉβ€˜π‘¦))    &   πΏ = {𝑀 ∈ (𝐴[,]𝐡) ∣ (πΉβ€˜π‘€) < π‘ˆ}    &   π‘… = {𝑀 ∈ (𝐴[,]𝐡) ∣ π‘ˆ < (πΉβ€˜π‘€)}    &   (πœ‘ β†’ 𝑄 ∈ 𝐿)    β‡’   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ 𝐿 𝑄 < π‘Ÿ)
 
5-Feb-2024ivthinc 14282 The intermediate value theorem, increasing case, for a strictly monotonic function. Theorem 5.5 of [Bauer], p. 494. This is Metamath 100 proof #79. (Contributed by Jim Kingdon, 5-Feb-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ π‘ˆ ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ (𝐴[,]𝐡) βŠ† 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐷–cnβ†’β„‚))    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) β†’ (πΉβ€˜π‘₯) ∈ ℝ)    &   (πœ‘ β†’ ((πΉβ€˜π΄) < π‘ˆ ∧ π‘ˆ < (πΉβ€˜π΅)))    &   (((πœ‘ ∧ π‘₯ ∈ (𝐴[,]𝐡)) ∧ (𝑦 ∈ (𝐴[,]𝐡) ∧ π‘₯ < 𝑦)) β†’ (πΉβ€˜π‘₯) < (πΉβ€˜π‘¦))    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ (𝐴(,)𝐡)(πΉβ€˜π‘) = π‘ˆ)
 
2-Feb-2024dedekindeulemuub 14256 Lemma for dedekindeu 14262. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-2024.)
(πœ‘ β†’ 𝐿 βŠ† ℝ)    &   (πœ‘ β†’ π‘ˆ βŠ† ℝ)    &   (πœ‘ β†’ βˆƒπ‘ž ∈ ℝ π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ ℝ π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ (π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ (π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ βˆ€π‘Ÿ ∈ ℝ (π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ βˆ€π‘§ ∈ 𝐿 𝑧 < 𝐴)
 
31-Jan-2024dedekindeulemeu 14261 Lemma for dedekindeu 14262. Part of proving uniqueness. (Contributed by Jim Kingdon, 31-Jan-2024.)
(πœ‘ β†’ 𝐿 βŠ† ℝ)    &   (πœ‘ β†’ π‘ˆ βŠ† ℝ)    &   (πœ‘ β†’ βˆƒπ‘ž ∈ ℝ π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ ℝ π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ (π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ (π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ βˆ€π‘Ÿ ∈ ℝ (π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ (βˆ€π‘ž ∈ 𝐿 π‘ž < 𝐴 ∧ βˆ€π‘Ÿ ∈ π‘ˆ 𝐴 < π‘Ÿ))    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (βˆ€π‘ž ∈ 𝐿 π‘ž < 𝐡 ∧ βˆ€π‘Ÿ ∈ π‘ˆ 𝐡 < π‘Ÿ))    &   (πœ‘ β†’ 𝐴 < 𝐡)    β‡’   (πœ‘ β†’ βŠ₯)
 
31-Jan-2024dedekindeulemlu 14260 Lemma for dedekindeu 14262. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 31-Jan-2024.)
(πœ‘ β†’ 𝐿 βŠ† ℝ)    &   (πœ‘ β†’ π‘ˆ βŠ† ℝ)    &   (πœ‘ β†’ βˆƒπ‘ž ∈ ℝ π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ ℝ π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ (π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ (π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ βˆ€π‘Ÿ ∈ ℝ (π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘ž ∈ 𝐿 π‘ž < π‘₯ ∧ βˆ€π‘Ÿ ∈ π‘ˆ π‘₯ < π‘Ÿ))
 
31-Jan-2024dedekindeulemlub 14259 Lemma for dedekindeu 14262. The set L has a least upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
(πœ‘ β†’ 𝐿 βŠ† ℝ)    &   (πœ‘ β†’ π‘ˆ βŠ† ℝ)    &   (πœ‘ β†’ βˆƒπ‘ž ∈ ℝ π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ ℝ π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ (π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ (π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ βˆ€π‘Ÿ ∈ ℝ (π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐿 Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ 𝐿 𝑦 < 𝑧)))
 
31-Jan-2024dedekindeulemloc 14258 Lemma for dedekindeu 14262. The set L is located. (Contributed by Jim Kingdon, 31-Jan-2024.)
(πœ‘ β†’ 𝐿 βŠ† ℝ)    &   (πœ‘ β†’ π‘ˆ βŠ† ℝ)    &   (πœ‘ β†’ βˆƒπ‘ž ∈ ℝ π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ ℝ π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ (π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ (π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ βˆ€π‘Ÿ ∈ ℝ (π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    β‡’   (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ < 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐿 π‘₯ < 𝑧 ∨ βˆ€π‘§ ∈ 𝐿 𝑧 < 𝑦)))
 
31-Jan-2024dedekindeulemub 14257 Lemma for dedekindeu 14262. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
(πœ‘ β†’ 𝐿 βŠ† ℝ)    &   (πœ‘ β†’ π‘ˆ βŠ† ℝ)    &   (πœ‘ β†’ βˆƒπ‘ž ∈ ℝ π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ ℝ π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ (π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ (π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ βˆ€π‘Ÿ ∈ ℝ (π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐿 𝑦 < π‘₯)
 
30-Jan-2024axsuploc 8033 An inhabited, bounded-above, located set of reals has a supremum. Axiom for real and complex numbers, derived from ZF set theory. (This restates ax-pre-suploc 7935 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.)
(((𝐴 βŠ† ℝ ∧ βˆƒπ‘₯ π‘₯ ∈ 𝐴) ∧ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 < π‘₯ ∧ βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ < 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯ < 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧 < 𝑦)))) β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 < 𝑧)))
 
30-Jan-2024iotam 5210 Representation of "the unique element such that πœ‘ " with a class expression 𝐴 which is inhabited (that means that "the unique element such that πœ‘ " exists). (Contributed by AV, 30-Jan-2024.)
(π‘₯ = 𝐴 β†’ (πœ‘ ↔ πœ“))    β‡’   ((𝐴 ∈ 𝑉 ∧ βˆƒπ‘€ 𝑀 ∈ 𝐴 ∧ 𝐴 = (β„©π‘₯πœ‘)) β†’ πœ“)
 
29-Jan-2024sgrpidmndm 12828 A semigroup with an identity element which is inhabited is a monoid. Of course there could be monoids with the empty set as identity element, but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.)
𝐡 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    β‡’   ((𝐺 ∈ Smgrp ∧ βˆƒπ‘’ ∈ 𝐡 (βˆƒπ‘€ 𝑀 ∈ 𝑒 ∧ 𝑒 = 0 )) β†’ 𝐺 ∈ Mnd)
 
24-Jan-2024axpre-suploclemres 7903 Lemma for axpre-suploc 7904. The result. The proof just needs to define 𝐡 as basically the same set as 𝐴 (but expressed as a subset of R rather than a subset of ℝ), and apply suplocsr 7811. (Contributed by Jim Kingdon, 24-Jan-2024.)
(πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ 𝐢 ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ <ℝ 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯ <ℝ 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧 <ℝ 𝑦)))    &   π΅ = {𝑀 ∈ R ∣ βŸ¨π‘€, 0R⟩ ∈ 𝐴}    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)))
 
23-Jan-2024ax-pre-suploc 7935 An inhabited, bounded-above, located set of reals has a supremum.

Locatedness here means that given π‘₯ < 𝑦, either there is an element of the set greater than π‘₯, or 𝑦 is an upper bound.

Although this and ax-caucvg 7934 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 7934.

(Contributed by Jim Kingdon, 23-Jan-2024.)

(((𝐴 βŠ† ℝ ∧ βˆƒπ‘₯ π‘₯ ∈ 𝐴) ∧ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯ ∧ βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ <ℝ 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯ <ℝ 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧 <ℝ 𝑦)))) β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)))
 
23-Jan-2024axpre-suploc 7904 An inhabited, bounded-above, located set of reals has a supremum.

Locatedness here means that given π‘₯ < 𝑦, either there is an element of the set greater than π‘₯, or 𝑦 is an upper bound.

This construction-dependent theorem should not be referenced directly; instead, use ax-pre-suploc 7935. (Contributed by Jim Kingdon, 23-Jan-2024.) (New usage is discouraged.)

(((𝐴 βŠ† ℝ ∧ βˆƒπ‘₯ π‘₯ ∈ 𝐴) ∧ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯ ∧ βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ <ℝ 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯ <ℝ 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧 <ℝ 𝑦)))) β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)))
 
22-Jan-2024suplocsr 7811 An inhabited, bounded, located set of signed reals has a supremum. (Contributed by Jim Kingdon, 22-Jan-2024.)
(πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ R βˆ€π‘¦ ∈ 𝐴 𝑦 <R π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ R βˆ€π‘¦ ∈ R (π‘₯ <R 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯ <R 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧 <R 𝑦)))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ R (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <R 𝑦 ∧ βˆ€π‘¦ ∈ R (𝑦 <R π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <R 𝑧)))
 
21-Jan-2024bj-el2oss1o 14687 Shorter proof of el2oss1o 6447 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ 2o β†’ 𝐴 βŠ† 1o)
 
21-Jan-2024ltm1sr 7779 Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.)
(𝐴 ∈ R β†’ (𝐴 +R -1R) <R 𝐴)
 
20-Jan-2024mndinvmod 12853 Uniqueness of an inverse element in a monoid, if it exists. (Contributed by AV, 20-Jan-2024.)
𝐡 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Mnd)    &   (πœ‘ β†’ 𝐴 ∈ 𝐡)    β‡’   (πœ‘ β†’ βˆƒ*𝑀 ∈ 𝐡 ((𝑀 + 𝐴) = 0 ∧ (𝐴 + 𝑀) = 0 ))
 
19-Jan-2024suplocsrlempr 7809 Lemma for suplocsr 7811. The set 𝐡 has a least upper bound. (Contributed by Jim Kingdon, 19-Jan-2024.)
𝐡 = {𝑀 ∈ P ∣ (𝐢 +R [βŸ¨π‘€, 1P⟩] ~R ) ∈ 𝐴}    &   (πœ‘ β†’ 𝐴 βŠ† R)    &   (πœ‘ β†’ 𝐢 ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ R βˆ€π‘¦ ∈ 𝐴 𝑦 <R π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ R βˆ€π‘¦ ∈ R (π‘₯ <R 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯ <R 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧 <R 𝑦)))    β‡’   (πœ‘ β†’ βˆƒπ‘£ ∈ P (βˆ€π‘€ ∈ 𝐡 Β¬ 𝑣<P 𝑀 ∧ βˆ€π‘€ ∈ P (𝑀<P 𝑣 β†’ βˆƒπ‘’ ∈ 𝐡 𝑀<P 𝑒)))
 
18-Jan-2024suplocsrlemb 7808 Lemma for suplocsr 7811. The set 𝐡 is located. (Contributed by Jim Kingdon, 18-Jan-2024.)
𝐡 = {𝑀 ∈ P ∣ (𝐢 +R [βŸ¨π‘€, 1P⟩] ~R ) ∈ 𝐴}    &   (πœ‘ β†’ 𝐴 βŠ† R)    &   (πœ‘ β†’ 𝐢 ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ R βˆ€π‘¦ ∈ 𝐴 𝑦 <R π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ R βˆ€π‘¦ ∈ R (π‘₯ <R 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯ <R 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧 <R 𝑦)))    β‡’   (πœ‘ β†’ βˆ€π‘’ ∈ P βˆ€π‘£ ∈ P (𝑒<P 𝑣 β†’ (βˆƒπ‘ž ∈ 𝐡 𝑒<P π‘ž ∨ βˆ€π‘ž ∈ 𝐡 π‘ž<P 𝑣)))
 
16-Jan-2024suplocsrlem 7810 Lemma for suplocsr 7811. The set 𝐴 has a least upper bound. (Contributed by Jim Kingdon, 16-Jan-2024.)
𝐡 = {𝑀 ∈ P ∣ (𝐢 +R [βŸ¨π‘€, 1P⟩] ~R ) ∈ 𝐴}    &   (πœ‘ β†’ 𝐴 βŠ† R)    &   (πœ‘ β†’ 𝐢 ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ R βˆ€π‘¦ ∈ 𝐴 𝑦 <R π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ R βˆ€π‘¦ ∈ R (π‘₯ <R 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯ <R 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧 <R 𝑦)))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ R (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <R 𝑦 ∧ βˆ€π‘¦ ∈ R (𝑦 <R π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <R 𝑧)))
 
14-Jan-2024suplocexprlemlub 7726 Lemma for suplocexpr 7727. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
(πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ P βˆ€π‘¦ ∈ 𝐴 𝑦<P π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ P βˆ€π‘¦ ∈ P (π‘₯<P 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯<P 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧<P 𝑦)))    &   π΅ = ⟨βˆͺ (1st β€œ 𝐴), {𝑒 ∈ Q ∣ βˆƒπ‘€ ∈ ∩ (2nd β€œ 𝐴)𝑀 <Q 𝑒}⟩    β‡’   (πœ‘ β†’ (𝑦<P 𝐡 β†’ βˆƒπ‘§ ∈ 𝐴 𝑦<P 𝑧))
 
14-Jan-2024suplocexprlemub 7725 Lemma for suplocexpr 7727. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
(πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ P βˆ€π‘¦ ∈ 𝐴 𝑦<P π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ P βˆ€π‘¦ ∈ P (π‘₯<P 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯<P 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧<P 𝑦)))    &   π΅ = ⟨βˆͺ (1st β€œ 𝐴), {𝑒 ∈ Q ∣ βˆƒπ‘€ ∈ ∩ (2nd β€œ 𝐴)𝑀 <Q 𝑒}⟩    β‡’   (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐴 Β¬ 𝐡<P 𝑦)
 
10-Jan-2024nfcsbw 3095 Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3096 with a disjoint variable condition. (Contributed by Mario Carneiro, 12-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    β‡’   β„²π‘₯⦋𝐴 / π‘¦β¦Œπ΅
 
10-Jan-2024nfsbcdw 3093 Version of nfsbcd 2984 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
β„²π‘¦πœ‘    &   (πœ‘ β†’ β„²π‘₯𝐴)    &   (πœ‘ β†’ β„²π‘₯πœ“)    β‡’   (πœ‘ β†’ β„²π‘₯[𝐴 / 𝑦]πœ“)
 
10-Jan-2024cbvcsbw 3063 Version of cbvcsb 3064 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝐢    &   β„²π‘₯𝐷    &   (π‘₯ = 𝑦 β†’ 𝐢 = 𝐷)    β‡’   β¦‹π΄ / π‘₯⦌𝐢 = ⦋𝐴 / π‘¦β¦Œπ·
 
10-Jan-2024cbvsbcw 2992 Version of cbvsbc 2993 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)
β„²π‘¦πœ‘    &   β„²π‘₯πœ“    &   (π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   ([𝐴 / π‘₯]πœ‘ ↔ [𝐴 / 𝑦]πœ“)
 
10-Jan-2024cbvrex2vw 2717 Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 2719 with a disjoint variable condition, which does not require ax-13 2150. (Contributed by FL, 2-Jul-2012.) (Revised by Gino Giotto, 10-Jan-2024.)
(π‘₯ = 𝑧 β†’ (πœ‘ ↔ πœ’))    &   (𝑦 = 𝑀 β†’ (πœ’ ↔ πœ“))    β‡’   (βˆƒπ‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘ ↔ βˆƒπ‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐡 πœ“)
 
10-Jan-2024cbvral2vw 2716 Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 2718 with a disjoint variable condition, which does not require ax-13 2150. (Contributed by NM, 10-Aug-2004.) (Revised by Gino Giotto, 10-Jan-2024.)
(π‘₯ = 𝑧 β†’ (πœ‘ ↔ πœ’))    &   (𝑦 = 𝑀 β†’ (πœ’ ↔ πœ“))    β‡’   (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 πœ‘ ↔ βˆ€π‘§ ∈ 𝐴 βˆ€π‘€ ∈ 𝐡 πœ“)
 
10-Jan-2024cbvrexw 2700 Rule used to change bound variables, using implicit substitution. Version of cbvrexfw 2696 with more disjoint variable conditions. Although we don't do so yet, we expect the disjoint variable conditions will allow us to remove reliance on ax-i12 1507 and ax-bndl 1509 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
β„²π‘¦πœ‘    &   β„²π‘₯πœ“    &   (π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆƒπ‘₯ ∈ 𝐴 πœ‘ ↔ βˆƒπ‘¦ ∈ 𝐴 πœ“)
 
10-Jan-2024cbvralw 2699 Rule used to change bound variables, using implicit substitution. Version of cbvral 2701 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1507 and ax-bndl 1509 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
β„²π‘¦πœ‘    &   β„²π‘₯πœ“    &   (π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆ€π‘₯ ∈ 𝐴 πœ‘ ↔ βˆ€π‘¦ ∈ 𝐴 πœ“)
 
10-Jan-2024cbvrexfw 2696 Rule used to change bound variables, using implicit substitution. Version of cbvrexf 2698 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1507 and ax-bndl 1509 in the proof. (Contributed by FL, 27-Apr-2008.) (Revised by Gino Giotto, 10-Jan-2024.)
β„²π‘₯𝐴    &   β„²π‘¦π΄    &   β„²π‘¦πœ‘    &   β„²π‘₯πœ“    &   (π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆƒπ‘₯ ∈ 𝐴 πœ‘ ↔ βˆƒπ‘¦ ∈ 𝐴 πœ“)
 
10-Jan-2024nfralw 2514 Bound-variable hypothesis builder for restricted quantification. See nfralya 2517 for a version with 𝑦 and 𝐴 distinct instead of π‘₯ and 𝑦. (Contributed by NM, 1-Sep-1999.) (Revised by Gino Giotto, 10-Jan-2024.)
β„²π‘₯𝐴    &   β„²π‘₯πœ‘    β‡’   β„²π‘₯βˆ€π‘¦ ∈ 𝐴 πœ‘
 
10-Jan-2024nfraldw 2509 Not-free for restricted universal quantification where π‘₯ and 𝑦 are distinct. See nfraldya 2512 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.)
β„²π‘¦πœ‘    &   (πœ‘ β†’ β„²π‘₯𝐴)    &   (πœ‘ β†’ β„²π‘₯πœ“)    β‡’   (πœ‘ β†’ β„²π‘₯βˆ€π‘¦ ∈ 𝐴 πœ“)
 
10-Jan-2024nfabdw 2338 Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2339 with a disjoint variable condition. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
β„²π‘¦πœ‘    &   (πœ‘ β†’ β„²π‘₯πœ“)    β‡’   (πœ‘ β†’ β„²π‘₯{𝑦 ∣ πœ“})
 
10-Jan-2024cbv2w 1750 Rule used to change bound variables, using implicit substitution. Version of cbv2 1749 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by Gino Giotto, 10-Jan-2024.)
β„²π‘₯πœ‘    &   β„²π‘¦πœ‘    &   (πœ‘ β†’ β„²π‘¦πœ“)    &   (πœ‘ β†’ β„²π‘₯πœ’)    &   (πœ‘ β†’ (π‘₯ = 𝑦 β†’ (πœ“ ↔ πœ’)))    β‡’   (πœ‘ β†’ (βˆ€π‘₯πœ“ ↔ βˆ€π‘¦πœ’))
 
9-Jan-2024suplocexprlemloc 7723 Lemma for suplocexpr 7727. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.)
(πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ P βˆ€π‘¦ ∈ 𝐴 𝑦<P π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ P βˆ€π‘¦ ∈ P (π‘₯<P 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯<P 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧<P 𝑦)))    &   π΅ = ⟨βˆͺ (1st β€œ 𝐴), {𝑒 ∈ Q ∣ βˆƒπ‘€ ∈ ∩ (2nd β€œ 𝐴)𝑀 <Q 𝑒}⟩    β‡’   (πœ‘ β†’ βˆ€π‘ž ∈ Q βˆ€π‘Ÿ ∈ Q (π‘ž <Q π‘Ÿ β†’ (π‘ž ∈ βˆͺ (1st β€œ 𝐴) ∨ π‘Ÿ ∈ (2nd β€˜π΅))))
 
9-Jan-2024suplocexprlemdisj 7722 Lemma for suplocexpr 7727. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.)
(πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ P βˆ€π‘¦ ∈ 𝐴 𝑦<P π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ P βˆ€π‘¦ ∈ P (π‘₯<P 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯<P 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧<P 𝑦)))    &   π΅ = ⟨βˆͺ (1st β€œ 𝐴), {𝑒 ∈ Q ∣ βˆƒπ‘€ ∈ ∩ (2nd β€œ 𝐴)𝑀 <Q 𝑒}⟩    β‡’   (πœ‘ β†’ βˆ€π‘ž ∈ Q Β¬ (π‘ž ∈ βˆͺ (1st β€œ 𝐴) ∧ π‘ž ∈ (2nd β€˜π΅)))
 
9-Jan-2024suplocexprlemru 7721 Lemma for suplocexpr 7727. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
(πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ P βˆ€π‘¦ ∈ 𝐴 𝑦<P π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ P βˆ€π‘¦ ∈ P (π‘₯<P 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯<P 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧<P 𝑦)))    &   π΅ = ⟨βˆͺ (1st β€œ 𝐴), {𝑒 ∈ Q ∣ βˆƒπ‘€ ∈ ∩ (2nd β€œ 𝐴)𝑀 <Q 𝑒}⟩    β‡’   (πœ‘ β†’ βˆ€π‘Ÿ ∈ Q (π‘Ÿ ∈ (2nd β€˜π΅) ↔ βˆƒπ‘ž ∈ Q (π‘ž <Q π‘Ÿ ∧ π‘ž ∈ (2nd β€˜π΅))))
 
9-Jan-2024suplocexprlemrl 7719 Lemma for suplocexpr 7727. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
(πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ P βˆ€π‘¦ ∈ 𝐴 𝑦<P π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ P βˆ€π‘¦ ∈ P (π‘₯<P 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯<P 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧<P 𝑦)))    β‡’   (πœ‘ β†’ βˆ€π‘ž ∈ Q (π‘ž ∈ βˆͺ (1st β€œ 𝐴) ↔ βˆƒπ‘Ÿ ∈ Q (π‘ž <Q π‘Ÿ ∧ π‘Ÿ ∈ βˆͺ (1st β€œ 𝐴))))
 
9-Jan-2024suplocexprlem2b 7716 Lemma for suplocexpr 7727. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
𝐡 = ⟨βˆͺ (1st β€œ 𝐴), {𝑒 ∈ Q ∣ βˆƒπ‘€ ∈ ∩ (2nd β€œ 𝐴)𝑀 <Q 𝑒}⟩    β‡’   (𝐴 βŠ† P β†’ (2nd β€˜π΅) = {𝑒 ∈ Q ∣ βˆƒπ‘€ ∈ ∩ (2nd β€œ 𝐴)𝑀 <Q 𝑒})
 
9-Jan-2024suplocexprlemell 7715 Lemma for suplocexpr 7727. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝐡 ∈ βˆͺ (1st β€œ 𝐴) ↔ βˆƒπ‘₯ ∈ 𝐴 𝐡 ∈ (1st β€˜π‘₯))
 
7-Jan-2024suplocexpr 7727 An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.)
(πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ P βˆ€π‘¦ ∈ 𝐴 𝑦<P π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ P βˆ€π‘¦ ∈ P (π‘₯<P 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯<P 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧<P 𝑦)))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ P (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯<P 𝑦 ∧ βˆ€π‘¦ ∈ P (𝑦<P π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦<P 𝑧)))
 
7-Jan-2024suplocexprlemex 7724 Lemma for suplocexpr 7727. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.)
(πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ P βˆ€π‘¦ ∈ 𝐴 𝑦<P π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ P βˆ€π‘¦ ∈ P (π‘₯<P 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯<P 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧<P 𝑦)))    &   π΅ = ⟨βˆͺ (1st β€œ 𝐴), {𝑒 ∈ Q ∣ βˆƒπ‘€ ∈ ∩ (2nd β€œ 𝐴)𝑀 <Q 𝑒}⟩    β‡’   (πœ‘ β†’ 𝐡 ∈ P)
 
7-Jan-2024suplocexprlemmu 7720 Lemma for suplocexpr 7727. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
(πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ P βˆ€π‘¦ ∈ 𝐴 𝑦<P π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ P βˆ€π‘¦ ∈ P (π‘₯<P 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯<P 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧<P 𝑦)))    &   π΅ = ⟨βˆͺ (1st β€œ 𝐴), {𝑒 ∈ Q ∣ βˆƒπ‘€ ∈ ∩ (2nd β€œ 𝐴)𝑀 <Q 𝑒}⟩    β‡’   (πœ‘ β†’ βˆƒπ‘  ∈ Q 𝑠 ∈ (2nd β€˜π΅))
 
7-Jan-2024suplocexprlemml 7718 Lemma for suplocexpr 7727. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
(πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ P βˆ€π‘¦ ∈ 𝐴 𝑦<P π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ P βˆ€π‘¦ ∈ P (π‘₯<P 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯<P 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧<P 𝑦)))    β‡’   (πœ‘ β†’ βˆƒπ‘  ∈ Q 𝑠 ∈ βˆͺ (1st β€œ 𝐴))
 
7-Jan-2024suplocexprlemss 7717 Lemma for suplocexpr 7727. 𝐴 is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.)
(πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ P βˆ€π‘¦ ∈ 𝐴 𝑦<P π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ P βˆ€π‘¦ ∈ P (π‘₯<P 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯<P 𝑧 ∨ βˆ€π‘§ ∈ 𝐴 𝑧<P 𝑦)))    β‡’   (πœ‘ β†’ 𝐴 βŠ† P)
 
5-Jan-2024dedekindicclemicc 14271 Lemma for dedekindicc 14272. Same as dedekindicc 14272, except that we merely show π‘₯ to be an element of (𝐴[,]𝐡). Later we will strengthen that to (𝐴(,)𝐡). (Contributed by Jim Kingdon, 5-Jan-2024.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐿 βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ π‘ˆ βŠ† (𝐴[,]𝐡))    &   (πœ‘ β†’ βˆƒπ‘ž ∈ (𝐴[,]𝐡)π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ (𝐴[,]𝐡)π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)(π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ (𝐴[,]𝐡)βˆ€π‘Ÿ ∈ (𝐴[,]𝐡)(π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    &   (πœ‘ β†’ 𝐴 < 𝐡)    β‡’   (πœ‘ β†’ βˆƒ!π‘₯ ∈ (𝐴[,]𝐡)(βˆ€π‘ž ∈ 𝐿 π‘ž < π‘₯ ∧ βˆ€π‘Ÿ ∈ π‘ˆ π‘₯ < π‘Ÿ))
 
5-Jan-2024dedekindeu 14262 A Dedekind cut identifies a unique real number. Similar to df-inp 7468 except that the the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 5-Jan-2024.)
(πœ‘ β†’ 𝐿 βŠ† ℝ)    &   (πœ‘ β†’ π‘ˆ βŠ† ℝ)    &   (πœ‘ β†’ βˆƒπ‘ž ∈ ℝ π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ ℝ π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ (π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ (π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ βˆ€π‘Ÿ ∈ ℝ (π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    β‡’   (πœ‘ β†’ βˆƒ!π‘₯ ∈ ℝ (βˆ€π‘ž ∈ 𝐿 π‘ž < π‘₯ ∧ βˆ€π‘Ÿ ∈ π‘ˆ π‘₯ < π‘Ÿ))
 
31-Dec-2023dvmptsubcn 14346 Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
((πœ‘ ∧ π‘₯ ∈ β„‚) β†’ 𝐴 ∈ β„‚)    &   ((πœ‘ ∧ π‘₯ ∈ β„‚) β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ 𝐴)) = (π‘₯ ∈ β„‚ ↦ 𝐡))    &   ((πœ‘ ∧ π‘₯ ∈ β„‚) β†’ 𝐢 ∈ β„‚)    &   ((πœ‘ ∧ π‘₯ ∈ β„‚) β†’ 𝐷 ∈ π‘Š)    &   (πœ‘ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ 𝐢)) = (π‘₯ ∈ β„‚ ↦ 𝐷))    β‡’   (πœ‘ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ (𝐴 βˆ’ 𝐢))) = (π‘₯ ∈ β„‚ ↦ (𝐡 βˆ’ 𝐷)))
 
31-Dec-2023dvmptnegcn 14345 Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
((πœ‘ ∧ π‘₯ ∈ β„‚) β†’ 𝐴 ∈ β„‚)    &   ((πœ‘ ∧ π‘₯ ∈ β„‚) β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ 𝐴)) = (π‘₯ ∈ β„‚ ↦ 𝐡))    β‡’   (πœ‘ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ -𝐴)) = (π‘₯ ∈ β„‚ ↦ -𝐡))
 
31-Dec-2023dvmptcmulcn 14344 Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
((πœ‘ ∧ π‘₯ ∈ β„‚) β†’ 𝐴 ∈ β„‚)    &   ((πœ‘ ∧ π‘₯ ∈ β„‚) β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ 𝐴)) = (π‘₯ ∈ β„‚ ↦ 𝐡))    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ (𝐢 Β· 𝐴))) = (π‘₯ ∈ β„‚ ↦ (𝐢 Β· 𝐡)))
 
31-Dec-2023rinvmod 13123 Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6071. (Contributed by AV, 31-Dec-2023.)
𝐡 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ CMnd)    &   (πœ‘ β†’ 𝐴 ∈ 𝐡)    β‡’   (πœ‘ β†’ βˆƒ*𝑀 ∈ 𝐡 (𝐴 + 𝑀) = 0 )
 
31-Dec-2023brm 4055 If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)
(𝐴𝑅𝐡 β†’ βˆƒπ‘₯ π‘₯ ∈ 𝑅)
 
30-Dec-2023dvmptccn 14340 Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ 𝐴)) = (π‘₯ ∈ β„‚ ↦ 0))
 
30-Dec-2023dvmptidcn 14339 Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
(β„‚ D (π‘₯ ∈ β„‚ ↦ π‘₯)) = (π‘₯ ∈ β„‚ ↦ 1)
 
29-Dec-2023mndbn0 12839 The base set of a monoid is not empty. (It is also inhabited, as seen at mndidcl 12838). Statement in [Lang] p. 3. (Contributed by AV, 29-Dec-2023.)
𝐡 = (Baseβ€˜πΊ)    β‡’   (𝐺 ∈ Mnd β†’ 𝐡 β‰  βˆ…)
 
26-Dec-2023lidrididd 12808 If there is a left and right identity element for any binary operation (group operation) +, the left identity element (and therefore also the right identity element according to lidrideqd 12807) is equal to the two-sided identity element. (Contributed by AV, 26-Dec-2023.)
(πœ‘ β†’ 𝐿 ∈ 𝐡)    &   (πœ‘ β†’ 𝑅 ∈ 𝐡)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 (𝐿 + π‘₯) = π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 (π‘₯ + 𝑅) = π‘₯)    &   π΅ = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    β‡’   (πœ‘ β†’ 𝐿 = 0 )
 
26-Dec-2023lidrideqd 12807 If there is a left and right identity element for any binary operation (group operation) +, both identity elements are equal. Generalization of statement in [Lang] p. 3: it is sufficient that "e" is a left identity element and "e`" is a right identity element instead of both being (two-sided) identity elements. (Contributed by AV, 26-Dec-2023.)
(πœ‘ β†’ 𝐿 ∈ 𝐡)    &   (πœ‘ β†’ 𝑅 ∈ 𝐡)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 (𝐿 + π‘₯) = π‘₯)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 (π‘₯ + 𝑅) = π‘₯)    β‡’   (πœ‘ β†’ 𝐿 = 𝑅)
 
25-Dec-2023ctfoex 7120 A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.)
(βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o) β†’ 𝐴 ∈ V)
 
23-Dec-2023enct 12437 Countability is invariant relative to equinumerosity. (Contributed by Jim Kingdon, 23-Dec-2023.)
(𝐴 β‰ˆ 𝐡 β†’ (βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o) ↔ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐡 βŠ” 1o)))
 
23-Dec-2023enctlem 12436 Lemma for enct 12437. One direction of the biconditional. (Contributed by Jim Kingdon, 23-Dec-2023.)
(𝐴 β‰ˆ 𝐡 β†’ (βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o) β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐡 βŠ” 1o)))
 
23-Dec-2023omct 7119 Ο‰ is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
βˆƒπ‘“ 𝑓:ω–ontoβ†’(Ο‰ βŠ” 1o)
 
21-Dec-2023dvcoapbr 14332 The chain rule for derivatives at a point. The 𝑒 # 𝐢 β†’ (πΊβ€˜π‘’) # (πΊβ€˜πΆ) hypothesis constrains what functions work for 𝐺. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 21-Dec-2023.)
(πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝑋 βŠ† 𝑆)    &   (πœ‘ β†’ 𝐺:π‘ŒβŸΆπ‘‹)    &   (πœ‘ β†’ π‘Œ βŠ† 𝑇)    &   (πœ‘ β†’ βˆ€π‘’ ∈ π‘Œ (𝑒 # 𝐢 β†’ (πΊβ€˜π‘’) # (πΊβ€˜πΆ)))    &   (πœ‘ β†’ 𝑆 βŠ† β„‚)    &   (πœ‘ β†’ 𝑇 βŠ† β„‚)    &   (πœ‘ β†’ (πΊβ€˜πΆ)(𝑆 D 𝐹)𝐾)    &   (πœ‘ β†’ 𝐢(𝑇 D 𝐺)𝐿)    &   π½ = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’   (πœ‘ β†’ 𝐢(𝑇 D (𝐹 ∘ 𝐺))(𝐾 Β· 𝐿))
 
19-Dec-2023apsscn 8607 The points apart from a given point are complex numbers. (Contributed by Jim Kingdon, 19-Dec-2023.)
{π‘₯ ∈ 𝐴 ∣ π‘₯ # 𝐡} βŠ† β„‚
 
19-Dec-2023aprcl 8606 Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.)
(𝐴 # 𝐡 β†’ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚))
 
18-Dec-2023limccoap 14308 Composition of two limits. This theorem is only usable in the case where π‘₯ # 𝑋 implies R(x) # 𝐢 so it is less general than might appear at first. (Contributed by Mario Carneiro, 29-Dec-2016.) (Revised by Jim Kingdon, 18-Dec-2023.)
((πœ‘ ∧ π‘₯ ∈ {𝑀 ∈ 𝐴 ∣ 𝑀 # 𝑋}) β†’ 𝑅 ∈ {𝑀 ∈ 𝐡 ∣ 𝑀 # 𝐢})    &   ((πœ‘ ∧ 𝑦 ∈ {𝑀 ∈ 𝐡 ∣ 𝑀 # 𝐢}) β†’ 𝑆 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 ∈ ((π‘₯ ∈ {𝑀 ∈ 𝐴 ∣ 𝑀 # 𝑋} ↦ 𝑅) limβ„‚ 𝑋))    &   (πœ‘ β†’ 𝐷 ∈ ((𝑦 ∈ {𝑀 ∈ 𝐡 ∣ 𝑀 # 𝐢} ↦ 𝑆) limβ„‚ 𝐢))    &   (𝑦 = 𝑅 β†’ 𝑆 = 𝑇)    β‡’   (πœ‘ β†’ 𝐷 ∈ ((π‘₯ ∈ {𝑀 ∈ 𝐴 ∣ 𝑀 # 𝑋} ↦ 𝑇) limβ„‚ 𝑋))
 
16-Dec-2023cnreim 10990 Complex apartness in terms of real and imaginary parts. See also apreim 8563 which is similar but with different notation. (Contributed by Jim Kingdon, 16-Dec-2023.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 # 𝐡 ↔ ((β„œβ€˜π΄) # (β„œβ€˜π΅) ∨ (β„‘β€˜π΄) # (β„‘β€˜π΅))))
 
14-Dec-2023cnopnap 14255 The complex numbers apart from a given complex number form an open set. (Contributed by Jim Kingdon, 14-Dec-2023.)
(𝐴 ∈ β„‚ β†’ {𝑀 ∈ β„‚ ∣ 𝑀 # 𝐴} ∈ (MetOpenβ€˜(abs ∘ βˆ’ )))
 
14-Dec-2023cnovex 13857 The class of all continuous functions from a topology to another is a set. (Contributed by Jim Kingdon, 14-Dec-2023.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top) β†’ (𝐽 Cn 𝐾) ∈ V)
 
13-Dec-2023reopnap 14199 The real numbers apart from a given real number form an open set. (Contributed by Jim Kingdon, 13-Dec-2023.)
(𝐴 ∈ ℝ β†’ {𝑀 ∈ ℝ ∣ 𝑀 # 𝐴} ∈ (topGenβ€˜ran (,)))
 
12-Dec-2023cnopncntop 14198 The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
β„‚ ∈ (MetOpenβ€˜(abs ∘ βˆ’ ))
 
12-Dec-2023unicntopcntop 14197 The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
β„‚ = βˆͺ (MetOpenβ€˜(abs ∘ βˆ’ ))
 
4-Dec-2023bj-pm2.18st 14663 Clavius law for stable formulas. See pm2.18dc 855. (Contributed by BJ, 4-Dec-2023.)
(STAB πœ‘ β†’ ((Β¬ πœ‘ β†’ πœ‘) β†’ πœ‘))
 
4-Dec-2023bj-nnclavius 14650 Clavius law with doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)
((Β¬ πœ‘ β†’ πœ‘) β†’ Β¬ Β¬ πœ‘)
 
2-Dec-2023dvmulxx 14329 The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 14327. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 2-Dec-2023.)
(πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝑋 βŠ† 𝑆)    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐢 ∈ dom (𝑆 D 𝐹))    &   (πœ‘ β†’ 𝐢 ∈ dom (𝑆 D 𝐺))    β‡’   (πœ‘ β†’ ((𝑆 D (𝐹 βˆ˜π‘“ Β· 𝐺))β€˜πΆ) = ((((𝑆 D 𝐹)β€˜πΆ) Β· (πΊβ€˜πΆ)) + (((𝑆 D 𝐺)β€˜πΆ) Β· (πΉβ€˜πΆ))))
 
1-Dec-2023dvmulxxbr 14327 The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmulxx 14329. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 1-Dec-2023.)
(πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝑋 βŠ† 𝑆)    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝑆 βŠ† β„‚)    &   (πœ‘ β†’ 𝐢(𝑆 D 𝐹)𝐾)    &   (πœ‘ β†’ 𝐢(𝑆 D 𝐺)𝐿)    &   π½ = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’   (πœ‘ β†’ 𝐢(𝑆 D (𝐹 βˆ˜π‘“ Β· 𝐺))((𝐾 Β· (πΊβ€˜πΆ)) + (𝐿 Β· (πΉβ€˜πΆ))))
 
29-Nov-2023subctctexmid 14912 If every subcountable set is countable and Markov's principle holds, excluded middle follows. Proposition 2.6 of [BauerSwan], p. 14:4. The proof is taken from that paper. (Contributed by Jim Kingdon, 29-Nov-2023.)
(πœ‘ β†’ βˆ€π‘₯(βˆƒπ‘ (𝑠 βŠ† Ο‰ ∧ βˆƒπ‘“ 𝑓:𝑠–ontoβ†’π‘₯) β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(π‘₯ βŠ” 1o)))    &   (πœ‘ β†’ Ο‰ ∈ Markov)    β‡’   (πœ‘ β†’ EXMID)
 
29-Nov-2023ismkvnex 7156 The predicate of being Markov stated in terms of double negation and comparison with 1o. (Contributed by Jim Kingdon, 29-Nov-2023.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Markov ↔ βˆ€π‘“ ∈ (2o β†‘π‘š 𝐴)(Β¬ Β¬ βˆƒπ‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1o β†’ βˆƒπ‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1o)))
 
28-Nov-2023ccfunen 7266 Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)
(πœ‘ β†’ CCHOICE)    &   (πœ‘ β†’ 𝐴 β‰ˆ Ο‰)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘€ 𝑀 ∈ π‘₯)    β‡’   (πœ‘ β†’ βˆƒπ‘“(𝑓 Fn 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) ∈ π‘₯))
 
28-Nov-2023exmid1stab 4210 If every proposition is stable, excluded middle follows. We are thinking of π‘₯ as a proposition and π‘₯ = {βˆ…} as "π‘₯ is true". (Contributed by Jim Kingdon, 28-Nov-2023.)
((πœ‘ ∧ π‘₯ βŠ† {βˆ…}) β†’ STAB π‘₯ = {βˆ…})    β‡’   (πœ‘ β†’ EXMID)
 
27-Nov-2023df-cc 7265 The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7208 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.)
(CCHOICE ↔ βˆ€π‘₯(dom π‘₯ β‰ˆ Ο‰ β†’ βˆƒπ‘“(𝑓 βŠ† π‘₯ ∧ 𝑓 Fn dom π‘₯)))
 
26-Nov-2023offeq 6099 Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) β†’ (π‘₯𝑅𝑦) ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐹:π΄βŸΆπ‘†)    &   (πœ‘ β†’ 𝐺:π΅βŸΆπ‘‡)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ π‘Š)    &   (𝐴 ∩ 𝐡) = 𝐢    &   (πœ‘ β†’ 𝐻:πΆβŸΆπ‘ˆ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ (πΉβ€˜π‘₯) = 𝐷)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (πΊβ€˜π‘₯) = 𝐸)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ (𝐷𝑅𝐸) = (π»β€˜π‘₯))    β‡’   (πœ‘ β†’ (𝐹 βˆ˜π‘“ 𝑅𝐺) = 𝐻)
 
25-Nov-2023dvaddxx 14328 The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 14326. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
(πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝑋 βŠ† 𝑆)    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐢 ∈ dom (𝑆 D 𝐹))    &   (πœ‘ β†’ 𝐢 ∈ dom (𝑆 D 𝐺))    β‡’   (πœ‘ β†’ ((𝑆 D (𝐹 βˆ˜π‘“ + 𝐺))β€˜πΆ) = (((𝑆 D 𝐹)β€˜πΆ) + ((𝑆 D 𝐺)β€˜πΆ)))
 
25-Nov-2023dvaddxxbr 14326 The sum rule for derivatives at a point. That is, if the derivative of 𝐹 at 𝐢 is 𝐾 and the derivative of 𝐺 at 𝐢 is 𝐿, then the derivative of the pointwise sum of those two functions at 𝐢 is 𝐾 + 𝐿. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
(πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝑋 βŠ† 𝑆)    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝑆 βŠ† β„‚)    &   (πœ‘ β†’ 𝐢(𝑆 D 𝐹)𝐾)    &   (πœ‘ β†’ 𝐢(𝑆 D 𝐺)𝐿)    &   π½ = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’   (πœ‘ β†’ 𝐢(𝑆 D (𝐹 βˆ˜π‘“ + 𝐺))(𝐾 + 𝐿))
 
25-Nov-2023dcnn 848 Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 842. The relation between dcn 842 and dcnn 848 is analogous to that between notnot 629 and notnotnot 634 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 848 means that a proposition is testable if and only if its negation is testable, and dcn 842 means that decidability implies testability. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof shortened by BJ, 25-Nov-2023.)
(DECID Β¬ πœ‘ ↔ DECID Β¬ Β¬ πœ‘)
 
24-Nov-2023bj-dcst 14674 Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.)
(DECID STAB πœ‘ ↔ STAB πœ‘)
 
24-Nov-2023bj-nnbidc 14670 If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 14657. (Contributed by BJ, 24-Nov-2023.)
(Β¬ Β¬ πœ‘ β†’ (DECID πœ‘ ↔ πœ‘))
 
24-Nov-2023bj-dcstab 14669 A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
(DECID πœ‘ β†’ STAB πœ‘)
 
24-Nov-2023bj-fadc 14667 A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
(Β¬ πœ‘ β†’ DECID πœ‘)
 
24-Nov-2023bj-trdc 14665 A provable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
(πœ‘ β†’ DECID πœ‘)
 
24-Nov-2023bj-stal 14662 The universal quantification of a stable formula is stable. See bj-stim 14659 for implication, stabnot 833 for negation, and bj-stan 14660 for conjunction. (Contributed by BJ, 24-Nov-2023.)
(βˆ€π‘₯STAB πœ‘ β†’ STAB βˆ€π‘₯πœ‘)
 
24-Nov-2023bj-stand 14661 The conjunction of two stable formulas is stable. Deduction form of bj-stan 14660. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 14660 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
(πœ‘ β†’ STAB πœ“)    &   (πœ‘ β†’ STAB πœ’)    β‡’   (πœ‘ β†’ STAB (πœ“ ∧ πœ’))
 
24-Nov-2023bj-stan 14660 The conjunction of two stable formulas is stable. See bj-stim 14659 for implication, stabnot 833 for negation, and bj-stal 14662 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
((STAB πœ‘ ∧ STAB πœ“) β†’ STAB (πœ‘ ∧ πœ“))
 
24-Nov-2023bj-stim 14659 A conjunction with a stable consequent is stable. See stabnot 833 for negation , bj-stan 14660 for conjunction , and bj-stal 14662 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
(STAB πœ“ β†’ STAB (πœ‘ β†’ πœ“))
 
24-Nov-2023bj-nnbist 14657 If a formula is not refutable, then it is stable if and only if it is provable. By double-negation translation, if πœ‘ is a classical tautology, then Β¬ Β¬ πœ‘ is an intuitionistic tautology. Therefore, if πœ‘ is a classical tautology, then πœ‘ is intuitionistically equivalent to its stability (and to its decidability, see bj-nnbidc 14670). (Contributed by BJ, 24-Nov-2023.)
(Β¬ Β¬ πœ‘ β†’ (STAB πœ‘ ↔ πœ‘))
 
24-Nov-2023bj-fast 14654 A refutable formula is stable. (Contributed by BJ, 24-Nov-2023.)
(Β¬ πœ‘ β†’ STAB πœ‘)
 
24-Nov-2023bj-trst 14652 A provable formula is stable. (Contributed by BJ, 24-Nov-2023.)
(πœ‘ β†’ STAB πœ‘)
 
24-Nov-2023bj-nnan 14649 The double negation of a conjunction implies the conjunction of the double negations. (Contributed by BJ, 24-Nov-2023.)
(Β¬ Β¬ (πœ‘ ∧ πœ“) β†’ (Β¬ Β¬ πœ‘ ∧ Β¬ Β¬ πœ“))
 
24-Nov-2023bj-nnim 14648 The double negation of an implication implies the implication with the consequent doubly negated. (Contributed by BJ, 24-Nov-2023.)
(Β¬ Β¬ (πœ‘ β†’ πœ“) β†’ (πœ‘ β†’ Β¬ Β¬ πœ“))
 
24-Nov-2023bj-nnsn 14646 As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.)
((πœ‘ β†’ Β¬ πœ“) ↔ (Β¬ Β¬ πœ‘ β†’ Β¬ πœ“))
 
24-Nov-2023nnal 1649 The double negation of a universal quantification implies the universal quantification of the double negation. (Contributed by BJ, 24-Nov-2023.)
(Β¬ Β¬ βˆ€π‘₯πœ‘ β†’ βˆ€π‘₯ Β¬ Β¬ πœ‘)
 
22-Nov-2023ofvalg 6095 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)
(πœ‘ β†’ 𝐹 Fn 𝐴)    &   (πœ‘ β†’ 𝐺 Fn 𝐡)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ π‘Š)    &   (𝐴 ∩ 𝐡) = 𝑆    &   ((πœ‘ ∧ 𝑋 ∈ 𝐴) β†’ (πΉβ€˜π‘‹) = 𝐢)    &   ((πœ‘ ∧ 𝑋 ∈ 𝐡) β†’ (πΊβ€˜π‘‹) = 𝐷)    &   ((πœ‘ ∧ 𝑋 ∈ 𝑆) β†’ (𝐢𝑅𝐷) ∈ π‘ˆ)    β‡’   ((πœ‘ ∧ 𝑋 ∈ 𝑆) β†’ ((𝐹 βˆ˜π‘“ 𝑅𝐺)β€˜π‘‹) = (𝐢𝑅𝐷))
 
21-Nov-2023exmidac 7211 The axiom of choice implies excluded middle. See acexmid 5877 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.)
(CHOICE β†’ EXMID)
 
21-Nov-2023exmidaclem 7210 Lemma for exmidac 7211. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.)
𝐴 = {π‘₯ ∈ {βˆ…, {βˆ…}} ∣ (π‘₯ = βˆ… ∨ 𝑦 = {βˆ…})}    &   π΅ = {π‘₯ ∈ {βˆ…, {βˆ…}} ∣ (π‘₯ = {βˆ…} ∨ 𝑦 = {βˆ…})}    &   πΆ = {𝐴, 𝐡}    β‡’   (CHOICE β†’ EXMID)
 
21-Nov-2023exmid1dc 4202 A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4195 or ordtriexmid 4522. In this context π‘₯ = {βˆ…} can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)
((πœ‘ ∧ π‘₯ βŠ† {βˆ…}) β†’ DECID π‘₯ = {βˆ…})    β‡’   (πœ‘ β†’ EXMID)
 
20-Nov-2023acfun 7209 A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.)
(πœ‘ β†’ CHOICE)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘€ 𝑀 ∈ π‘₯)    β‡’   (πœ‘ β†’ βˆƒπ‘“(𝑓 Fn 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) ∈ π‘₯))
 
18-Nov-2023condc 853 Contraposition of a decidable proposition.

This theorem swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky". This theorem (without the decidability condition, of course) is called Transp or "the principle of transposition" in Principia Mathematica (Theorem *2.17 of [WhiteheadRussell] p. 103) and is Axiom A3 of [Margaris] p. 49. We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning.

(Contributed by Jim Kingdon, 13-Mar-2018.) (Proof shortened by BJ, 18-Nov-2023.)

(DECID πœ‘ β†’ ((Β¬ πœ‘ β†’ Β¬ πœ“) β†’ (πœ“ β†’ πœ‘)))
 
18-Nov-2023stdcn 847 A formula is stable if and only if the decidability of its negation implies its decidability. Note that the right-hand side of this biconditional is the converse of dcn 842. (Contributed by BJ, 18-Nov-2023.)
(STAB πœ‘ ↔ (DECID Β¬ πœ‘ β†’ DECID πœ‘))
 
17-Nov-2023cnplimclemr 14299 Lemma for cnplimccntop 14300. The reverse direction. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
𝐾 = (MetOpenβ€˜(abs ∘ βˆ’ ))    &   π½ = (𝐾 β†Ύt 𝐴)    &   (πœ‘ β†’ 𝐴 βŠ† β„‚)    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    &   (πœ‘ β†’ (πΉβ€˜π΅) ∈ (𝐹 limβ„‚ 𝐡))    β‡’   (πœ‘ β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΅))
 
17-Nov-2023cnplimclemle 14298 Lemma for cnplimccntop 14300. Satisfying the epsilon condition for continuity. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
𝐾 = (MetOpenβ€˜(abs ∘ βˆ’ ))    &   π½ = (𝐾 β†Ύt 𝐴)    &   (πœ‘ β†’ 𝐴 βŠ† β„‚)    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    &   (πœ‘ β†’ (πΉβ€˜π΅) ∈ (𝐹 limβ„‚ 𝐡))    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝐷 ∈ ℝ+)    &   (πœ‘ β†’ 𝑍 ∈ 𝐴)    &   ((πœ‘ ∧ 𝑍 # 𝐡 ∧ (absβ€˜(𝑍 βˆ’ 𝐡)) < 𝐷) β†’ (absβ€˜((πΉβ€˜π‘) βˆ’ (πΉβ€˜π΅))) < (𝐸 / 2))    &   (πœ‘ β†’ (absβ€˜(𝑍 βˆ’ 𝐡)) < 𝐷)    β‡’   (πœ‘ β†’ (absβ€˜((πΉβ€˜π‘) βˆ’ (πΉβ€˜π΅))) < 𝐸)
 
14-Nov-2023limccnp2cntop 14307 The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Nov-2023.)
((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝑅 ∈ 𝑋)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝑆 ∈ π‘Œ)    &   (πœ‘ β†’ 𝑋 βŠ† β„‚)    &   (πœ‘ β†’ π‘Œ βŠ† β„‚)    &   πΎ = (MetOpenβ€˜(abs ∘ βˆ’ ))    &   π½ = ((𝐾 Γ—t 𝐾) β†Ύt (𝑋 Γ— π‘Œ))    &   (πœ‘ β†’ 𝐢 ∈ ((π‘₯ ∈ 𝐴 ↦ 𝑅) limβ„‚ 𝐡))    &   (πœ‘ β†’ 𝐷 ∈ ((π‘₯ ∈ 𝐴 ↦ 𝑆) limβ„‚ 𝐡))    &   (πœ‘ β†’ 𝐻 ∈ ((𝐽 CnP 𝐾)β€˜βŸ¨πΆ, 𝐷⟩))    β‡’   (πœ‘ β†’ (𝐢𝐻𝐷) ∈ ((π‘₯ ∈ 𝐴 ↦ (𝑅𝐻𝑆)) limβ„‚ 𝐡))
 
10-Nov-2023rpmaxcl 11235 The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.)
((𝐴 ∈ ℝ+ ∧ 𝐡 ∈ ℝ+) β†’ sup({𝐴, 𝐡}, ℝ, < ) ∈ ℝ+)
 
9-Nov-2023limccnp2lem 14306 Lemma for limccnp2cntop 14307. This is most of the result, expressed in epsilon-delta form, with a large number of hypotheses so that lengthy expressions do not need to be repeated. (Contributed by Jim Kingdon, 9-Nov-2023.)
((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝑅 ∈ 𝑋)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝑆 ∈ π‘Œ)    &   (πœ‘ β†’ 𝑋 βŠ† β„‚)    &   (πœ‘ β†’ π‘Œ βŠ† β„‚)    &   πΎ = (MetOpenβ€˜(abs ∘ βˆ’ ))    &   π½ = ((𝐾 Γ—t 𝐾) β†Ύt (𝑋 Γ— π‘Œ))    &   (πœ‘ β†’ 𝐢 ∈ ((π‘₯ ∈ 𝐴 ↦ 𝑅) limβ„‚ 𝐡))    &   (πœ‘ β†’ 𝐷 ∈ ((π‘₯ ∈ 𝐴 ↦ 𝑆) limβ„‚ 𝐡))    &   (πœ‘ β†’ 𝐻 ∈ ((𝐽 CnP 𝐾)β€˜βŸ¨πΆ, 𝐷⟩))    &   β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝐿 ∈ ℝ+)    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ 𝑋 βˆ€π‘  ∈ π‘Œ (((𝐢((abs ∘ βˆ’ ) β†Ύ (𝑋 Γ— 𝑋))π‘Ÿ) < 𝐿 ∧ (𝐷((abs ∘ βˆ’ ) β†Ύ (π‘Œ Γ— π‘Œ))𝑠) < 𝐿) β†’ ((𝐢𝐻𝐷)(abs ∘ βˆ’ )(π‘Ÿπ»π‘ )) < 𝐸))    &   (πœ‘ β†’ 𝐹 ∈ ℝ+)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 ((π‘₯ # 𝐡 ∧ (absβ€˜(π‘₯ βˆ’ 𝐡)) < 𝐹) β†’ (absβ€˜(𝑅 βˆ’ 𝐢)) < 𝐿))    &   (πœ‘ β†’ 𝐺 ∈ ℝ+)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 ((π‘₯ # 𝐡 ∧ (absβ€˜(π‘₯ βˆ’ 𝐡)) < 𝐺) β†’ (absβ€˜(𝑆 βˆ’ 𝐷)) < 𝐿))    β‡’   (πœ‘ β†’ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘₯ ∈ 𝐴 ((π‘₯ # 𝐡 ∧ (absβ€˜(π‘₯ βˆ’ 𝐡)) < 𝑑) β†’ (absβ€˜((𝑅𝐻𝑆) βˆ’ (𝐢𝐻𝐷))) < 𝐸))
 
4-Nov-2023ellimc3apf 14290 Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 4-Nov-2023.)
(πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   β„²π‘§πΉ    β‡’   (πœ‘ β†’ (𝐢 ∈ (𝐹 limβ„‚ 𝐡) ↔ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ βˆ€π‘§ ∈ 𝐴 ((𝑧 # 𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))))
 
3-Nov-2023limcmpted 14293 Express the limit operator for a function defined by a mapping, via epsilon-delta. (Contributed by Jim Kingdon, 3-Nov-2023.)
(πœ‘ β†’ 𝐴 βŠ† β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ 𝐷 ∈ β„‚)    β‡’   (πœ‘ β†’ (𝐢 ∈ ((𝑧 ∈ 𝐴 ↦ 𝐷) limβ„‚ 𝐡) ↔ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ βˆ€π‘§ ∈ 𝐴 ((𝑧 # 𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜(𝐷 βˆ’ 𝐢)) < π‘₯))))
 
1-Nov-2023unct 12446 The union of two countable sets is countable. Corollary 8.1.20 of [AczelRathjen], p. 75. (Contributed by Jim Kingdon, 1-Nov-2023.)
((βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o) ∧ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐡 βŠ” 1o)) β†’ βˆƒβ„Ž β„Ž:ω–ontoβ†’((𝐴 βˆͺ 𝐡) βŠ” 1o))
 
31-Oct-2023ctiunct 12444 A sequence of enumerations gives an enumeration of the union. We refer to "sequence of enumerations" rather than "countably many countable sets" because the hypothesis provides more than countability for each 𝐡(π‘₯): it refers to 𝐡(π‘₯) together with the 𝐺(π‘₯) which enumerates it. Theorem 8.1.19 of [AczelRathjen], p. 74.

For "countably many countable sets" the key hypothesis would be (πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ βˆƒπ‘”π‘”:ω–ontoβ†’(𝐡 βŠ” 1o). This is almost omiunct 12448 (which uses countable choice) although that is for a countably infinite collection not any countable collection.

Compare with the case of two sets instead of countably many, as seen at unct 12446, which says that the union of two countable sets is countable .

The proof proceeds by mapping a natural number to a pair of natural numbers (by xpomen 12399) and using the first number to map to an element π‘₯ of 𝐴 and the second number to map to an element of B(x) . In this way we are able to map to every element of βˆͺ π‘₯ ∈ 𝐴𝐡. Although it would be possible to work directly with countability expressed as 𝐹:ω–ontoβ†’(𝐴 βŠ” 1o), we instead use functions from subsets of the natural numbers via ctssdccl 7113 and ctssdc 7115.

(Contributed by Jim Kingdon, 31-Oct-2023.)

(πœ‘ β†’ 𝐹:ω–ontoβ†’(𝐴 βŠ” 1o))    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐺:ω–ontoβ†’(𝐡 βŠ” 1o))    β‡’   (πœ‘ β†’ βˆƒβ„Ž β„Ž:ω–ontoβ†’(βˆͺ π‘₯ ∈ 𝐴 𝐡 βŠ” 1o))
 
30-Oct-2023ctssdccl 7113 A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7115 but expressed in terms of classes rather than βˆƒ. (Contributed by Jim Kingdon, 30-Oct-2023.)
(πœ‘ β†’ 𝐹:ω–ontoβ†’(𝐴 βŠ” 1o))    &   π‘† = {π‘₯ ∈ Ο‰ ∣ (πΉβ€˜π‘₯) ∈ (inl β€œ 𝐴)}    &   πΊ = (β—‘inl ∘ 𝐹)    β‡’   (πœ‘ β†’ (𝑆 βŠ† Ο‰ ∧ 𝐺:𝑆–onto→𝐴 ∧ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆))
 
28-Oct-2023ctiunctlemfo 12443 Lemma for ctiunct 12444. (Contributed by Jim Kingdon, 28-Oct-2023.)
(πœ‘ β†’ 𝑆 βŠ† Ο‰)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆)    &   (πœ‘ β†’ 𝐹:𝑆–onto→𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝑇 βŠ† Ο‰)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑇)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐺:𝑇–onto→𝐡)    &   (πœ‘ β†’ 𝐽:ω–1-1-ontoβ†’(Ο‰ Γ— Ο‰))    &   π‘ˆ = {𝑧 ∈ Ο‰ ∣ ((1st β€˜(π½β€˜π‘§)) ∈ 𝑆 ∧ (2nd β€˜(π½β€˜π‘§)) ∈ ⦋(πΉβ€˜(1st β€˜(π½β€˜π‘§))) / π‘₯β¦Œπ‘‡)}    &   π» = (𝑛 ∈ π‘ˆ ↦ (⦋(πΉβ€˜(1st β€˜(π½β€˜π‘›))) / π‘₯β¦ŒπΊβ€˜(2nd β€˜(π½β€˜π‘›))))    &   β„²π‘₯𝐻    &   β„²π‘₯π‘ˆ    β‡’   (πœ‘ β†’ 𝐻:π‘ˆβ€“ontoβ†’βˆͺ π‘₯ ∈ 𝐴 𝐡)
 
28-Oct-2023ctiunctlemf 12442 Lemma for ctiunct 12444. (Contributed by Jim Kingdon, 28-Oct-2023.)
(πœ‘ β†’ 𝑆 βŠ† Ο‰)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆)    &   (πœ‘ β†’ 𝐹:𝑆–onto→𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝑇 βŠ† Ο‰)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑇)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐺:𝑇–onto→𝐡)    &   (πœ‘ β†’ 𝐽:ω–1-1-ontoβ†’(Ο‰ Γ— Ο‰))    &   π‘ˆ = {𝑧 ∈ Ο‰ ∣ ((1st β€˜(π½β€˜π‘§)) ∈ 𝑆 ∧ (2nd β€˜(π½β€˜π‘§)) ∈ ⦋(πΉβ€˜(1st β€˜(π½β€˜π‘§))) / π‘₯β¦Œπ‘‡)}    &   π» = (𝑛 ∈ π‘ˆ ↦ (⦋(πΉβ€˜(1st β€˜(π½β€˜π‘›))) / π‘₯β¦ŒπΊβ€˜(2nd β€˜(π½β€˜π‘›))))    β‡’   (πœ‘ β†’ 𝐻:π‘ˆβŸΆβˆͺ π‘₯ ∈ 𝐴 𝐡)
 
28-Oct-2023ctiunctlemudc 12441 Lemma for ctiunct 12444. (Contributed by Jim Kingdon, 28-Oct-2023.)
(πœ‘ β†’ 𝑆 βŠ† Ο‰)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆)    &   (πœ‘ β†’ 𝐹:𝑆–onto→𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝑇 βŠ† Ο‰)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑇)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐺:𝑇–onto→𝐡)    &   (πœ‘ β†’ 𝐽:ω–1-1-ontoβ†’(Ο‰ Γ— Ο‰))    &   π‘ˆ = {𝑧 ∈ Ο‰ ∣ ((1st β€˜(π½β€˜π‘§)) ∈ 𝑆 ∧ (2nd β€˜(π½β€˜π‘§)) ∈ ⦋(πΉβ€˜(1st β€˜(π½β€˜π‘§))) / π‘₯β¦Œπ‘‡)}    β‡’   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ π‘ˆ)
 
28-Oct-2023ctiunctlemuom 12440 Lemma for ctiunct 12444. (Contributed by Jim Kingdon, 28-Oct-2023.)
(πœ‘ β†’ 𝑆 βŠ† Ο‰)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆)    &   (πœ‘ β†’ 𝐹:𝑆–onto→𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝑇 βŠ† Ο‰)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑇)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐺:𝑇–onto→𝐡)    &   (πœ‘ β†’ 𝐽:ω–1-1-ontoβ†’(Ο‰ Γ— Ο‰))    &   π‘ˆ = {𝑧 ∈ Ο‰ ∣ ((1st β€˜(π½β€˜π‘§)) ∈ 𝑆 ∧ (2nd β€˜(π½β€˜π‘§)) ∈ ⦋(πΉβ€˜(1st β€˜(π½β€˜π‘§))) / π‘₯β¦Œπ‘‡)}    β‡’   (πœ‘ β†’ π‘ˆ βŠ† Ο‰)
 
28-Oct-2023ctiunctlemu2nd 12439 Lemma for ctiunct 12444. (Contributed by Jim Kingdon, 28-Oct-2023.)
(πœ‘ β†’ 𝑆 βŠ† Ο‰)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆)    &   (πœ‘ β†’ 𝐹:𝑆–onto→𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝑇 βŠ† Ο‰)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑇)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐺:𝑇–onto→𝐡)    &   (πœ‘ β†’ 𝐽:ω–1-1-ontoβ†’(Ο‰ Γ— Ο‰))    &   π‘ˆ = {𝑧 ∈ Ο‰ ∣ ((1st β€˜(π½β€˜π‘§)) ∈ 𝑆 ∧ (2nd β€˜(π½β€˜π‘§)) ∈ ⦋(πΉβ€˜(1st β€˜(π½β€˜π‘§))) / π‘₯β¦Œπ‘‡)}    &   (πœ‘ β†’ 𝑁 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ (2nd β€˜(π½β€˜π‘)) ∈ ⦋(πΉβ€˜(1st β€˜(π½β€˜π‘))) / π‘₯β¦Œπ‘‡)
 
28-Oct-2023ctiunctlemu1st 12438 Lemma for ctiunct 12444. (Contributed by Jim Kingdon, 28-Oct-2023.)
(πœ‘ β†’ 𝑆 βŠ† Ο‰)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆)    &   (πœ‘ β†’ 𝐹:𝑆–onto→𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝑇 βŠ† Ο‰)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑇)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐺:𝑇–onto→𝐡)    &   (πœ‘ β†’ 𝐽:ω–1-1-ontoβ†’(Ο‰ Γ— Ο‰))    &   π‘ˆ = {𝑧 ∈ Ο‰ ∣ ((1st β€˜(π½β€˜π‘§)) ∈ 𝑆 ∧ (2nd β€˜(π½β€˜π‘§)) ∈ ⦋(πΉβ€˜(1st β€˜(π½β€˜π‘§))) / π‘₯β¦Œπ‘‡)}    &   (πœ‘ β†’ 𝑁 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ (1st β€˜(π½β€˜π‘)) ∈ 𝑆)
 
28-Oct-2023pm2.521gdc 868 A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107, under a decidability condition. (Contributed by BJ, 28-Oct-2023.)
(DECID πœ‘ β†’ (Β¬ (πœ‘ β†’ πœ“) β†’ (πœ’ β†’ πœ‘)))
 
28-Oct-2023stdcndc 845 A formula is decidable if and only if its negation is decidable and it is stable (that is, it is testable and stable). (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof shortened by BJ, 28-Oct-2023.)
((STAB πœ‘ ∧ DECID Β¬ πœ‘) ↔ DECID πœ‘)
 
28-Oct-2023conax1k 654 Weakening of conax1 653. General instance of pm2.51 655 and of pm2.52 656. (Contributed by BJ, 28-Oct-2023.)
(Β¬ (πœ‘ β†’ πœ“) β†’ (πœ’ β†’ Β¬ πœ“))
 
28-Oct-2023conax1 653 Contrapositive of ax-1 6. (Contributed by BJ, 28-Oct-2023.)
(Β¬ (πœ‘ β†’ πœ“) β†’ Β¬ πœ“)
 
25-Oct-2023divcnap 14216 Complex number division is a continuous function, when the second argument is apart from zero. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Jim Kingdon, 25-Oct-2023.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    &   πΎ = (𝐽 β†Ύt {π‘₯ ∈ β„‚ ∣ π‘₯ # 0})    β‡’   (𝑦 ∈ β„‚, 𝑧 ∈ {π‘₯ ∈ β„‚ ∣ π‘₯ # 0} ↦ (𝑦 / 𝑧)) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽)
 
23-Oct-2023cnm 7834 A complex number is an inhabited set. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by Jim Kingdon, 23-Oct-2023.) (New usage is discouraged.)
(𝐴 ∈ β„‚ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)
 
23-Oct-2023oprssdmm 6175 Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.)
((πœ‘ ∧ 𝑒 ∈ 𝑆) β†’ βˆƒπ‘£ 𝑣 ∈ 𝑒)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯𝐹𝑦) ∈ 𝑆)    &   (πœ‘ β†’ Rel 𝐹)    β‡’   (πœ‘ β†’ (𝑆 Γ— 𝑆) βŠ† dom 𝐹)
 
22-Oct-2023addcncntoplem 14212 Lemma for addcncntop 14213, subcncntop 14214, and mulcncntop 14215. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 22-Oct-2023.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    &    + :(β„‚ Γ— β„‚)βŸΆβ„‚    &   ((π‘Ž ∈ ℝ+ ∧ 𝑏 ∈ β„‚ ∧ 𝑐 ∈ β„‚) β†’ βˆƒπ‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ+ βˆ€π‘’ ∈ β„‚ βˆ€π‘£ ∈ β„‚ (((absβ€˜(𝑒 βˆ’ 𝑏)) < 𝑦 ∧ (absβ€˜(𝑣 βˆ’ 𝑐)) < 𝑧) β†’ (absβ€˜((𝑒 + 𝑣) βˆ’ (𝑏 + 𝑐))) < π‘Ž))    β‡’    + ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)
 
22-Oct-2023txmetcnp 14179 Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by Jim Kingdon, 22-Oct-2023.)
𝐽 = (MetOpenβ€˜πΆ)    &   πΎ = (MetOpenβ€˜π·)    &   πΏ = (MetOpenβ€˜πΈ)    β‡’   (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝐷 ∈ (∞Metβ€˜π‘Œ) ∧ 𝐸 ∈ (∞Metβ€˜π‘)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ)) β†’ (𝐹 ∈ (((𝐽 Γ—t 𝐾) CnP 𝐿)β€˜βŸ¨π΄, 𝐡⟩) ↔ (𝐹:(𝑋 Γ— π‘Œ)βŸΆπ‘ ∧ βˆ€π‘§ ∈ ℝ+ βˆƒπ‘€ ∈ ℝ+ βˆ€π‘’ ∈ 𝑋 βˆ€π‘£ ∈ π‘Œ (((𝐴𝐢𝑒) < 𝑀 ∧ (𝐡𝐷𝑣) < 𝑀) β†’ ((𝐴𝐹𝐡)𝐸(𝑒𝐹𝑣)) < 𝑧))))
 
22-Oct-2023xmetxpbl 14169 The maximum metric (Chebyshev distance) on the product of two sets, expressed in terms of balls centered on a point 𝐢 with radius 𝑅. (Contributed by Jim Kingdon, 22-Oct-2023.)
𝑃 = (𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ sup({((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))}, ℝ*, < ))    &   (πœ‘ β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))    &   (πœ‘ β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))    &   (πœ‘ β†’ 𝑅 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝑋 Γ— π‘Œ))    β‡’   (πœ‘ β†’ (𝐢(ballβ€˜π‘ƒ)𝑅) = (((1st β€˜πΆ)(ballβ€˜π‘€)𝑅) Γ— ((2nd β€˜πΆ)(ballβ€˜π‘)𝑅)))
 
15-Oct-2023xmettxlem 14170 Lemma for xmettx 14171. (Contributed by Jim Kingdon, 15-Oct-2023.)
𝑃 = (𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ sup({((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))}, ℝ*, < ))    &   (πœ‘ β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))    &   (πœ‘ β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))    &   π½ = (MetOpenβ€˜π‘€)    &   πΎ = (MetOpenβ€˜π‘)    &   πΏ = (MetOpenβ€˜π‘ƒ)    β‡’   (πœ‘ β†’ 𝐿 βŠ† (𝐽 Γ—t 𝐾))
 
11-Oct-2023xmettx 14171 The maximum metric (Chebyshev distance) on the product of two sets, expressed as a binary topological product. (Contributed by Jim Kingdon, 11-Oct-2023.)
𝑃 = (𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ sup({((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))}, ℝ*, < ))    &   (πœ‘ β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))    &   (πœ‘ β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))    &   π½ = (MetOpenβ€˜π‘€)    &   πΎ = (MetOpenβ€˜π‘)    &   πΏ = (MetOpenβ€˜π‘ƒ)    β‡’   (πœ‘ β†’ 𝐿 = (𝐽 Γ—t 𝐾))
 
11-Oct-2023xmetxp 14168 The maximum metric (Chebyshev distance) on the product of two sets. (Contributed by Jim Kingdon, 11-Oct-2023.)
𝑃 = (𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ sup({((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))}, ℝ*, < ))    &   (πœ‘ β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))    &   (πœ‘ β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))    β‡’   (πœ‘ β†’ 𝑃 ∈ (∞Metβ€˜(𝑋 Γ— π‘Œ)))
 
7-Oct-2023df-iress 12473 Define a multifunction restriction operator for extensible structures, which can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the Base set while leaving operators alone; individual kinds of structures will need to handle this behavior, by ignoring operators' values outside the range, defining a function using the base set and applying that, or explicitly truncating the slot before use.

(Credit for this operator, as well as the 2023 modification for iset.mm, goes to Mario Carneiro.)

(Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 7-Oct-2023.)

β†Ύs = (𝑀 ∈ V, π‘₯ ∈ V ↦ (𝑀 sSet ⟨(Baseβ€˜ndx), (π‘₯ ∩ (Baseβ€˜π‘€))⟩))
 
29-Sep-2023syl2anc2 412 Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023.)
(πœ‘ β†’ πœ“)    &   (πœ“ β†’ πœ’)    &   ((πœ“ ∧ πœ’) β†’ πœƒ)    β‡’   (πœ‘ β†’ πœƒ)
 
27-Sep-2023fnpr2ob 12766 Biconditional version of fnpr2o 12765. (Contributed by Jim Kingdon, 27-Sep-2023.)
((𝐴 ∈ V ∧ 𝐡 ∈ V) ↔ {βŸ¨βˆ…, 𝐴⟩, ⟨1o, 𝐡⟩} Fn 2o)
 
25-Sep-2023xpsval 12778 Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
𝑇 = (𝑅 Γ—s 𝑆)    &   π‘‹ = (Baseβ€˜π‘…)    &   π‘Œ = (Baseβ€˜π‘†)    &   (πœ‘ β†’ 𝑅 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ π‘Š)    &   πΉ = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©})    &   πΊ = (Scalarβ€˜π‘…)    &   π‘ˆ = (𝐺Xs{βŸ¨βˆ…, π‘…βŸ©, ⟨1o, π‘†βŸ©})    β‡’   (πœ‘ β†’ 𝑇 = (◑𝐹 β€œs π‘ˆ))
 
25-Sep-2023fvpr1o 12768 The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
(𝐡 ∈ 𝑉 β†’ ({βŸ¨βˆ…, 𝐴⟩, ⟨1o, 𝐡⟩}β€˜1o) = 𝐡)
 
25-Sep-2023fvpr0o 12767 The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
(𝐴 ∈ 𝑉 β†’ ({βŸ¨βˆ…, 𝐴⟩, ⟨1o, 𝐡⟩}β€˜βˆ…) = 𝐴)
 
25-Sep-2023fnpr2o 12765 Function with a domain of 2o. (Contributed by Jim Kingdon, 25-Sep-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ {βŸ¨βˆ…, 𝐴⟩, ⟨1o, 𝐡⟩} Fn 2o)
 
25-Sep-2023df-xps 12731 Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
Γ—s = (π‘Ÿ ∈ V, 𝑠 ∈ V ↦ (β—‘(π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Baseβ€˜π‘ ) ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©}) β€œs ((Scalarβ€˜π‘Ÿ)Xs{βŸ¨βˆ…, π‘ŸβŸ©, ⟨1o, π‘ βŸ©})))
 
12-Sep-2023pwntru 4201 A slight strengthening of pwtrufal 14909. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)
((𝐴 βŠ† {βˆ…} ∧ 𝐴 β‰  {βˆ…}) β†’ 𝐴 = βˆ…)
 
11-Sep-2023pwtrufal 14909 A subset of the singleton {βˆ…} cannot be anything other than βˆ… or {βˆ…}. Removing the double negation would change the meaning, as seen at exmid01 4200. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4198), then this theorem states there are no truth values other than true and false, as described in section 1.1 of [Bauer], p. 481. (Contributed by Mario Carneiro and Jim Kingdon, 11-Sep-2023.)
(𝐴 βŠ† {βˆ…} β†’ Β¬ Β¬ (𝐴 = βˆ… ∨ 𝐴 = {βˆ…}))
 
9-Sep-2023mathbox 14645 (This theorem is a dummy placeholder for these guidelines. The label of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of iset.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of iset.mm.

Guidelines:

Mathboxes in iset.mm follow the same practices as in set.mm, so refer to the mathbox guidelines there for more details.

(Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (New usage is discouraged.)

πœ‘    β‡’   πœ‘
 
6-Sep-2023djuexb 7046 The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.)
((𝐴 ∈ V ∧ 𝐡 ∈ V) ↔ (𝐴 βŠ” 𝐡) ∈ V)
 
3-Sep-2023pwf1oexmid 14911 An exercise related to 𝑁 copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
𝑇 = βˆͺ π‘₯ ∈ 𝑁 ({π‘₯} Γ— 1o)    β‡’   ((𝑁 ∈ Ο‰ ∧ 𝐺:𝑇–1-1→𝒫 1o) β†’ (ran 𝐺 = 𝒫 1o ↔ (𝑁 = 2o ∧ EXMID)))
 
3-Sep-2023pwle2 14910 An exercise related to 𝑁 copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
𝑇 = βˆͺ π‘₯ ∈ 𝑁 ({π‘₯} Γ— 1o)    β‡’   ((𝑁 ∈ Ο‰ ∧ 𝐺:𝑇–1-1→𝒫 1o) β†’ 𝑁 βŠ† 2o)
 
30-Aug-2023isomninn 14941 Omniscience stated in terms of natural numbers. Similar to isomnimap 7138 but it will sometimes be more convenient to use 0 and 1 rather than βˆ… and 1o. (Contributed by Jim Kingdon, 30-Aug-2023.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Omni ↔ βˆ€π‘“ ∈ ({0, 1} β†‘π‘š 𝐴)(βˆƒπ‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 0 ∨ βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1)))
 
30-Aug-2023isomninnlem 14940 Lemma for isomninn 14941. The result, with a hypothesis to provide a convenient notation. (Contributed by Jim Kingdon, 30-Aug-2023.)
𝐺 = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    β‡’   (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Omni ↔ βˆ€π‘“ ∈ ({0, 1} β†‘π‘š 𝐴)(βˆƒπ‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 0 ∨ βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1)))
 
28-Aug-2023trilpolemisumle 14948 Lemma for trilpo 14953. An upper bound for the sum of the digits beyond a certain point. (Contributed by Jim Kingdon, 28-Aug-2023.)
(πœ‘ β†’ 𝐹:β„•βŸΆ{0, 1})    &   π΄ = Σ𝑖 ∈ β„• ((1 / (2↑𝑖)) Β· (πΉβ€˜π‘–))    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    β‡’   (πœ‘ β†’ Σ𝑖 ∈ 𝑍 ((1 / (2↑𝑖)) Β· (πΉβ€˜π‘–)) ≀ Σ𝑖 ∈ 𝑍 (1 / (2↑𝑖)))
 
25-Aug-2023cvgcmp2n 14943 A comparison test for convergence of a real infinite series. (Contributed by Jim Kingdon, 25-Aug-2023.)
((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΊβ€˜π‘˜) ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ 0 ≀ (πΊβ€˜π‘˜))    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΊβ€˜π‘˜) ≀ (1 / (2β†‘π‘˜)))    β‡’   (πœ‘ β†’ seq1( + , 𝐺) ∈ dom ⇝ )
 
25-Aug-2023cvgcmp2nlemabs 14942 Lemma for cvgcmp2n 14943. The partial sums get closer to each other as we go further out. The proof proceeds by rewriting (seq1( + , 𝐺)β€˜π‘) as the sum of (seq1( + , 𝐺)β€˜π‘€) and a term which gets smaller as 𝑀 gets large. (Contributed by Jim Kingdon, 25-Aug-2023.)
((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΊβ€˜π‘˜) ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ 0 ≀ (πΊβ€˜π‘˜))    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΊβ€˜π‘˜) ≀ (1 / (2β†‘π‘˜)))    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    β‡’   (πœ‘ β†’ (absβ€˜((seq1( + , 𝐺)β€˜π‘) βˆ’ (seq1( + , 𝐺)β€˜π‘€))) < (2 / 𝑀))
 
24-Aug-2023trilpolemclim 14946 Lemma for trilpo 14953. Convergence of the series. (Contributed by Jim Kingdon, 24-Aug-2023.)
(πœ‘ β†’ 𝐹:β„•βŸΆ{0, 1})    &   πΊ = (𝑛 ∈ β„• ↦ ((1 / (2↑𝑛)) Β· (πΉβ€˜π‘›)))    β‡’   (πœ‘ β†’ seq1( + , 𝐺) ∈ dom ⇝ )
 
23-Aug-2023trilpo 14953 Real number trichotomy implies the Limited Principle of Omniscience (LPO). We expect that we'd need some form of countable choice to prove the converse.

Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence contains a zero or it is all ones. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. Compare it with one using trichotomy. The three cases from trichotomy are trilpolemlt1 14951 (which means the sequence contains a zero), trilpolemeq1 14950 (which means the sequence is all ones), and trilpolemgt1 14949 (which is not possible).

Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 14939) or that the real numbers are a discrete field (see trirec0 14954).

LPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qtri3or 10246 for real numbers. (Contributed by Jim Kingdon, 23-Aug-2023.)

(βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ < 𝑦 ∨ π‘₯ = 𝑦 ∨ 𝑦 < π‘₯) β†’ Ο‰ ∈ Omni)
 
23-Aug-2023trilpolemres 14952 Lemma for trilpo 14953. The result. (Contributed by Jim Kingdon, 23-Aug-2023.)
(πœ‘ β†’ 𝐹:β„•βŸΆ{0, 1})    &   π΄ = Σ𝑖 ∈ β„• ((1 / (2↑𝑖)) Β· (πΉβ€˜π‘–))    &   (πœ‘ β†’ (𝐴 < 1 ∨ 𝐴 = 1 ∨ 1 < 𝐴))    β‡’   (πœ‘ β†’ (βˆƒπ‘₯ ∈ β„• (πΉβ€˜π‘₯) = 0 ∨ βˆ€π‘₯ ∈ β„• (πΉβ€˜π‘₯) = 1))
 
23-Aug-2023trilpolemlt1 14951 Lemma for trilpo 14953. The 𝐴 < 1 case. We can use the distance between 𝐴 and one (that is, 1 βˆ’ 𝐴) to find a position in the sequence 𝑛 where terms after that point will not add up to as much as 1 βˆ’ 𝐴. By finomni 7141 we know the terms up to 𝑛 either contain a zero or are all one. But if they are all one that contradicts the way we constructed 𝑛, so we know that the sequence contains a zero. (Contributed by Jim Kingdon, 23-Aug-2023.)
(πœ‘ β†’ 𝐹:β„•βŸΆ{0, 1})    &   π΄ = Σ𝑖 ∈ β„• ((1 / (2↑𝑖)) Β· (πΉβ€˜π‘–))    &   (πœ‘ β†’ 𝐴 < 1)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ β„• (πΉβ€˜π‘₯) = 0)
 
23-Aug-2023trilpolemeq1 14950 Lemma for trilpo 14953. The 𝐴 = 1 case. This is proved by noting that if any (πΉβ€˜π‘₯) is zero, then the infinite sum 𝐴 is less than one based on the term which is zero. We are using the fact that the 𝐹 sequence is decidable (in the sense that each element is either zero or one). (Contributed by Jim Kingdon, 23-Aug-2023.)
(πœ‘ β†’ 𝐹:β„•βŸΆ{0, 1})    &   π΄ = Σ𝑖 ∈ β„• ((1 / (2↑𝑖)) Β· (πΉβ€˜π‘–))    &   (πœ‘ β†’ 𝐴 = 1)    β‡’   (πœ‘ β†’ βˆ€π‘₯ ∈ β„• (πΉβ€˜π‘₯) = 1)
 
23-Aug-2023trilpolemgt1 14949 Lemma for trilpo 14953. The 1 < 𝐴 case. (Contributed by Jim Kingdon, 23-Aug-2023.)
(πœ‘ β†’ 𝐹:β„•βŸΆ{0, 1})    &   π΄ = Σ𝑖 ∈ β„• ((1 / (2↑𝑖)) Β· (πΉβ€˜π‘–))    β‡’   (πœ‘ β†’ Β¬ 1 < 𝐴)
 
23-Aug-2023trilpolemcl 14947 Lemma for trilpo 14953. The sum exists. (Contributed by Jim Kingdon, 23-Aug-2023.)
(πœ‘ β†’ 𝐹:β„•βŸΆ{0, 1})    &   π΄ = Σ𝑖 ∈ β„• ((1 / (2↑𝑖)) Β· (πΉβ€˜π‘–))    β‡’   (πœ‘ β†’ 𝐴 ∈ ℝ)
 
23-Aug-2023triap 14939 Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ ((𝐴 < 𝐡 ∨ 𝐴 = 𝐡 ∨ 𝐡 < 𝐴) ↔ DECID 𝐴 # 𝐡))
 
19-Aug-2023djuenun 7214 Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.)
((𝐴 β‰ˆ 𝐡 ∧ 𝐢 β‰ˆ 𝐷 ∧ (𝐡 ∩ 𝐷) = βˆ…) β†’ (𝐴 βŠ” 𝐢) β‰ˆ (𝐡 βˆͺ 𝐷))
 
16-Aug-2023ctssdclemr 7114 Lemma for ctssdc 7115. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.)
(βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o) β†’ βˆƒπ‘ (𝑠 βŠ† Ο‰ ∧ βˆƒπ‘“ 𝑓:𝑠–onto→𝐴 ∧ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑠))
 
16-Aug-2023ctssdclemn0 7112 Lemma for ctssdc 7115. The Β¬ βˆ… ∈ 𝑆 case. (Contributed by Jim Kingdon, 16-Aug-2023.)
(πœ‘ β†’ 𝑆 βŠ† Ο‰)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆)    &   (πœ‘ β†’ 𝐹:𝑆–onto→𝐴)    &   (πœ‘ β†’ Β¬ βˆ… ∈ 𝑆)    β‡’   (πœ‘ β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐴 βŠ” 1o))
 
15-Aug-2023ctssexmid 7151 The decidability condition in ctssdc 7115 is needed. More specifically, ctssdc 7115 minus that condition, plus the Limited Principle of Omniscience (LPO), implies excluded middle. (Contributed by Jim Kingdon, 15-Aug-2023.)
((𝑦 βŠ† Ο‰ ∧ βˆƒπ‘“ 𝑓:𝑦–ontoβ†’π‘₯) β†’ βˆƒπ‘“ 𝑓:ω–ontoβ†’(π‘₯ βŠ” 1o))    &   Ο‰ ∈ Omni    β‡’   (πœ‘ ∨ Β¬ πœ‘)
 
15-Aug-2023ctssdc 7115 A set is countable iff there is a surjection from a decidable subset of the natural numbers onto it. The decidability condition is needed as shown at ctssexmid 7151. (Contributed by Jim Kingdon, 15-Aug-2023.)
(βˆƒπ‘ (𝑠 βŠ† Ο‰ ∧ βˆƒπ‘“ 𝑓:𝑠–onto→𝐴 ∧ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑠) ↔ βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o))
 
14-Aug-2023mpoexw 6217 Weak version of mpoex 6218 that holds without ax-coll 4120. If the domain and codomain of an operation given by maps-to notation are sets, the operation is a set. (Contributed by Rohan Ridenour, 14-Aug-2023.)
𝐴 ∈ V    &   π΅ ∈ V    &   π· ∈ V    &   βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 𝐢 ∈ 𝐷    β‡’   (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐡 ↦ 𝐢) ∈ V
 
13-Aug-2023grpinvfvalg 12922 The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) (Revised by Rohan Ridenour, 13-Aug-2023.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   π‘ = (invgβ€˜πΊ)    β‡’   (𝐺 ∈ 𝑉 β†’ 𝑁 = (π‘₯ ∈ 𝐡 ↦ (℩𝑦 ∈ 𝐡 (𝑦 + π‘₯) = 0 )))
 
13-Aug-2023ltntri 8088 Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy, 𝐴 < 𝐡 ∨ 𝐴 = 𝐡 ∨ 𝐡 < 𝐴. Does that mean there is a pair of real numbers where none of those hold (that is, where we can refute each of those three relationships)? Actually, no, as shown here. This is another example of distinguishing between being unable to prove something, or being able to refute it. (Contributed by Jim Kingdon, 13-Aug-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ Β¬ (Β¬ 𝐴 < 𝐡 ∧ Β¬ 𝐴 = 𝐡 ∧ Β¬ 𝐡 < 𝐴))
 
13-Aug-2023mptexw 6117 Weak version of mptex 5745 that holds without ax-coll 4120. If the domain and codomain of a function given by maps-to notation are sets, the function is a set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝐴 ∈ V    &   πΆ ∈ V    &   βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ 𝐢    β‡’   (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ V
 
13-Aug-2023funexw 6116 Weak version of funex 5742 that holds without ax-coll 4120. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.)
((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐡 ∧ ran 𝐹 ∈ 𝐢) β†’ 𝐹 ∈ V)
 
11-Aug-2023qnnen 12435 The rational numbers are countably infinite. Corollary 8.1.23 of [AczelRathjen], p. 75. This is Metamath 100 proof #3. (Contributed by Jim Kingdon, 11-Aug-2023.)
β„š β‰ˆ β„•
 
10-Aug-2023ctinfomlemom 12431 Lemma for ctinfom 12432. Converting between Ο‰ and β„•0. (Contributed by Jim Kingdon, 10-Aug-2023.)
𝑁 = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   πΊ = (𝐹 ∘ ◑𝑁)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ Β¬ (πΉβ€˜π‘˜) ∈ (𝐹 β€œ 𝑛))    β‡’   (πœ‘ β†’ (𝐺:β„•0–onto→𝐴 ∧ βˆ€π‘š ∈ β„•0 βˆƒπ‘— ∈ β„•0 βˆ€π‘– ∈ (0...π‘š)(πΊβ€˜π‘—) β‰  (πΊβ€˜π‘–)))
 
9-Aug-2023difinfsnlem 7101 Lemma for difinfsn 7102. The case where we need to swap 𝐡 and (inrβ€˜βˆ…) in building the mapping 𝐺. (Contributed by Jim Kingdon, 9-Aug-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    &   (πœ‘ β†’ 𝐹:(Ο‰ βŠ” 1o)–1-1→𝐴)    &   (πœ‘ β†’ (πΉβ€˜(inrβ€˜βˆ…)) β‰  𝐡)    &   πΊ = (𝑛 ∈ Ο‰ ↦ if((πΉβ€˜(inlβ€˜π‘›)) = 𝐡, (πΉβ€˜(inrβ€˜βˆ…)), (πΉβ€˜(inlβ€˜π‘›))))    β‡’   (πœ‘ β†’ 𝐺:ω–1-1β†’(𝐴 βˆ– {𝐡}))
 
8-Aug-2023difinfinf 7103 An infinite set minus a finite subset is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)
(((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴) ∧ (𝐡 βŠ† 𝐴 ∧ 𝐡 ∈ Fin)) β†’ Ο‰ β‰Ό (𝐴 βˆ– 𝐡))
 
8-Aug-2023difinfsn 7102 An infinite set minus one element is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)
((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ Ο‰ β‰Ό (𝐴 βˆ– {𝐡}))
 
7-Aug-2023ctinf 12434 A set is countably infinite if and only if it has decidable equality, is countable, and is infinite. (Contributed by Jim Kingdon, 7-Aug-2023.)
(𝐴 β‰ˆ β„• ↔ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ βˆƒπ‘“ 𝑓:ω–onto→𝐴 ∧ Ο‰ β‰Ό 𝐴))
 
7-Aug-2023inffinp1 12433 An infinite set contains an element not contained in a given finite subset. (Contributed by Jim Kingdon, 7-Aug-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ Ο‰ β‰Ό 𝐴)    &   (πœ‘ β†’ 𝐡 βŠ† 𝐴)    &   (πœ‘ β†’ 𝐡 ∈ Fin)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 Β¬ π‘₯ ∈ 𝐡)
 
7-Aug-2023ctinfom 12432 A condition for a set being countably infinite. Restates ennnfone 12429 in terms of Ο‰ and function image. Like ennnfone 12429 the condition can be summarized as 𝐴 being countable, infinite, and having decidable equality. (Contributed by Jim Kingdon, 7-Aug-2023.)
(𝐴 β‰ˆ β„• ↔ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ βˆƒπ‘“(𝑓:ω–onto→𝐴 ∧ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ Β¬ (π‘“β€˜π‘˜) ∈ (𝑓 β€œ 𝑛))))
 
6-Aug-2023rerestcntop 14211 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    &   π‘… = (topGenβ€˜ran (,))    β‡’   (𝐴 βŠ† ℝ β†’ (𝐽 β†Ύt 𝐴) = (𝑅 β†Ύt 𝐴))
 
6-Aug-2023tgioo2cntop 14210 The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’   (topGenβ€˜ran (,)) = (𝐽 β†Ύt ℝ)
 
4-Aug-2023nninffeq 14931 Equality of two functions on β„•βˆž which agree at every integer and at the point at infinity. From an online post by Martin Escardo. Remark: the last two hypotheses can be grouped into one, (πœ‘ β†’ βˆ€π‘› ∈ suc Ο‰...). (Contributed by Jim Kingdon, 4-Aug-2023.)
(πœ‘ β†’ 𝐹:β„•βˆžβŸΆβ„•0)    &   (πœ‘ β†’ 𝐺:β„•βˆžβŸΆβ„•0)    &   (πœ‘ β†’ (πΉβ€˜(π‘₯ ∈ Ο‰ ↦ 1o)) = (πΊβ€˜(π‘₯ ∈ Ο‰ ↦ 1o)))    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ (πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))))    β‡’   (πœ‘ β†’ 𝐹 = 𝐺)
 
3-Aug-2023txvalex 13915 Existence of the binary topological product. If 𝑅 and 𝑆 are known to be topologies, see txtop 13921. (Contributed by Jim Kingdon, 3-Aug-2023.)
((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ π‘Š) β†’ (𝑅 Γ—t 𝑆) ∈ V)
 
3-Aug-2023ablgrpd 13105 An Abelian group is a group, deduction form of ablgrp 13104. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(πœ‘ β†’ 𝐺 ∈ Abel)    β‡’   (πœ‘ β†’ 𝐺 ∈ Grp)
 
3-Aug-20231nsgtrivd 13089 A group with exactly one normal subgroup is trivial. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐡 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Grp)    &   (πœ‘ β†’ (NrmSGrpβ€˜πΊ) β‰ˆ 1o)    β‡’   (πœ‘ β†’ 𝐡 = { 0 })
 
3-Aug-2023triv1nsgd 13088 A trivial group has exactly one normal subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐡 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Grp)    &   (πœ‘ β†’ 𝐡 = { 0 })    β‡’   (πœ‘ β†’ (NrmSGrpβ€˜πΊ) β‰ˆ 1o)
 
3-Aug-2023trivnsgd 13087 The only normal subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐡 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Grp)    &   (πœ‘ β†’ 𝐡 = { 0 })    β‡’   (πœ‘ β†’ (NrmSGrpβ€˜πΊ) = {𝐡})
 
3-Aug-20230idnsgd 13086 The whole group and the zero subgroup are normal subgroups of a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐡 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Grp)    β‡’   (πœ‘ β†’ {{ 0 }, 𝐡} βŠ† (NrmSGrpβ€˜πΊ))
 
3-Aug-2023trivsubgsnd 13071 The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐡 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Grp)    &   (πœ‘ β†’ 𝐡 = { 0 })    β‡’   (πœ‘ β†’ (SubGrpβ€˜πΊ) = {𝐡})
 
3-Aug-2023trivsubgd 13070 The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐡 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Grp)    &   (πœ‘ β†’ 𝐡 = { 0 })    &   (πœ‘ β†’ 𝐴 ∈ (SubGrpβ€˜πΊ))    β‡’   (πœ‘ β†’ 𝐴 = 𝐡)
 
3-Aug-2023mulgcld 13015 Deduction associated with mulgcl 13010. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐡 = (Baseβ€˜πΊ)    &    Β· = (.gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Grp)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑁 Β· 𝑋) ∈ 𝐡)
 
3-Aug-2023hashfingrpnn 12916 A finite group has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐡 = (Baseβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Grp)    &   (πœ‘ β†’ 𝐡 ∈ Fin)    β‡’   (πœ‘ β†’ (β™―β€˜π΅) ∈ β„•)
 
3-Aug-2023hashfinmndnn 12840 A finite monoid has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐡 = (Baseβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Mnd)    &   (πœ‘ β†’ 𝐡 ∈ Fin)    β‡’   (πœ‘ β†’ (β™―β€˜π΅) ∈ β„•)
 
3-Aug-2023dvdsgcdidd 11998 The greatest common divisor of a positive integer and another integer it divides is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝑀 βˆ₯ 𝑁)    β‡’   (πœ‘ β†’ (𝑀 gcd 𝑁) = 𝑀)
 
3-Aug-2023gcdmultipled 11997 The greatest common divisor of a nonnegative integer 𝑀 and a multiple of it is 𝑀 itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(πœ‘ β†’ 𝑀 ∈ β„•0)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    β‡’   (πœ‘ β†’ (𝑀 gcd (𝑁 Β· 𝑀)) = 𝑀)
 
3-Aug-2023fihashelne0d 10780 A finite set with an element has nonzero size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(πœ‘ β†’ 𝐡 ∈ 𝐴)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    β‡’   (πœ‘ β†’ Β¬ (β™―β€˜π΄) = 0)
 
3-Aug-2023phpeqd 6935 Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 6868 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐡 βŠ† 𝐴)    &   (πœ‘ β†’ 𝐴 β‰ˆ 𝐡)    β‡’   (πœ‘ β†’ 𝐴 = 𝐡)
 
3-Aug-2023enpr2d 6820 A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(πœ‘ β†’ 𝐴 ∈ 𝐢)    &   (πœ‘ β†’ 𝐡 ∈ 𝐷)    &   (πœ‘ β†’ Β¬ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ {𝐴, 𝐡} β‰ˆ 2o)
 
3-Aug-2023elrnmpt2d 4884 Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐹 = (π‘₯ ∈ 𝐴 ↦ 𝐡)    &   (πœ‘ β†’ 𝐢 ∈ ran 𝐹)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 𝐢 = 𝐡)
 
3-Aug-2023elrnmptdv 4883 Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐹 = (π‘₯ ∈ 𝐴 ↦ 𝐡)    &   (πœ‘ β†’ 𝐢 ∈ 𝐴)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   ((πœ‘ ∧ π‘₯ = 𝐢) β†’ 𝐷 = 𝐡)    β‡’   (πœ‘ β†’ 𝐷 ∈ ran 𝐹)
 
3-Aug-2023rspcime 2850 Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023.)
((πœ‘ ∧ π‘₯ = 𝐴) β†’ πœ“)    &   (πœ‘ β†’ 𝐴 ∈ 𝐡)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐡 πœ“)
 
3-Aug-2023neqcomd 2182 Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(πœ‘ β†’ Β¬ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ Β¬ 𝐡 = 𝐴)
 
2-Aug-2023dvid 14323 Derivative of the identity function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
(β„‚ D ( I β†Ύ β„‚)) = (β„‚ Γ— {1})
 
2-Aug-2023dvconst 14322 Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
(𝐴 ∈ β„‚ β†’ (β„‚ D (β„‚ Γ— {𝐴})) = (β„‚ Γ— {0}))
 
2-Aug-2023dvidlemap 14321 Lemma for dvid 14323 and dvconst 14322. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
(πœ‘ β†’ 𝐹:β„‚βŸΆβ„‚)    &   ((πœ‘ ∧ (π‘₯ ∈ β„‚ ∧ 𝑧 ∈ β„‚ ∧ 𝑧 # π‘₯)) β†’ (((πΉβ€˜π‘§) βˆ’ (πΉβ€˜π‘₯)) / (𝑧 βˆ’ π‘₯)) = 𝐡)    &   π΅ ∈ β„‚    β‡’   (πœ‘ β†’ (β„‚ D 𝐹) = (β„‚ Γ— {𝐡}))
 
2-Aug-2023diveqap1bd 8796 If two complex numbers are equal, their quotient is one. One-way deduction form of diveqap1 8665. Converse of diveqap1d 8758. (Contributed by David Moews, 28-Feb-2017.) (Revised by Jim Kingdon, 2-Aug-2023.)
(πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 # 0)    &   (πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ (𝐴 / 𝐡) = 1)
 
31-Jul-2023mul0inf 11252 Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11074 and mulap0bd 8617 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((𝐴 Β· 𝐡) = 0 ↔ inf({(absβ€˜π΄), (absβ€˜π΅)}, ℝ, < ) = 0))
 
31-Jul-2023mul0eqap 8630 If two numbers are apart from each other and their product is zero, one of them must be zero. (Contributed by Jim Kingdon, 31-Jul-2023.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐴 # 𝐡)    &   (πœ‘ β†’ (𝐴 Β· 𝐡) = 0)    β‡’   (πœ‘ β†’ (𝐴 = 0 ∨ 𝐡 = 0))
 
31-Jul-2023apcon4bid 8584 Contrapositive law deduction for apartness. (Contributed by Jim Kingdon, 31-Jul-2023.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐷 ∈ β„‚)    &   (πœ‘ β†’ (𝐴 # 𝐡 ↔ 𝐢 # 𝐷))    β‡’   (πœ‘ β†’ (𝐴 = 𝐡 ↔ 𝐢 = 𝐷))
 
30-Jul-2023uzennn 10439 An upper integer set is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 30-Jul-2023.)
(𝑀 ∈ β„€ β†’ (β„€β‰₯β€˜π‘€) β‰ˆ β„•)
 
30-Jul-2023djuen 7213 Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.)
((𝐴 β‰ˆ 𝐡 ∧ 𝐢 β‰ˆ 𝐷) β†’ (𝐴 βŠ” 𝐢) β‰ˆ (𝐡 βŠ” 𝐷))
 
30-Jul-2023endjudisj 7212 Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by Jim Kingdon, 30-Jul-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (𝐴 βŠ” 𝐡) β‰ˆ (𝐴 βˆͺ 𝐡))
 
30-Jul-2023eninr 7100 Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
(𝐴 ∈ 𝑉 β†’ (inr β€œ 𝐴) β‰ˆ 𝐴)
 
30-Jul-2023eninl 7099 Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
(𝐴 ∈ 𝑉 β†’ (inl β€œ 𝐴) β‰ˆ 𝐴)
 
29-Jul-2023exmidunben 12430 If any unbounded set of positive integers is equinumerous to β„•, then the Limited Principle of Omniscience (LPO) implies excluded middle. (Contributed by Jim Kingdon, 29-Jul-2023.)
((βˆ€π‘₯((π‘₯ βŠ† β„• ∧ βˆ€π‘š ∈ β„• βˆƒπ‘› ∈ π‘₯ π‘š < 𝑛) β†’ π‘₯ β‰ˆ β„•) ∧ Ο‰ ∈ Omni) β†’ EXMID)
 
29-Jul-2023exmidsssnc 4205 Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4200 but lets you choose any set as the element of the singleton rather than just βˆ…. It is similar to exmidsssn 4204 but for a particular set 𝐡 rather than all sets. (Contributed by Jim Kingdon, 29-Jul-2023.)
(𝐡 ∈ 𝑉 β†’ (EXMID ↔ βˆ€π‘₯(π‘₯ βŠ† {𝐡} β†’ (π‘₯ = βˆ… ∨ π‘₯ = {𝐡}))))
 
28-Jul-2023dvfcnpm 14320 The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jul-2023.)
(𝐹 ∈ (β„‚ ↑pm β„‚) β†’ (β„‚ D 𝐹):dom (β„‚ D 𝐹)βŸΆβ„‚)
 
28-Jul-2023dvfpm 14319 The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 28-Jul-2023.)
(𝐹 ∈ (β„‚ ↑pm ℝ) β†’ (ℝ D 𝐹):dom (ℝ D 𝐹)βŸΆβ„‚)
 
24-Jul-2023sraring 13547 Condition for a subring algebra to be a ring. (Contributed by Thierry Arnoux, 24-Jul-2023.)
𝐴 = ((subringAlg β€˜π‘…)β€˜π‘‰)    &   π΅ = (Baseβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑉 βŠ† 𝐡) β†’ 𝐴 ∈ Ring)
 
23-Jul-2023ennnfonelemhdmp1 12413 Lemma for ennnfone 12429. Domain at a successor where we need to add an element to the sequence. (Contributed by Jim Kingdon, 23-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    &   (πœ‘ β†’ 𝑃 ∈ β„•0)    &   (πœ‘ β†’ Β¬ (πΉβ€˜(β—‘π‘β€˜π‘ƒ)) ∈ (𝐹 β€œ (β—‘π‘β€˜π‘ƒ)))    β‡’   (πœ‘ β†’ dom (π»β€˜(𝑃 + 1)) = suc dom (π»β€˜π‘ƒ))
 
23-Jul-2023ennnfonelemp1 12410 Lemma for ennnfone 12429. Value of 𝐻 at a successor. (Contributed by Jim Kingdon, 23-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    &   (πœ‘ β†’ 𝑃 ∈ β„•0)    β‡’   (πœ‘ β†’ (π»β€˜(𝑃 + 1)) = if((πΉβ€˜(β—‘π‘β€˜π‘ƒ)) ∈ (𝐹 β€œ (β—‘π‘β€˜π‘ƒ)), (π»β€˜π‘ƒ), ((π»β€˜π‘ƒ) βˆͺ {⟨dom (π»β€˜π‘ƒ), (πΉβ€˜(β—‘π‘β€˜π‘ƒ))⟩})))
 
22-Jul-2023nntr2 6507 Transitive law for natural numbers. (Contributed by Jim Kingdon, 22-Jul-2023.)
((𝐴 ∈ Ο‰ ∧ 𝐢 ∈ Ο‰) β†’ ((𝐴 βŠ† 𝐡 ∧ 𝐡 ∈ 𝐢) β†’ 𝐴 ∈ 𝐢))
 
22-Jul-2023nnsssuc 6506 A natural number is a subset of another natural number if and only if it belongs to its successor. (Contributed by Jim Kingdon, 22-Jul-2023.)
((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) β†’ (𝐴 βŠ† 𝐡 ↔ 𝐴 ∈ suc 𝐡))
 
21-Jul-2023ennnfoneleminc 12415 Lemma for ennnfone 12429. We only add elements to 𝐻 as the index increases. (Contributed by Jim Kingdon, 21-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    &   (πœ‘ β†’ 𝑃 ∈ β„•0)    &   (πœ‘ β†’ 𝑄 ∈ β„•0)    &   (πœ‘ β†’ 𝑃 ≀ 𝑄)    β‡’   (πœ‘ β†’ (π»β€˜π‘ƒ) βŠ† (π»β€˜π‘„))
 
20-Jul-2023ennnfonelemg 12407 Lemma for ennnfone 12429. Closure for 𝐺. (Contributed by Jim Kingdon, 20-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    β‡’   ((πœ‘ ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑pm Ο‰) ∣ dom 𝑔 ∈ Ο‰} ∧ 𝑗 ∈ Ο‰)) β†’ (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴 ↑pm Ο‰) ∣ dom 𝑔 ∈ Ο‰})
 
20-Jul-2023ennnfonelemjn 12406 Lemma for ennnfone 12429. Non-initial state for 𝐽. (Contributed by Jim Kingdon, 20-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    β‡’   ((πœ‘ ∧ 𝑓 ∈ (β„€β‰₯β€˜(0 + 1))) β†’ (π½β€˜π‘“) ∈ Ο‰)
 
20-Jul-2023ennnfonelemj0 12405 Lemma for ennnfone 12429. Initial state for 𝐽. (Contributed by Jim Kingdon, 20-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    β‡’   (πœ‘ β†’ (π½β€˜0) ∈ {𝑔 ∈ (𝐴 ↑pm Ο‰) ∣ dom 𝑔 ∈ Ο‰})
 
20-Jul-2023seqp1cd 10469 Value of the sequence builder function at a successor. A version of seq3p1 10465 which provides two classes 𝐷 and 𝐢 for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 20-Jul-2023.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ (πΉβ€˜π‘€) ∈ 𝐢)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐷)) β†’ (π‘₯ + 𝑦) ∈ 𝐢)    &   ((πœ‘ ∧ π‘₯ ∈ (β„€β‰₯β€˜(𝑀 + 1))) β†’ (πΉβ€˜π‘₯) ∈ 𝐷)    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜(𝑁 + 1)) = ((seq𝑀( + , 𝐹)β€˜π‘) + (πΉβ€˜(𝑁 + 1))))
 
20-Jul-2023seqovcd 10466 A closure law for the recursive sequence builder. This is a lemma for theorems such as seqf2 10467 and seq1cd 10468 and is unlikely to be needed once such theorems are proved. (Contributed by Jim Kingdon, 20-Jul-2023.)
((πœ‘ ∧ π‘₯ ∈ (β„€β‰₯β€˜(𝑀 + 1))) β†’ (πΉβ€˜π‘₯) ∈ 𝐷)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐷)) β†’ (π‘₯ + 𝑦) ∈ 𝐢)    β‡’   ((πœ‘ ∧ (π‘₯ ∈ (β„€β‰₯β€˜π‘€) ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯(𝑧 ∈ (β„€β‰₯β€˜π‘€), 𝑀 ∈ 𝐢 ↦ (𝑀 + (πΉβ€˜(𝑧 + 1))))𝑦) ∈ 𝐢)
 
19-Jul-2023ennnfonelemhom 12419 Lemma for ennnfone 12429. The sequences in 𝐻 increase in length without bound if you go out far enough. (Contributed by Jim Kingdon, 19-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    &   (πœ‘ β†’ 𝑀 ∈ Ο‰)    β‡’   (πœ‘ β†’ βˆƒπ‘– ∈ β„•0 𝑀 ∈ dom (π»β€˜π‘–))
 
19-Jul-2023ennnfonelemex 12418 Lemma for ennnfone 12429. Extending the sequence (π»β€˜π‘ƒ) to include an additional element. (Contributed by Jim Kingdon, 19-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    &   (πœ‘ β†’ 𝑃 ∈ β„•0)    β‡’   (πœ‘ β†’ βˆƒπ‘– ∈ β„•0 dom (π»β€˜π‘ƒ) ∈ dom (π»β€˜π‘–))
 
19-Jul-2023ennnfonelemkh 12416 Lemma for ennnfone 12429. Because we add zero or one entries for each new index, the length of each sequence is no greater than its index. (Contributed by Jim Kingdon, 19-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    &   (πœ‘ β†’ 𝑃 ∈ β„•0)    β‡’   (πœ‘ β†’ dom (π»β€˜π‘ƒ) βŠ† (β—‘π‘β€˜π‘ƒ))
 
19-Jul-2023ennnfonelemom 12412 Lemma for ennnfone 12429. 𝐻 yields finite sequences. (Contributed by Jim Kingdon, 19-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    &   (πœ‘ β†’ 𝑃 ∈ β„•0)    β‡’   (πœ‘ β†’ dom (π»β€˜π‘ƒ) ∈ Ο‰)
 
19-Jul-2023ennnfonelem1 12411 Lemma for ennnfone 12429. Second value. (Contributed by Jim Kingdon, 19-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    β‡’   (πœ‘ β†’ (π»β€˜1) = {βŸ¨βˆ…, (πΉβ€˜βˆ…)⟩})
 
19-Jul-2023seq1cd 10468 Initial value of the recursive sequence builder. A version of seq3-1 10463 which provides two classes 𝐷 and 𝐢 for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 19-Jul-2023.)
(πœ‘ β†’ (πΉβ€˜π‘€) ∈ 𝐢)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐷)) β†’ (π‘₯ + 𝑦) ∈ 𝐢)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   ((πœ‘ ∧ π‘₯ ∈ (β„€β‰₯β€˜(𝑀 + 1))) β†’ (πΉβ€˜π‘₯) ∈ 𝐷)    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘€) = (πΉβ€˜π‘€))
 
17-Jul-2023ennnfonelemhf1o 12417 Lemma for ennnfone 12429. Each of the functions in 𝐻 is one to one and onto an image of 𝐹. (Contributed by Jim Kingdon, 17-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    &   (πœ‘ β†’ 𝑃 ∈ β„•0)    β‡’   (πœ‘ β†’ (π»β€˜π‘ƒ):dom (π»β€˜π‘ƒ)–1-1-ontoβ†’(𝐹 β€œ (β—‘π‘β€˜π‘ƒ)))
 
16-Jul-2023ennnfonelemen 12425 Lemma for ennnfone 12429. The result. (Contributed by Jim Kingdon, 16-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    &   πΏ = βˆͺ 𝑖 ∈ β„•0 (π»β€˜π‘–)    β‡’   (πœ‘ β†’ 𝐴 β‰ˆ β„•)
 
16-Jul-2023ennnfonelemdm 12424 Lemma for ennnfone 12429. The function 𝐿 is defined everywhere. (Contributed by Jim Kingdon, 16-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    &   πΏ = βˆͺ 𝑖 ∈ β„•0 (π»β€˜π‘–)    β‡’   (πœ‘ β†’ dom 𝐿 = Ο‰)
 
16-Jul-2023ennnfonelemrn 12423 Lemma for ennnfone 12429. 𝐿 is onto 𝐴. (Contributed by Jim Kingdon, 16-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    &   πΏ = βˆͺ 𝑖 ∈ β„•0 (π»β€˜π‘–)    β‡’   (πœ‘ β†’ ran 𝐿 = 𝐴)
 
16-Jul-2023ennnfonelemf1 12422 Lemma for ennnfone 12429. 𝐿 is one-to-one. (Contributed by Jim Kingdon, 16-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    &   πΏ = βˆͺ 𝑖 ∈ β„•0 (π»β€˜π‘–)    β‡’   (πœ‘ β†’ 𝐿:dom 𝐿–1-1→𝐴)
 
16-Jul-2023ennnfonelemfun 12421 Lemma for ennnfone 12429. 𝐿 is a function. (Contributed by Jim Kingdon, 16-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    &   πΏ = βˆͺ 𝑖 ∈ β„•0 (π»β€˜π‘–)    β‡’   (πœ‘ β†’ Fun 𝐿)
 
16-Jul-2023ennnfonelemrnh 12420 Lemma for ennnfone 12429. A consequence of ennnfonelemss 12414. (Contributed by Jim Kingdon, 16-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    &   (πœ‘ β†’ 𝑋 ∈ ran 𝐻)    &   (πœ‘ β†’ π‘Œ ∈ ran 𝐻)    β‡’   (πœ‘ β†’ (𝑋 βŠ† π‘Œ ∨ π‘Œ βŠ† 𝑋))
 
15-Jul-2023ennnfonelemss 12414 Lemma for ennnfone 12429. We only add elements to 𝐻 as the index increases. (Contributed by Jim Kingdon, 15-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    &   (πœ‘ β†’ 𝑃 ∈ β„•0)    β‡’   (πœ‘ β†’ (π»β€˜π‘ƒ) βŠ† (π»β€˜(𝑃 + 1)))
 
15-Jul-2023ennnfonelem0 12409 Lemma for ennnfone 12429. Initial value. (Contributed by Jim Kingdon, 15-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    β‡’   (πœ‘ β†’ (π»β€˜0) = βˆ…)
 
15-Jul-2023ennnfonelemk 12404 Lemma for ennnfone 12429. (Contributed by Jim Kingdon, 15-Jul-2023.)
(πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ 𝐾 ∈ Ο‰)    &   (πœ‘ β†’ 𝑁 ∈ Ο‰)    &   (πœ‘ β†’ βˆ€π‘— ∈ suc 𝑁(πΉβ€˜πΎ) β‰  (πΉβ€˜π‘—))    β‡’   (πœ‘ β†’ 𝑁 ∈ 𝐾)
 
15-Jul-2023ennnfonelemdc 12403 Lemma for ennnfone 12429. A direct consequence of fidcenumlemrk 6956. (Contributed by Jim Kingdon, 15-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ 𝑃 ∈ Ο‰)    β‡’   (πœ‘ β†’ DECID (πΉβ€˜π‘ƒ) ∈ (𝐹 β€œ 𝑃))
 
14-Jul-2023djur 7071 A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) Upgrade implication to biconditional and shorten proof. (Revised by BJ, 14-Jul-2023.)
(𝐢 ∈ (𝐴 βŠ” 𝐡) ↔ (βˆƒπ‘₯ ∈ 𝐴 𝐢 = (inlβ€˜π‘₯) ∨ βˆƒπ‘₯ ∈ 𝐡 𝐢 = (inrβ€˜π‘₯)))
 
13-Jul-2023sbthomlem 14935 Lemma for sbthom 14936. (Contributed by Mario Carneiro and Jim Kingdon, 13-Jul-2023.)
(πœ‘ β†’ Ο‰ ∈ Omni)    &   (πœ‘ β†’ π‘Œ βŠ† {βˆ…})    &   (πœ‘ β†’ 𝐹:ω–1-1-ontoβ†’(π‘Œ βŠ” Ο‰))    β‡’   (πœ‘ β†’ (π‘Œ = βˆ… ∨ π‘Œ = {βˆ…}))
 
12-Jul-2023caseinr 7094 Applying the "case" construction to a right injection. (Contributed by Jim Kingdon, 12-Jul-2023.)
(πœ‘ β†’ Fun 𝐹)    &   (πœ‘ β†’ 𝐺 Fn 𝐡)    &   (πœ‘ β†’ 𝐴 ∈ 𝐡)    β‡’   (πœ‘ β†’ (case(𝐹, 𝐺)β€˜(inrβ€˜π΄)) = (πΊβ€˜π΄))
 
12-Jul-2023inl11 7067 Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ((inlβ€˜π΄) = (inlβ€˜π΅) ↔ 𝐴 = 𝐡))
 
11-Jul-2023djudomr 7222 A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐡 β‰Ό (𝐴 βŠ” 𝐡))
 
11-Jul-2023djudoml 7221 A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐴 β‰Ό (𝐴 βŠ” 𝐡))
 
11-Jul-2023omp1eomlem 7096 Lemma for omp1eom 7097. (Contributed by Jim Kingdon, 11-Jul-2023.)
𝐹 = (π‘₯ ∈ Ο‰ ↦ if(π‘₯ = βˆ…, (inrβ€˜π‘₯), (inlβ€˜βˆͺ π‘₯)))    &   π‘† = (π‘₯ ∈ Ο‰ ↦ suc π‘₯)    &   πΊ = case(𝑆, ( I β†Ύ 1o))    β‡’   πΉ:ω–1-1-ontoβ†’(Ο‰ βŠ” 1o)
 
11-Jul-2023xp01disjl 6438 Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)
(({βˆ…} Γ— 𝐴) ∩ ({1o} Γ— 𝐢)) = βˆ…
 
10-Jul-2023sbthom 14936 Schroeder-Bernstein is not possible even for Ο‰. We know by exmidsbth 14934 that full Schroeder-Bernstein will not be provable but what about the case where one of the sets is Ο‰? That case plus the Limited Principle of Omniscience (LPO) implies excluded middle, so we will not be able to prove it. (Contributed by Mario Carneiro and Jim Kingdon, 10-Jul-2023.)
((βˆ€π‘₯((π‘₯ β‰Ό Ο‰ ∧ Ο‰ β‰Ό π‘₯) β†’ π‘₯ β‰ˆ Ο‰) ∧ Ο‰ ∈ Omni) β†’ EXMID)
 
10-Jul-2023endjusym 7098 Reversing right and left operands of a disjoint union produces an equinumerous result. (Contributed by Jim Kingdon, 10-Jul-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐴 βŠ” 𝐡) β‰ˆ (𝐡 βŠ” 𝐴))
 
10-Jul-2023omp1eom 7097 Adding one to Ο‰. (Contributed by Jim Kingdon, 10-Jul-2023.)
(Ο‰ βŠ” 1o) β‰ˆ Ο‰
 
9-Jul-2023refeq 14938 Equality of two real functions which agree at negative numbers, positive numbers, and zero. This holds even without real trichotomy. From an online post by Martin Escardo. (Contributed by Jim Kingdon, 9-Jul-2023.)
(πœ‘ β†’ 𝐹:β„βŸΆβ„)    &   (πœ‘ β†’ 𝐺:β„βŸΆβ„)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ (π‘₯ < 0 β†’ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯)))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ (0 < π‘₯ β†’ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯)))    &   (πœ‘ β†’ (πΉβ€˜0) = (πΊβ€˜0))    β‡’   (πœ‘ β†’ 𝐹 = 𝐺)
 
9-Jul-2023seqvalcd 10462 Value of the sequence builder function. Similar to seq3val 10461 but the classes 𝐷 (type of each term) and 𝐢 (type of the value we are accumulating) do not need to be the same. (Contributed by Jim Kingdon, 9-Jul-2023.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘… = frec((π‘₯ ∈ (β„€β‰₯β€˜π‘€), 𝑦 ∈ V ↦ ⟨(π‘₯ + 1), (π‘₯(𝑧 ∈ (β„€β‰₯β€˜π‘€), 𝑀 ∈ 𝐢 ↦ (𝑀 + (πΉβ€˜(𝑧 + 1))))𝑦)⟩), βŸ¨π‘€, (πΉβ€˜π‘€)⟩)    &   (πœ‘ β†’ (πΉβ€˜π‘€) ∈ 𝐢)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐷)) β†’ (π‘₯ + 𝑦) ∈ 𝐢)    &   ((πœ‘ ∧ π‘₯ ∈ (β„€β‰₯β€˜(𝑀 + 1))) β†’ (πΉβ€˜π‘₯) ∈ 𝐷)    β‡’   (πœ‘ β†’ seq𝑀( + , 𝐹) = ran 𝑅)
 
9-Jul-2023djuun 7069 The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)
((inl β€œ 𝐴) βˆͺ (inr β€œ 𝐡)) = (𝐴 βŠ” 𝐡)
 
9-Jul-2023djuin 7066 The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)
((inl β€œ 𝐴) ∩ (inr β€œ 𝐡)) = βˆ…
 
8-Jul-2023limcimo 14295 Conditions which ensure there is at most one limit value of 𝐹 at 𝐡. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 8-Jul-2023.)
(πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ 𝐢)    &   (πœ‘ β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ 𝐢 ∈ (𝐾 β†Ύt 𝑆))    &   (πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ {π‘ž ∈ 𝐢 ∣ π‘ž # 𝐡} βŠ† 𝐴)    &   πΎ = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’   (πœ‘ β†’ βˆƒ*π‘₯ π‘₯ ∈ (𝐹 limβ„‚ 𝐡))
 
8-Jul-2023ennnfonelemh 12408 Lemma for ennnfone 12429. (Contributed by Jim Kingdon, 8-Jul-2023.)
(πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)    &   (πœ‘ β†’ 𝐹:ω–onto→𝐴)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ βˆƒπ‘˜ ∈ Ο‰ βˆ€π‘— ∈ suc 𝑛(πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘—))    &   πΊ = (π‘₯ ∈ (𝐴 ↑pm Ο‰), 𝑦 ∈ Ο‰ ↦ if((πΉβ€˜π‘¦) ∈ (𝐹 β€œ 𝑦), π‘₯, (π‘₯ βˆͺ {⟨dom π‘₯, (πΉβ€˜π‘¦)⟩})))    &   π‘ = frec((π‘₯ ∈ β„€ ↦ (π‘₯ + 1)), 0)    &   π½ = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, βˆ…, (β—‘π‘β€˜(π‘₯ βˆ’ 1))))    &   π» = seq0(𝐺, 𝐽)    β‡’   (πœ‘ β†’ 𝐻:β„•0⟢(𝐴 ↑pm Ο‰))
 
7-Jul-2023seqf2 10467 Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 7-Jul-2023.)
(πœ‘ β†’ (πΉβ€˜π‘€) ∈ 𝐢)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐷)) β†’ (π‘₯ + 𝑦) ∈ 𝐢)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   ((πœ‘ ∧ π‘₯ ∈ (β„€β‰₯β€˜(𝑀 + 1))) β†’ (πΉβ€˜π‘₯) ∈ 𝐷)    β‡’   (πœ‘ β†’ seq𝑀( + , 𝐹):π‘βŸΆπΆ)
 
3-Jul-2023limcimolemlt 14294 Lemma for limcimo 14295. (Contributed by Jim Kingdon, 3-Jul-2023.)
(πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ 𝐢)    &   (πœ‘ β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ 𝐢 ∈ (𝐾 β†Ύt 𝑆))    &   (πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ {π‘ž ∈ 𝐢 ∣ π‘ž # 𝐡} βŠ† 𝐴)    &   πΎ = (MetOpenβ€˜(abs ∘ βˆ’ ))    &   (πœ‘ β†’ 𝐷 ∈ ℝ+)    &   (πœ‘ β†’ 𝑋 ∈ (𝐹 limβ„‚ 𝐡))    &   (πœ‘ β†’ π‘Œ ∈ (𝐹 limβ„‚ 𝐡))    &   (πœ‘ β†’ βˆ€π‘§ ∈ 𝐴 ((𝑧 # 𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝐷) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝑋)) < ((absβ€˜(𝑋 βˆ’ π‘Œ)) / 2)))    &   (πœ‘ β†’ 𝐺 ∈ ℝ+)    &   (πœ‘ β†’ βˆ€π‘€ ∈ 𝐴 ((𝑀 # 𝐡 ∧ (absβ€˜(𝑀 βˆ’ 𝐡)) < 𝐺) β†’ (absβ€˜((πΉβ€˜π‘€) βˆ’ π‘Œ)) < ((absβ€˜(𝑋 βˆ’ π‘Œ)) / 2)))    β‡’   (πœ‘ β†’ (absβ€˜(𝑋 βˆ’ π‘Œ)) < (absβ€˜(𝑋 βˆ’ π‘Œ)))
 
28-Jun-2023dvfgg 14318 Explicitly write out the functionality condition on derivative for 𝑆 = ℝ and β„‚. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm 𝑆)) β†’ (𝑆 D 𝐹):dom (𝑆 D 𝐹)βŸΆβ„‚)
 
28-Jun-2023dvbsssg 14316 The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
((𝑆 βŠ† β„‚ ∧ 𝐹 ∈ (β„‚ ↑pm 𝑆)) β†’ dom (𝑆 D 𝐹) βŠ† 𝑆)
 
27-Jun-2023dvbssntrcntop 14314 The set of differentiable points is a subset of the interior of the domain of the function. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
(πœ‘ β†’ 𝑆 βŠ† β„‚)    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    &   π½ = (𝐾 β†Ύt 𝑆)    &   πΎ = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’   (πœ‘ β†’ dom (𝑆 D 𝐹) βŠ† ((intβ€˜π½)β€˜π΄))
 
27-Jun-2023eldvap 14312 The differentiable predicate. A function 𝐹 is differentiable at 𝐡 with derivative 𝐢 iff 𝐹 is defined in a neighborhood of 𝐡 and the difference quotient has limit 𝐢 at 𝐡. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
𝑇 = (𝐾 β†Ύt 𝑆)    &   πΎ = (MetOpenβ€˜(abs ∘ βˆ’ ))    &   πΊ = (𝑧 ∈ {𝑀 ∈ 𝐴 ∣ 𝑀 # 𝐡} ↦ (((πΉβ€˜π‘§) βˆ’ (πΉβ€˜π΅)) / (𝑧 βˆ’ 𝐡)))    &   (πœ‘ β†’ 𝑆 βŠ† β„‚)    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    β‡’   (πœ‘ β†’ (𝐡(𝑆 D 𝐹)𝐢 ↔ (𝐡 ∈ ((intβ€˜π‘‡)β€˜π΄) ∧ 𝐢 ∈ (𝐺 limβ„‚ 𝐡))))
 
27-Jun-2023dvfvalap 14311 Value and set bounds on the derivative operator. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
𝑇 = (𝐾 β†Ύt 𝑆)    &   πΎ = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’   ((𝑆 βŠ† β„‚ ∧ 𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† 𝑆) β†’ ((𝑆 D 𝐹) = βˆͺ π‘₯ ∈ ((intβ€˜π‘‡)β€˜π΄)({π‘₯} Γ— ((𝑧 ∈ {𝑀 ∈ 𝐴 ∣ 𝑀 # π‘₯} ↦ (((πΉβ€˜π‘§) βˆ’ (πΉβ€˜π‘₯)) / (𝑧 βˆ’ π‘₯))) limβ„‚ π‘₯)) ∧ (𝑆 D 𝐹) βŠ† (((intβ€˜π‘‡)β€˜π΄) Γ— β„‚)))
 
27-Jun-2023dvlemap 14310 Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
(πœ‘ β†’ 𝐹:π·βŸΆβ„‚)    &   (πœ‘ β†’ 𝐷 βŠ† β„‚)    &   (πœ‘ β†’ 𝐡 ∈ 𝐷)    β‡’   ((πœ‘ ∧ 𝐴 ∈ {𝑀 ∈ 𝐷 ∣ 𝑀 # 𝐡}) β†’ (((πΉβ€˜π΄) βˆ’ (πΉβ€˜π΅)) / (𝐴 βˆ’ 𝐡)) ∈ β„‚)
 
25-Jun-2023reldvg 14309 The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
((𝑆 βŠ† β„‚ ∧ 𝐹 ∈ (β„‚ ↑pm 𝑆)) β†’ Rel (𝑆 D 𝐹))
 
25-Jun-2023df-dvap 14287 Define the derivative operator. This acts on functions to produce a function that is defined where the original function is differentiable, with value the derivative of the function at these points. The set 𝑠 here is the ambient topological space under which we are evaluating the continuity of the difference quotient. Although the definition is valid for any subset of β„‚ and is well-behaved when 𝑠 contains no isolated points, we will restrict our attention to the cases 𝑠 = ℝ or 𝑠 = β„‚ for the majority of the development, these corresponding respectively to real and complex differentiation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
D = (𝑠 ∈ 𝒫 β„‚, 𝑓 ∈ (β„‚ ↑pm 𝑠) ↦ βˆͺ π‘₯ ∈ ((intβ€˜((MetOpenβ€˜(abs ∘ βˆ’ )) β†Ύt 𝑠))β€˜dom 𝑓)({π‘₯} Γ— ((𝑧 ∈ {𝑀 ∈ dom 𝑓 ∣ 𝑀 # π‘₯} ↦ (((π‘“β€˜π‘§) βˆ’ (π‘“β€˜π‘₯)) / (𝑧 βˆ’ π‘₯))) limβ„‚ π‘₯)))
 
18-Jun-2023limccnpcntop 14305 If the limit of 𝐹 at 𝐡 is 𝐢 and 𝐺 is continuous at 𝐢, then the limit of 𝐺 ∘ 𝐹 at 𝐡 is 𝐺(𝐢). (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 18-Jun-2023.)
(πœ‘ β†’ 𝐹:𝐴⟢𝐷)    &   (πœ‘ β†’ 𝐷 βŠ† β„‚)    &   πΎ = (MetOpenβ€˜(abs ∘ βˆ’ ))    &   π½ = (𝐾 β†Ύt 𝐷)    &   (πœ‘ β†’ 𝐢 ∈ (𝐹 limβ„‚ 𝐡))    &   (πœ‘ β†’ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜πΆ))    β‡’   (πœ‘ β†’ (πΊβ€˜πΆ) ∈ ((𝐺 ∘ 𝐹) limβ„‚ 𝐡))
 
18-Jun-2023r19.30dc 2624 Restricted quantifier version of 19.30dc 1627. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.)
((βˆ€π‘₯ ∈ 𝐴 (πœ‘ ∨ πœ“) ∧ DECID βˆƒπ‘₯ ∈ 𝐴 πœ“) β†’ (βˆ€π‘₯ ∈ 𝐴 πœ‘ ∨ βˆƒπ‘₯ ∈ 𝐴 πœ“))
 
17-Jun-2023r19.28v 2605 Restricted quantifier version of one direction of 19.28 1563. (The other direction holds when 𝐴 is inhabited, see r19.28mv 3517.) (Contributed by NM, 2-Apr-2004.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
((πœ‘ ∧ βˆ€π‘₯ ∈ 𝐴 πœ“) β†’ βˆ€π‘₯ ∈ 𝐴 (πœ‘ ∧ πœ“))
 
17-Jun-2023r19.27v 2604 Restricted quantitifer version of one direction of 19.27 1561. (The other direction holds when 𝐴 is inhabited, see r19.27mv 3521.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
((βˆ€π‘₯ ∈ 𝐴 πœ‘ ∧ πœ“) β†’ βˆ€π‘₯ ∈ 𝐴 (πœ‘ ∧ πœ“))
 
16-Jun-2023cnlimcim 14301 If 𝐹 is a continuous function, the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 16-Jun-2023.)
(𝐴 βŠ† β„‚ β†’ (𝐹 ∈ (𝐴–cnβ†’β„‚) β†’ (𝐹:π΄βŸΆβ„‚ ∧ βˆ€π‘₯ ∈ 𝐴 (πΉβ€˜π‘₯) ∈ (𝐹 limβ„‚ π‘₯))))
 
16-Jun-2023cncfcn1cntop 14242 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.) (Revised by Jim Kingdon, 16-Jun-2023.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’   (ℂ–cnβ†’β„‚) = (𝐽 Cn 𝐽)
 
14-Jun-2023cnplimcim 14297 If a function is continuous at 𝐡, its limit at 𝐡 equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Jun-2023.)
𝐾 = (MetOpenβ€˜(abs ∘ βˆ’ ))    &   π½ = (𝐾 β†Ύt 𝐴)    β‡’   ((𝐴 βŠ† β„‚ ∧ 𝐡 ∈ 𝐴) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΅) β†’ (𝐹:π΄βŸΆβ„‚ ∧ (πΉβ€˜π΅) ∈ (𝐹 limβ„‚ 𝐡))))
 
14-Jun-2023metcnpd 14181 Two ways to say a mapping from metric 𝐢 to metric 𝐷 is continuous at point 𝑃. (Contributed by Jim Kingdon, 14-Jun-2023.)
(πœ‘ β†’ 𝐽 = (MetOpenβ€˜πΆ))    &   (πœ‘ β†’ 𝐾 = (MetOpenβ€˜π·))    &   (πœ‘ β†’ 𝐢 ∈ (∞Metβ€˜π‘‹))    &   (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π‘Œ))    &   (πœ‘ β†’ 𝑃 ∈ 𝑋)    β‡’   (πœ‘ β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ+ βˆ€π‘€ ∈ 𝑋 ((𝑃𝐢𝑀) < 𝑧 β†’ ((πΉβ€˜π‘ƒ)𝐷(πΉβ€˜π‘€)) < 𝑦))))
 
6-Jun-2023cntoptop 14194 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’   π½ ∈ Top
 
6-Jun-2023cntoptopon 14193 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’   π½ ∈ (TopOnβ€˜β„‚)
 
3-Jun-2023limcdifap 14292 It suffices to consider functions which are not defined at 𝐡 to define the limit of a function. In particular, the value of the original function 𝐹 at 𝐡 does not affect the limit of 𝐹. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)
(πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† β„‚)    β‡’   (πœ‘ β†’ (𝐹 limβ„‚ 𝐡) = ((𝐹 β†Ύ {π‘₯ ∈ 𝐴 ∣ π‘₯ # 𝐡}) limβ„‚ 𝐡))
 
3-Jun-2023ellimc3ap 14291 Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) Use apartness. (Revised by Jim Kingdon, 3-Jun-2023.)
(πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (𝐢 ∈ (𝐹 limβ„‚ 𝐡) ↔ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ βˆ€π‘§ ∈ 𝐴 ((𝑧 # 𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))))
 
3-Jun-2023df-limced 14286 Define the set of limits of a complex function at a point. Under normal circumstances, this will be a singleton or empty, depending on whether the limit exists. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)
limβ„‚ = (𝑓 ∈ (β„‚ ↑pm β„‚), π‘₯ ∈ β„‚ ↦ {𝑦 ∈ β„‚ ∣ ((𝑓:dom π‘“βŸΆβ„‚ ∧ dom 𝑓 βŠ† β„‚) ∧ (π‘₯ ∈ β„‚ ∧ βˆ€π‘’ ∈ ℝ+ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘§ ∈ dom 𝑓((𝑧 # π‘₯ ∧ (absβ€˜(𝑧 βˆ’ π‘₯)) < 𝑑) β†’ (absβ€˜((π‘“β€˜π‘§) βˆ’ 𝑦)) < 𝑒)))})
 
30-May-2023modprm1div 12250 A prime number divides an integer minus 1 iff the integer modulo the prime number is 1. (Contributed by Alexander van der Vekens, 17-May-2018.) (Proof shortened by AV, 30-May-2023.)
((𝑃 ∈ β„™ ∧ 𝐴 ∈ β„€) β†’ ((𝐴 mod 𝑃) = 1 ↔ 𝑃 βˆ₯ (𝐴 βˆ’ 1)))
 
30-May-2023modm1div 11810 An integer greater than one divides another integer minus one iff the second integer modulo the first integer is one. (Contributed by AV, 30-May-2023.)
((𝑁 ∈ (β„€β‰₯β€˜2) ∧ 𝐴 ∈ β„€) β†’ ((𝐴 mod 𝑁) = 1 ↔ 𝑁 βˆ₯ (𝐴 βˆ’ 1)))
 
30-May-2023eluz4nn 9571 An integer greater than or equal to 4 is a positive integer. (Contributed by AV, 30-May-2023.)
(𝑋 ∈ (β„€β‰₯β€˜4) β†’ 𝑋 ∈ β„•)
 
30-May-2023eluz4eluz2 9570 An integer greater than or equal to 4 is an integer greater than or equal to 2. (Contributed by AV, 30-May-2023.)
(𝑋 ∈ (β„€β‰₯β€˜4) β†’ 𝑋 ∈ (β„€β‰₯β€˜2))
 
29-May-2023mulcncflem 14251 Lemma for mulcncf 14252. (Contributed by Jim Kingdon, 29-May-2023.)
(πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ 𝑉 ∈ 𝑋)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝐹 ∈ ℝ+)    &   (πœ‘ β†’ 𝐺 ∈ ℝ+)    &   (πœ‘ β†’ 𝑆 ∈ ℝ+)    &   (πœ‘ β†’ 𝑇 ∈ ℝ+)    &   (πœ‘ β†’ βˆ€π‘’ ∈ 𝑋 ((absβ€˜(𝑒 βˆ’ 𝑉)) < 𝑆 β†’ (absβ€˜(((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘’) βˆ’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘‰))) < 𝐹))    &   (πœ‘ β†’ βˆ€π‘’ ∈ 𝑋 ((absβ€˜(𝑒 βˆ’ 𝑉)) < 𝑇 β†’ (absβ€˜(((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘’) βˆ’ ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘‰))) < 𝐺))    &   (πœ‘ β†’ βˆ€π‘’ ∈ 𝑋 (((absβ€˜(⦋𝑒 / π‘₯⦌𝐴 βˆ’ ⦋𝑉 / π‘₯⦌𝐴)) < 𝐹 ∧ (absβ€˜(⦋𝑒 / π‘₯⦌𝐡 βˆ’ ⦋𝑉 / π‘₯⦌𝐡)) < 𝐺) β†’ (absβ€˜((⦋𝑒 / π‘₯⦌𝐴 Β· ⦋𝑒 / π‘₯⦌𝐡) βˆ’ (⦋𝑉 / π‘₯⦌𝐴 Β· ⦋𝑉 / π‘₯⦌𝐡))) < 𝐸))    β‡’   (πœ‘ β†’ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘’ ∈ 𝑋 ((absβ€˜(𝑒 βˆ’ 𝑉)) < 𝑑 β†’ (absβ€˜(((π‘₯ ∈ 𝑋 ↦ (𝐴 Β· 𝐡))β€˜π‘’) βˆ’ ((π‘₯ ∈ 𝑋 ↦ (𝐴 Β· 𝐡))β€˜π‘‰))) < 𝐸))
 
26-May-2023cdivcncfap 14248 Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 26-May-2023.)
𝐹 = (π‘₯ ∈ {𝑦 ∈ β„‚ ∣ 𝑦 # 0} ↦ (𝐴 / π‘₯))    β‡’   (𝐴 ∈ β„‚ β†’ 𝐹 ∈ ({𝑦 ∈ β„‚ ∣ 𝑦 # 0}–cnβ†’β„‚))
 
26-May-2023reccn2ap 11324 The reciprocal function is continuous. The class 𝑇 is just for convenience in writing the proof and typically would be passed in as an instance of eqid 2177. (Contributed by Mario Carneiro, 9-Feb-2014.) Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
𝑇 = (inf({1, ((absβ€˜π΄) Β· 𝐡)}, ℝ, < ) Β· ((absβ€˜π΄) / 2))    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐴 # 0 ∧ 𝐡 ∈ ℝ+) β†’ βˆƒπ‘¦ ∈ ℝ+ βˆ€π‘§ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ((absβ€˜(𝑧 βˆ’ 𝐴)) < 𝑦 β†’ (absβ€˜((1 / 𝑧) βˆ’ (1 / 𝐴))) < 𝐡))
 
23-May-2023iooretopg 14189 Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.) (Revised by Jim Kingdon, 23-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴(,)𝐡) ∈ (topGenβ€˜ran (,)))
 
23-May-2023minclpr 11248 The minimum of two real numbers is one of those numbers if and only if dichotomy (𝐴 ≀ 𝐡 ∨ 𝐡 ≀ 𝐴) holds. For example, this can be combined with zletric 9300 if one is dealing with integers, but real number dichotomy in general does not follow from our axioms. (Contributed by Jim Kingdon, 23-May-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (inf({𝐴, 𝐡}, ℝ, < ) ∈ {𝐴, 𝐡} ↔ (𝐴 ≀ 𝐡 ∨ 𝐡 ≀ 𝐴)))
 
22-May-2023qtopbasss 14182 The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Jim Kingdon, 22-May-2023.)
𝑆 βŠ† ℝ*    &   ((π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) β†’ sup({π‘₯, 𝑦}, ℝ*, < ) ∈ 𝑆)    &   ((π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) β†’ inf({π‘₯, 𝑦}, ℝ*, < ) ∈ 𝑆)    β‡’   ((,) β€œ (𝑆 Γ— 𝑆)) ∈ TopBases
 
22-May-2023iooinsup 11288 Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) ∧ (𝐢 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) β†’ ((𝐴(,)𝐡) ∩ (𝐢(,)𝐷)) = (sup({𝐴, 𝐢}, ℝ*, < )(,)inf({𝐡, 𝐷}, ℝ*, < )))
 
21-May-2023df-sumdc 11365 Define the sum of a series with an index set of integers 𝐴. The variable π‘˜ is normally a free variable in 𝐡, i.e., 𝐡 can be thought of as 𝐡(π‘˜). This definition is the result of a collection of discussions over the most general definition for a sum that does not need the index set to have a specified ordering. This definition is in two parts, one for finite sums and one for subsets of the upper integers. When summing over a subset of the upper integers, we extend the index set to the upper integers by adding zero outside the domain, and then sum the set in order, setting the result to the limit of the partial sums, if it exists. This means that conditionally convergent sums can be evaluated meaningfully. For finite sums, we are explicitly order-independent, by picking any bijection to a 1-based finite sequence and summing in the induced order. In both cases we have an if expression so that we only need 𝐡 to be defined where π‘˜ ∈ 𝐴. In the infinite case, we also require that the indexing set be a decidable subset of an upperset of integers (that is, membership of integers in it is decidable). These two methods of summation produce the same result on their common region of definition (i.e., finite sets of integers). Examples: Ξ£π‘˜ ∈ {1, 2, 4}π‘˜ means 1 + 2 + 4 = 7, and Ξ£π‘˜ ∈ β„•(1 / (2β†‘π‘˜)) = 1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 11533). (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 21-May-2023.)
Ξ£π‘˜ ∈ 𝐴 𝐡 = (β„©π‘₯(βˆƒπ‘š ∈ β„€ (𝐴 βŠ† (β„€β‰₯β€˜π‘š) ∧ βˆ€π‘— ∈ (β„€β‰₯β€˜π‘š)DECID 𝑗 ∈ 𝐴 ∧ seqπ‘š( + , (𝑛 ∈ β„€ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / π‘˜β¦Œπ΅, 0))) ⇝ π‘₯) ∨ βˆƒπ‘š ∈ β„• βˆƒπ‘“(𝑓:(1...π‘š)–1-1-onto→𝐴 ∧ π‘₯ = (seq1( + , (𝑛 ∈ β„• ↦ if(𝑛 ≀ π‘š, ⦋(π‘“β€˜π‘›) / π‘˜β¦Œπ΅, 0)))β€˜π‘š))))
 
19-May-2023bdmopn 14165 The standard bounded metric corresponding to 𝐢 generates the same topology as 𝐢. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(π‘₯𝐢𝑦), 𝑅}, ℝ*, < ))    &   π½ = (MetOpenβ€˜πΆ)    β‡’   ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) β†’ 𝐽 = (MetOpenβ€˜π·))
 
19-May-2023bdbl 14164 The standard bounded metric corresponding to 𝐢 generates the same balls as 𝐢 for radii less than 𝑅. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(π‘₯𝐢𝑦), 𝑅}, ℝ*, < ))    β‡’   (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≀ 𝑅)) β†’ (𝑃(ballβ€˜π·)𝑆) = (𝑃(ballβ€˜πΆ)𝑆))
 
19-May-2023bdmet 14163 The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(π‘₯𝐢𝑦), 𝑅}, ℝ*, < ))    β‡’   ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑅 ∈ ℝ+) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
 
19-May-2023xrminltinf 11283 Two ways of saying an extended real is greater than the minimum of two others. (Contributed by Jim Kingdon, 19-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ (inf({𝐡, 𝐢}, ℝ*, < ) < 𝐴 ↔ (𝐡 < 𝐴 ∨ 𝐢 < 𝐴)))
 
19-May-2023clel5 2876 Alternate definition of class membership: a class 𝑋 is an element of another class 𝐴 iff there is an element of 𝐴 equal to 𝑋. (Contributed by AV, 13-Nov-2020.) (Revised by Steven Nguyen, 19-May-2023.)
(𝑋 ∈ 𝐴 ↔ βˆƒπ‘₯ ∈ 𝐴 𝑋 = π‘₯)
 
18-May-2023xrminrecl 11284 The minimum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 18-May-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ inf({𝐴, 𝐡}, ℝ*, < ) = inf({𝐴, 𝐡}, ℝ, < ))
 
18-May-2023ralnex2 2616 Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 18-May-2023.)
(βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 Β¬ πœ‘ ↔ Β¬ βˆƒπ‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘)
 
17-May-2023bdtrilem 11250 Lemma for bdtri 11251. (Contributed by Steven Nguyen and Jim Kingdon, 17-May-2023.)
(((𝐴 ∈ ℝ ∧ 0 ≀ 𝐴) ∧ (𝐡 ∈ ℝ ∧ 0 ≀ 𝐡) ∧ 𝐢 ∈ ℝ+) β†’ ((absβ€˜(𝐴 βˆ’ 𝐢)) + (absβ€˜(𝐡 βˆ’ 𝐢))) ≀ (𝐢 + (absβ€˜((𝐴 + 𝐡) βˆ’ 𝐢))))
 
15-May-2023xrbdtri 11287 Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
(((𝐴 ∈ ℝ* ∧ 0 ≀ 𝐴) ∧ (𝐡 ∈ ℝ* ∧ 0 ≀ 𝐡) ∧ (𝐢 ∈ ℝ* ∧ 0 < 𝐢)) β†’ inf({(𝐴 +𝑒 𝐡), 𝐢}, ℝ*, < ) ≀ (inf({𝐴, 𝐢}, ℝ*, < ) +𝑒 inf({𝐡, 𝐢}, ℝ*, < )))
 
15-May-2023bdtri 11251 Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
(((𝐴 ∈ ℝ ∧ 0 ≀ 𝐴) ∧ (𝐡 ∈ ℝ ∧ 0 ≀ 𝐡) ∧ 𝐢 ∈ ℝ+) β†’ inf({(𝐴 + 𝐡), 𝐢}, ℝ, < ) ≀ (inf({𝐴, 𝐢}, ℝ, < ) + inf({𝐡, 𝐢}, ℝ, < )))
 
15-May-2023minabs 11247 The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ inf({𝐴, 𝐡}, ℝ, < ) = (((𝐴 + 𝐡) βˆ’ (absβ€˜(𝐴 βˆ’ 𝐡))) / 2))
 
11-May-2023xrmaxadd 11272 Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ sup({(𝐴 +𝑒 𝐡), (𝐴 +𝑒 𝐢)}, ℝ*, < ) = (𝐴 +𝑒 sup({𝐡, 𝐢}, ℝ*, < )))
 
11-May-2023xrmaxaddlem 11271 Lemma for xrmaxadd 11272. The case where 𝐴 is real. (Contributed by Jim Kingdon, 11-May-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ sup({(𝐴 +𝑒 𝐡), (𝐴 +𝑒 𝐢)}, ℝ*, < ) = (𝐴 +𝑒 sup({𝐡, 𝐢}, ℝ*, < )))
 
10-May-2023xrminadd 11286 Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ inf({(𝐴 +𝑒 𝐡), (𝐴 +𝑒 𝐢)}, ℝ*, < ) = (𝐴 +𝑒 inf({𝐡, 𝐢}, ℝ*, < )))
 
10-May-2023xrmaxlesup 11270 Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 10-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ (sup({𝐴, 𝐡}, ℝ*, < ) ≀ 𝐢 ↔ (𝐴 ≀ 𝐢 ∧ 𝐡 ≀ 𝐢)))
 
10-May-2023xrltmaxsup 11268 The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ (𝐢 < sup({𝐴, 𝐡}, ℝ*, < ) ↔ (𝐢 < 𝐴 ∨ 𝐢 < 𝐡)))
 
9-May-2023bdxmet 14162 The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(π‘₯𝐢𝑦), 𝑅}, ℝ*, < ))    β‡’   ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
 
9-May-2023bdmetval 14161 Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(π‘₯𝐢𝑦), 𝑅}, ℝ*, < ))    β‡’   (((𝐢:(𝑋 Γ— 𝑋)βŸΆβ„* ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐷𝐡) = inf({(𝐴𝐢𝐡), 𝑅}, ℝ*, < ))
 
7-May-2023setsmstsetg 14142 The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Revised by Jim Kingdon, 7-May-2023.)
(πœ‘ β†’ 𝑋 = (Baseβ€˜π‘€))    &   (πœ‘ β†’ 𝐷 = ((distβ€˜π‘€) β†Ύ (𝑋 Γ— 𝑋)))    &   (πœ‘ β†’ 𝐾 = (𝑀 sSet ⟨(TopSetβ€˜ndx), (MetOpenβ€˜π·)⟩))    &   (πœ‘ β†’ 𝑀 ∈ 𝑉)    &   (πœ‘ β†’ (MetOpenβ€˜π·) ∈ π‘Š)    β‡’   (πœ‘ β†’ (MetOpenβ€˜π·) = (TopSetβ€˜πΎ))
 
6-May-2023dsslid 12674 Slot property of dist. (Contributed by Jim Kingdon, 6-May-2023.)
(dist = Slot (distβ€˜ndx) ∧ (distβ€˜ndx) ∈ β„•)
 
5-May-2023mopnrel 14102 The class of open sets of a metric space is a relation. (Contributed by Jim Kingdon, 5-May-2023.)
Rel MetOpen
 
5-May-2023fsumsersdc 11406 Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Jim Kingdon, 5-May-2023.)
((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) = if(π‘˜ ∈ 𝐴, 𝐡, 0))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ DECID π‘˜ ∈ 𝐴)    &   (πœ‘ β†’ 𝐴 βŠ† (𝑀...𝑁))    β‡’   (πœ‘ β†’ Ξ£π‘˜ ∈ 𝐴 𝐡 = (seq𝑀( + , 𝐹)β€˜π‘))
 
4-May-2023blex 14048 A ball is a set. (Contributed by Jim Kingdon, 4-May-2023.)
(𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (ballβ€˜π·) ∈ V)
 
4-May-2023summodc 11394 A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 0))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   πΊ = (𝑛 ∈ β„• ↦ if(𝑛 ≀ (β™―β€˜π΄), ⦋(π‘“β€˜π‘›) / π‘˜β¦Œπ΅, 0))    &   πΊ = (𝑛 ∈ β„• ↦ if(𝑛 ≀ (β™―β€˜π΄), ⦋(π‘“β€˜π‘›) / π‘˜β¦Œπ΅, 0))    β‡’   (πœ‘ β†’ βˆƒ*π‘₯(βˆƒπ‘š ∈ β„€ (𝐴 βŠ† (β„€β‰₯β€˜π‘š) ∧ βˆ€π‘— ∈ (β„€β‰₯β€˜π‘š)DECID 𝑗 ∈ 𝐴 ∧ seqπ‘š( + , 𝐹) ⇝ π‘₯) ∨ βˆƒπ‘š ∈ β„• βˆƒπ‘“(𝑓:(1...π‘š)–1-1-onto→𝐴 ∧ π‘₯ = (seq1( + , 𝐺)β€˜π‘š))))
 
4-May-2023summodclem2 11393 Lemma for summodc 11394. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 0))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   πΊ = (𝑛 ∈ β„• ↦ if(𝑛 ≀ (β™―β€˜π΄), ⦋(π‘“β€˜π‘›) / π‘˜β¦Œπ΅, 0))    β‡’   ((πœ‘ ∧ βˆƒπ‘š ∈ β„€ (𝐴 βŠ† (β„€β‰₯β€˜π‘š) ∧ βˆ€π‘— ∈ (β„€β‰₯β€˜π‘š)DECID 𝑗 ∈ 𝐴 ∧ seqπ‘š( + , 𝐹) ⇝ π‘₯)) β†’ (βˆƒπ‘š ∈ β„• βˆƒπ‘“(𝑓:(1...π‘š)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)β€˜π‘š)) β†’ π‘₯ = 𝑦))
 
4-May-2023xrminrpcl 11285 The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.)
((𝐴 ∈ ℝ+ ∧ 𝐡 ∈ ℝ+) β†’ inf({𝐴, 𝐡}, ℝ*, < ) ∈ ℝ+)
 
4-May-2023xrlemininf 11282 Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 4-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ (𝐴 ≀ inf({𝐡, 𝐢}, ℝ*, < ) ↔ (𝐴 ≀ 𝐡 ∧ 𝐴 ≀ 𝐢)))
 
3-May-2023xrltmininf 11281 Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 3-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ (𝐴 < inf({𝐡, 𝐢}, ℝ*, < ) ↔ (𝐴 < 𝐡 ∧ 𝐴 < 𝐢)))
 
3-May-2023xrmineqinf 11280 The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) (Revised by Jim Kingdon, 3-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐡 ≀ 𝐴) β†’ inf({𝐴, 𝐡}, ℝ*, < ) = 𝐡)
 
3-May-2023xrmin2inf 11279 The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ inf({𝐴, 𝐡}, ℝ*, < ) ≀ 𝐡)
 
3-May-2023xrmin1inf 11278 The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ inf({𝐴, 𝐡}, ℝ*, < ) ≀ 𝐴)
 
3-May-2023xrmincl 11277 The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ inf({𝐴, 𝐡}, ℝ*, < ) ∈ ℝ*)
 
2-May-2023xrminmax 11276 Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ inf({𝐴, 𝐡}, ℝ*, < ) = -𝑒sup({-𝑒𝐴, -𝑒𝐡}, ℝ*, < ))
 
2-May-2023xrnegcon1d 11275 Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    β‡’   (πœ‘ β†’ (-𝑒𝐴 = 𝐡 ↔ -𝑒𝐡 = 𝐴))
 
2-May-2023infxrnegsupex 11274 The infimum of a set of extended reals 𝐴 is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 2-May-2023.)
(πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ* (βˆ€π‘¦ ∈ 𝐴 Β¬ 𝑦 < π‘₯ ∧ βˆ€π‘¦ ∈ ℝ* (π‘₯ < 𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 𝑧 < 𝑦)))    &   (πœ‘ β†’ 𝐴 βŠ† ℝ*)    β‡’   (πœ‘ β†’ inf(𝐴, ℝ*, < ) = -𝑒sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧 ∈ 𝐴}, ℝ*, < ))
 
2-May-2023xrnegiso 11273 Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
𝐹 = (π‘₯ ∈ ℝ* ↦ -𝑒π‘₯)    β‡’   (𝐹 Isom < , β—‘ < (ℝ*, ℝ*) ∧ ◑𝐹 = 𝐹)
 
30-Apr-2023xrmaxltsup 11269 Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ (sup({𝐴, 𝐡}, ℝ*, < ) < 𝐢 ↔ (𝐴 < 𝐢 ∧ 𝐡 < 𝐢)))
 
30-Apr-2023xrmaxrecl 11266 The maximum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 30-Apr-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ sup({𝐴, 𝐡}, ℝ*, < ) = sup({𝐴, 𝐡}, ℝ, < ))
 
30-Apr-2023xrmax2sup 11265 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 30-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ 𝐡 ≀ sup({𝐴, 𝐡}, ℝ*, < ))
 
30-Apr-2023xrmax1sup 11264 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 30-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ 𝐴 ≀ sup({𝐴, 𝐡}, ℝ*, < ))
 
29-Apr-2023xrmaxcl 11263 The maximum of two extended reals is an extended real. (Contributed by Jim Kingdon, 29-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ sup({𝐴, 𝐡}, ℝ*, < ) ∈ ℝ*)
 
29-Apr-2023xrmaxiflemval 11261 Lemma for xrmaxif 11262. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
𝑀 = if(𝐡 = +∞, +∞, if(𝐡 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐡, sup({𝐴, 𝐡}, ℝ, < )))))    β‡’   ((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝑀 ∈ ℝ* ∧ βˆ€π‘₯ ∈ {𝐴, 𝐡} Β¬ 𝑀 < π‘₯ ∧ βˆ€π‘₯ ∈ ℝ* (π‘₯ < 𝑀 β†’ βˆƒπ‘§ ∈ {𝐴, 𝐡}π‘₯ < 𝑧)))
 
29-Apr-2023xrmaxiflemcom 11260 Lemma for xrmaxif 11262. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ if(𝐡 = +∞, +∞, if(𝐡 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐡, sup({𝐴, 𝐡}, ℝ, < ))))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐡, if(𝐡 = +∞, +∞, if(𝐡 = -∞, 𝐴, sup({𝐡, 𝐴}, ℝ, < ))))))
 
29-Apr-2023xrmaxiflemcl 11256 Lemma for xrmaxif 11262. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ if(𝐡 = +∞, +∞, if(𝐡 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐡, sup({𝐴, 𝐡}, ℝ, < ))))) ∈ ℝ*)
 
29-Apr-2023sbco2v 1948 Version of sbco2 1965 with disjoint variable conditions. (Contributed by Wolf Lammen, 29-Apr-2023.)
β„²π‘§πœ‘    β‡’   ([𝑦 / 𝑧][𝑧 / π‘₯]πœ‘ ↔ [𝑦 / π‘₯]πœ‘)
 
28-Apr-2023xrmaxiflemlub 11259 Lemma for xrmaxif 11262. A least upper bound for {𝐴, 𝐡}. (Contributed by Jim Kingdon, 28-Apr-2023.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 < if(𝐡 = +∞, +∞, if(𝐡 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐡, sup({𝐴, 𝐡}, ℝ, < ))))))    β‡’   (πœ‘ β†’ (𝐢 < 𝐴 ∨ 𝐢 < 𝐡))
 
26-Apr-2023xrmaxif 11262 Maximum of two extended reals in terms of if expressions. (Contributed by Jim Kingdon, 26-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ sup({𝐴, 𝐡}, ℝ*, < ) = if(𝐡 = +∞, +∞, if(𝐡 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐡, sup({𝐴, 𝐡}, ℝ, < ))))))
 
26-Apr-2023xrmaxiflemab 11258 Lemma for xrmaxif 11262. A variation of xrmaxleim 11255- that is, if we know which of two real numbers is larger, we know the maximum of the two. (Contributed by Jim Kingdon, 26-Apr-2023.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 < 𝐡)    β‡’   (πœ‘ β†’ if(𝐡 = +∞, +∞, if(𝐡 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐡, sup({𝐴, 𝐡}, ℝ, < ))))) = 𝐡)
 
26-Apr-2023xrmaxifle 11257 An upper bound for {𝐴, 𝐡} in the extended reals. (Contributed by Jim Kingdon, 26-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ 𝐴 ≀ if(𝐡 = +∞, +∞, if(𝐡 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐡, sup({𝐴, 𝐡}, ℝ, < ))))))
 
25-Apr-2023xrmaxleim 11255 Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴 ≀ 𝐡 β†’ sup({𝐴, 𝐡}, ℝ*, < ) = 𝐡))
 
25-Apr-2023rpmincl 11249 The minumum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
((𝐴 ∈ ℝ+ ∧ 𝐡 ∈ ℝ+) β†’ inf({𝐴, 𝐡}, ℝ, < ) ∈ ℝ+)
 
25-Apr-2023mincl 11242 The minumum of two real numbers is a real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ inf({𝐴, 𝐡}, ℝ, < ) ∈ ℝ)
 
24-Apr-2023psmetrel 13983 The class of pseudometrics is a relation. (Contributed by Jim Kingdon, 24-Apr-2023.)
Rel PsMet
 
23-Apr-2023bcval5 10746 Write out the top and bottom parts of the binomial coefficient (𝑁C𝐾) = (𝑁 Β· (𝑁 βˆ’ 1) Β· ... Β· ((𝑁 βˆ’ 𝐾) + 1)) / 𝐾! explicitly. In this form, it is valid even for 𝑁 < 𝐾, although it is no longer valid for nonpositive 𝐾. (Contributed by Mario Carneiro, 22-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
((𝑁 ∈ β„•0 ∧ 𝐾 ∈ β„•) β†’ (𝑁C𝐾) = ((seq((𝑁 βˆ’ 𝐾) + 1)( Β· , I )β€˜π‘) / (!β€˜πΎ)))
 
23-Apr-2023ser3le 10521 Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΊβ€˜π‘˜) ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) ≀ (πΊβ€˜π‘˜))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘) ≀ (seq𝑀( + , 𝐺)β€˜π‘))
 
23-Apr-2023seq3z 10514 If the operation + has an absorbing element 𝑍 (a.k.a. zero element), then any sequence containing a 𝑍 evaluates to 𝑍. (Contributed by Mario Carneiro, 27-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘₯) ∈ 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑆) β†’ (𝑍 + π‘₯) = 𝑍)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑆) β†’ (π‘₯ + 𝑍) = 𝑍)    &   (πœ‘ β†’ 𝐾 ∈ (𝑀...𝑁))    &   (πœ‘ β†’ (πΉβ€˜πΎ) = 𝑍)    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘) = 𝑍)
 
23-Apr-2023seq3caopr 10486 The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) = (𝑦 + π‘₯))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) β†’ ((π‘₯ + 𝑦) + 𝑧) = (π‘₯ + (𝑦 + 𝑧)))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) ∈ 𝑆)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΊβ€˜π‘˜) ∈ 𝑆)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (π»β€˜π‘˜) = ((πΉβ€˜π‘˜) + (πΊβ€˜π‘˜)))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐻)β€˜π‘) = ((seq𝑀( + , 𝐹)β€˜π‘) + (seq𝑀( + , 𝐺)β€˜π‘)))
 
23-Apr-2023seq3caopr2 10485 The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯𝑄𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ ((π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑀 ∈ 𝑆))) β†’ ((π‘₯𝑄𝑧) + (𝑦𝑄𝑀)) = ((π‘₯ + 𝑦)𝑄(𝑧 + 𝑀)))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) ∈ 𝑆)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΊβ€˜π‘˜) ∈ 𝑆)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (π»β€˜π‘˜) = ((πΉβ€˜π‘˜)𝑄(πΊβ€˜π‘˜)))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐻)β€˜π‘) = ((seq𝑀( + , 𝐹)β€˜π‘)𝑄(seq𝑀( + , 𝐺)β€˜π‘)))
 
22-Apr-2023ser3sub 10509 The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΊβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (π»β€˜π‘˜) = ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐻)β€˜π‘) = ((seq𝑀( + , 𝐹)β€˜π‘) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘)))
 
22-Apr-2023seq3caopr3 10484 Lemma for seq3caopr2 10485. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯𝑄𝑦) ∈ 𝑆)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) ∈ 𝑆)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΊβ€˜π‘˜) ∈ 𝑆)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (π»β€˜π‘˜) = ((πΉβ€˜π‘˜)𝑄(πΊβ€˜π‘˜)))    &   ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁)) β†’ (((seq𝑀( + , 𝐹)β€˜π‘›)𝑄(seq𝑀( + , 𝐺)β€˜π‘›)) + ((πΉβ€˜(𝑛 + 1))𝑄(πΊβ€˜(𝑛 + 1)))) = (((seq𝑀( + , 𝐹)β€˜π‘›) + (πΉβ€˜(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)β€˜π‘›) + (πΊβ€˜(𝑛 + 1)))))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐻)β€˜π‘) = ((seq𝑀( + , 𝐹)β€˜π‘)𝑄(seq𝑀( + , 𝐺)β€˜π‘)))
 
22-Apr-2023ser3mono 10481 The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
(πœ‘ β†’ 𝐾 ∈ (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜πΎ))    &   ((πœ‘ ∧ π‘₯ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘₯) ∈ ℝ)    &   ((πœ‘ ∧ π‘₯ ∈ ((𝐾 + 1)...𝑁)) β†’ 0 ≀ (πΉβ€˜π‘₯))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜πΎ) ≀ (seq𝑀( + , 𝐹)β€˜π‘))
 
21-Apr-2023metrtri 14038 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 21-Apr-2023.)
((𝐷 ∈ (Metβ€˜π‘‹) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (absβ€˜((𝐴𝐷𝐢) βˆ’ (𝐡𝐷𝐢))) ≀ (𝐴𝐷𝐡))
 
21-Apr-2023sqxpeq0 5054 A Cartesian square is empty iff its member is empty. (Contributed by Jim Kingdon, 21-Apr-2023.)
((𝐴 Γ— 𝐴) = βˆ… ↔ 𝐴 = βˆ…)
 
20-Apr-2023xmetrel 14004 The class of extended metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
Rel ∞Met
 
20-Apr-2023metrel 14003 The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
Rel Met
 
19-Apr-2023psmetge0 13992 The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) (Revised by Jim Kingdon, 19-Apr-2023.)
((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ 0 ≀ (𝐴𝐷𝐡))
 
18-Apr-2023xleaddadd 9890 Cancelling a factor of two in ≀ (expressed as addition rather than as a factor to avoid extended real multiplication). (Contributed by Jim Kingdon, 18-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴 ≀ 𝐡 ↔ (𝐴 +𝑒 𝐴) ≀ (𝐡 +𝑒 𝐡)))
 
17-Apr-2023xposdif 9885 Extended real version of posdif 8415. (Contributed by Mario Carneiro, 24-Aug-2015.) (Revised by Jim Kingdon, 17-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴 < 𝐡 ↔ 0 < (𝐡 +𝑒 -𝑒𝐴)))
 
17-Apr-2023nmnfgt 9821 An extended real is greater than minus infinite iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
(𝐴 ∈ ℝ* β†’ (-∞ < 𝐴 ↔ 𝐴 β‰  -∞))
 
17-Apr-2023npnflt 9818 An extended real is less than plus infinity iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
(𝐴 ∈ ℝ* β†’ (𝐴 < +∞ ↔ 𝐴 β‰  +∞))
 
16-Apr-2023xltadd1 9879 Extended real version of ltadd1 8389. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Jim Kingdon, 16-Apr-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ) β†’ (𝐴 < 𝐡 ↔ (𝐴 +𝑒 𝐢) < (𝐡 +𝑒 𝐢)))
 
13-Apr-2023xrmnfdc 9846 An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
(𝐴 ∈ ℝ* β†’ DECID 𝐴 = -∞)
 
13-Apr-2023xrpnfdc 9845 An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
(𝐴 ∈ ℝ* β†’ DECID 𝐴 = +∞)
 
11-Apr-2023dmxpid 4850 The domain of a square Cartesian product. (Contributed by NM, 28-Jul-1995.) (Revised by Jim Kingdon, 11-Apr-2023.)
dom (𝐴 Γ— 𝐴) = 𝐴
 
9-Apr-2023isumz 11400 Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
(((𝑀 ∈ β„€ ∧ 𝐴 βŠ† (β„€β‰₯β€˜π‘€) ∧ βˆ€π‘— ∈ (β„€β‰₯β€˜π‘€)DECID 𝑗 ∈ 𝐴) ∨ 𝐴 ∈ Fin) β†’ Ξ£π‘˜ ∈ 𝐴 0 = 0)
 
9-Apr-2023summodclem2a 11392 Lemma for summodc 11394. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 0))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ DECID π‘˜ ∈ 𝐴)    &   πΊ = (𝑛 ∈ β„• ↦ if(𝑛 ≀ (β™―β€˜π΄), ⦋(π‘“β€˜π‘›) / π‘˜β¦Œπ΅, 0))    &   π» = (𝑛 ∈ β„• ↦ if(𝑛 ≀ 𝑁, ⦋(πΎβ€˜π‘›) / π‘˜β¦Œπ΅, 0))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐴 βŠ† (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ 𝑓:(1...𝑁)–1-1-onto→𝐴)    &   (πœ‘ β†’ 𝐾 Isom < , < ((1...(β™―β€˜π΄)), 𝐴))    β‡’   (πœ‘ β†’ seq𝑀( + , 𝐹) ⇝ (seq1( + , 𝐺)β€˜π‘))
 
9-Apr-2023summodclem3 11391 Lemma for summodc 11394. (Contributed by Mario Carneiro, 29-Mar-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 0))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ (𝑀 ∈ β„• ∧ 𝑁 ∈ β„•))    &   (πœ‘ β†’ 𝑓:(1...𝑀)–1-1-onto→𝐴)    &   (πœ‘ β†’ 𝐾:(1...𝑁)–1-1-onto→𝐴)    &   πΊ = (𝑛 ∈ β„• ↦ if(𝑛 ≀ 𝑀, ⦋(π‘“β€˜π‘›) / π‘˜β¦Œπ΅, 0))    &   π» = (𝑛 ∈ β„• ↦ if(𝑛 ≀ 𝑁, ⦋(πΎβ€˜π‘›) / π‘˜β¦Œπ΅, 0))    β‡’   (πœ‘ β†’ (seq1( + , 𝐺)β€˜π‘€) = (seq1( + , 𝐻)β€˜π‘))
 
9-Apr-2023sumrbdc 11390 Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 0))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝐴 βŠ† (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ 𝐴 βŠ† (β„€β‰₯β€˜π‘))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ DECID π‘˜ ∈ 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ DECID π‘˜ ∈ 𝐴)    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹) ⇝ 𝐢 ↔ seq𝑁( + , 𝐹) ⇝ 𝐢))
 
9-Apr-2023seq3coll 10825 The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 2-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
((πœ‘ ∧ π‘˜ ∈ 𝑆) β†’ (𝑍 + π‘˜) = π‘˜)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑆) β†’ (π‘˜ + 𝑍) = π‘˜)    &   ((πœ‘ ∧ (π‘˜ ∈ 𝑆 ∧ 𝑛 ∈ 𝑆)) β†’ (π‘˜ + 𝑛) ∈ 𝑆)    &   (πœ‘ β†’ 𝑍 ∈ 𝑆)    &   (πœ‘ β†’ 𝐺 Isom < , < ((1...(β™―β€˜π΄)), 𝐴))    &   (πœ‘ β†’ 𝑁 ∈ (1...(β™―β€˜π΄)))    &   (πœ‘ β†’ 𝐴 βŠ† (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) ∈ 𝑆)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜1)) β†’ (π»β€˜π‘˜) ∈ 𝑆)    &   ((πœ‘ ∧ π‘˜ ∈ ((𝑀...(πΊβ€˜(β™―β€˜π΄))) βˆ– 𝐴)) β†’ (πΉβ€˜π‘˜) = 𝑍)    &   ((πœ‘ ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ (π»β€˜π‘›) = (πΉβ€˜(πΊβ€˜π‘›)))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜(πΊβ€˜π‘)) = (seq1( + , 𝐻)β€˜π‘))
 
8-Apr-2023zsumdc 11395 Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.) (Revised by Jim Kingdon, 8-Apr-2023.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑍)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) = if(π‘˜ ∈ 𝐴, 𝐡, 0))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑍 DECID π‘₯ ∈ 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ Ξ£π‘˜ ∈ 𝐴 𝐡 = ( ⇝ β€˜seq𝑀( + , 𝐹)))
 
8-Apr-2023sumrbdclem 11388 Lemma for sumrbdc 11390. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 8-Apr-2023.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 0))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ DECID π‘˜ ∈ 𝐴)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    β‡’   ((πœ‘ ∧ 𝐴 βŠ† (β„€β‰₯β€˜π‘)) β†’ (seq𝑀( + , 𝐹) β†Ύ (β„€β‰₯β€˜π‘)) = seq𝑁( + , 𝐹))
 
8-Apr-2023isermulc2 11351 Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Jim Kingdon, 8-Apr-2023.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ seq𝑀( + , 𝐹) ⇝ 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΊβ€˜π‘˜) = (𝐢 Β· (πΉβ€˜π‘˜)))    β‡’   (πœ‘ β†’ seq𝑀( + , 𝐺) ⇝ (𝐢 Β· 𝐴))
 
8-Apr-2023seq3id 10511 Discarding the first few terms of a sequence that starts with all zeroes (or any element which is a left-identity for +) has no effect on its sum. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 8-Apr-2023.)
((πœ‘ ∧ π‘₯ ∈ 𝑆) β†’ (𝑍 + π‘₯) = π‘₯)    &   (πœ‘ β†’ 𝑍 ∈ 𝑆)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ (πΉβ€˜π‘) ∈ 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜π‘₯) = 𝑍)    &   ((πœ‘ ∧ π‘₯ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘₯) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹) β†Ύ (β„€β‰₯β€˜π‘)) = seq𝑁( + , 𝐹))
 
8-Apr-2023seq3id3 10510 A sequence that consists entirely of "zeroes" sums to "zero". More precisely, a constant sequence with value an element which is a + -idempotent sums (or "+'s") to that element. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Jim Kingdon, 8-Apr-2023.)
(πœ‘ β†’ (𝑍 + 𝑍) = 𝑍)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘₯ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘₯) = 𝑍)    &   (πœ‘ β†’ 𝑍 ∈ 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘₯) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘) = 𝑍)
 
7-Apr-2023seq3shft2 10476 Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ 𝐾 ∈ β„€)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) = (πΊβ€˜(π‘˜ + 𝐾)))    &   ((πœ‘ ∧ π‘₯ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘₯) ∈ 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ (β„€β‰₯β€˜(𝑀 + 𝐾))) β†’ (πΊβ€˜π‘₯) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘) = (seq(𝑀 + 𝐾)( + , 𝐺)β€˜(𝑁 + 𝐾)))
 
7-Apr-2023seq3feq 10475 Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   ((πœ‘ ∧ π‘₯ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘₯) ∈ 𝑆)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) = (πΊβ€˜π‘˜))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    β‡’   (πœ‘ β†’ seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))
 
7-Apr-2023r19.2m 3511 Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1638). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.)
((βˆƒπ‘¦ 𝑦 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 πœ‘) β†’ βˆƒπ‘₯ ∈ 𝐴 πœ‘)
 
6-Apr-2023lmtopcnp 13911 The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)
(πœ‘ β†’ 𝐹(β‡π‘‘β€˜π½)𝑃)    &   (πœ‘ β†’ 𝐾 ∈ Top)    &   (πœ‘ β†’ 𝐺 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))    β‡’   (πœ‘ β†’ (𝐺 ∘ 𝐹)(β‡π‘‘β€˜πΎ)(πΊβ€˜π‘ƒ))
 
6-Apr-2023cnptoprest2 13901 Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)
𝑋 = βˆͺ 𝐽    &   π‘Œ = βˆͺ 𝐾    β‡’   (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:π‘‹βŸΆπ΅ ∧ 𝐡 βŠ† π‘Œ)) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾 β†Ύt 𝐡))β€˜π‘ƒ)))
 
5-Apr-2023cnptoprest 13900 Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 5-Apr-2023.)
𝑋 = βˆͺ 𝐽    &   π‘Œ = βˆͺ 𝐾    β‡’   (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 βŠ† 𝑋) ∧ (𝑃 ∈ ((intβ€˜π½)β€˜π΄) ∧ 𝐹:π‘‹βŸΆπ‘Œ)) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ↔ (𝐹 β†Ύ 𝐴) ∈ (((𝐽 β†Ύt 𝐴) CnP 𝐾)β€˜π‘ƒ)))
 
4-Apr-2023exmidmp 7158 Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.)
(EXMID β†’ Ο‰ ∈ Markov)
 
2-Apr-2023sup3exmid 8917 If any inhabited set of real numbers bounded from above has a supremum, excluded middle follows. (Contributed by Jim Kingdon, 2-Apr-2023.)
((𝑒 βŠ† ℝ ∧ βˆƒπ‘€ 𝑀 ∈ 𝑒 ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝑒 𝑦 ≀ π‘₯) β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝑒 Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ 𝑒 𝑦 < 𝑧)))    β‡’   DECID πœ‘
 
31-Mar-2023cnptopresti 13899 One direction of cnptoprest 13900 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 31-Mar-2023.)
(((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ (𝐹 β†Ύ 𝐴) ∈ (((𝐽 β†Ύt 𝐴) CnP 𝐾)β€˜π‘ƒ))
 
30-Mar-2023cncnp2m 13892 A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Jim Kingdon, 30-Mar-2023.)
𝑋 = βˆͺ 𝐽    &   π‘Œ = βˆͺ 𝐾    β‡’   ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ βˆƒπ‘¦ 𝑦 ∈ 𝑋) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ βˆ€π‘₯ ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘₯)))
 
29-Mar-2023exmidlpo 7144 Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.)
(EXMID β†’ Ο‰ ∈ Omni)
 
28-Mar-2023icnpimaex 13872 Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.) (Revised by Jim Kingdon, 28-Mar-2023.)
(((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝐴 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝐴)) β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝐴))
 
28-Mar-2023cnpf2 13868 A continuous function at point 𝑃 is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
 
28-Mar-2023cnprcl2k 13867 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝑃 ∈ 𝑋)
 
27-Mar-2023mptrcl 5601 Reverse closure for a mapping: If the function value of a mapping has a member, the argument belongs to the base class of the mapping. (Contributed by AV, 4-Apr-2020.) (Revised by Jim Kingdon, 27-Mar-2023.)
𝐹 = (π‘₯ ∈ 𝐴 ↦ 𝐡)    β‡’   (𝐼 ∈ (πΉβ€˜π‘‹) β†’ 𝑋 ∈ 𝐴)
 
25-Mar-2023lmreltop 13854 The topological space convergence relation is a relation. (Contributed by Jim Kingdon, 25-Mar-2023.)
(𝐽 ∈ Top β†’ Rel (β‡π‘‘β€˜π½))
 
25-Mar-2023fodjumkv 7161 A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.)
(πœ‘ β†’ 𝑀 ∈ Markov)    &   (πœ‘ β†’ 𝐹:𝑀–ontoβ†’(𝐴 βŠ” 𝐡))    β‡’   (πœ‘ β†’ (𝐴 β‰  βˆ… β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴))
 
25-Mar-2023fodjumkvlemres 7160 Lemma for fodjumkv 7161. The final result with 𝑃 expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.)
(πœ‘ β†’ 𝑀 ∈ Markov)    &   (πœ‘ β†’ 𝐹:𝑀–ontoβ†’(𝐴 βŠ” 𝐡))    &   π‘ƒ = (𝑦 ∈ 𝑀 ↦ if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘¦) = (inlβ€˜π‘§), βˆ…, 1o))    β‡’   (πœ‘ β†’ (𝐴 β‰  βˆ… β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴))
 
25-Mar-2023fodju0 7148 Lemma for fodjuomni 7150 and fodjumkv 7161. A condition which shows that 𝐴 is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
(πœ‘ β†’ 𝐹:𝑂–ontoβ†’(𝐴 βŠ” 𝐡))    &   π‘ƒ = (𝑦 ∈ 𝑂 ↦ if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘¦) = (inlβ€˜π‘§), βˆ…, 1o))    &   (πœ‘ β†’ βˆ€π‘€ ∈ 𝑂 (π‘ƒβ€˜π‘€) = 1o)    β‡’   (πœ‘ β†’ 𝐴 = βˆ…)
 
25-Mar-2023fodjum 7147 Lemma for fodjuomni 7150 and fodjumkv 7161. A condition which shows that 𝐴 is inhabited. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
(πœ‘ β†’ 𝐹:𝑂–ontoβ†’(𝐴 βŠ” 𝐡))    &   π‘ƒ = (𝑦 ∈ 𝑂 ↦ if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘¦) = (inlβ€˜π‘§), βˆ…, 1o))    &   (πœ‘ β†’ βˆƒπ‘€ ∈ 𝑂 (π‘ƒβ€˜π‘€) = βˆ…)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)
 
25-Mar-2023fodjuf 7146 Lemma for fodjuomni 7150 and fodjumkv 7161. Domain and range of 𝑃. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
(πœ‘ β†’ 𝐹:𝑂–ontoβ†’(𝐴 βŠ” 𝐡))    &   π‘ƒ = (𝑦 ∈ 𝑂 ↦ if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘¦) = (inlβ€˜π‘§), βˆ…, 1o))    &   (πœ‘ β†’ 𝑂 ∈ 𝑉)    β‡’   (πœ‘ β†’ 𝑃 ∈ (2o β†‘π‘š 𝑂))
 
23-Mar-2023restrcl 13828 Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Jim Kingdon, 23-Mar-2023.)
((𝐽 β†Ύt 𝐴) ∈ Top β†’ (𝐽 ∈ V ∧ 𝐴 ∈ V))
 
22-Mar-2023neipsm 13815 A neighborhood of a set is a neighborhood of every point in the set. Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.) (Revised by Jim Kingdon, 22-Mar-2023.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ βˆƒπ‘₯ π‘₯ ∈ 𝑆) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝})))
 
19-Mar-2023mkvprop 7159 Markov's Principle expressed in terms of propositions (or more precisely, the 𝐴 = Ο‰ case is Markov's Principle). (Contributed by Jim Kingdon, 19-Mar-2023.)
((𝐴 ∈ Markov ∧ βˆ€π‘› ∈ 𝐴 DECID πœ‘ ∧ Β¬ βˆ€π‘› ∈ 𝐴 Β¬ πœ‘) β†’ βˆƒπ‘› ∈ 𝐴 πœ‘)
 
18-Mar-2023omnimkv 7157 An omniscient set is Markov. In particular, the case where 𝐴 is Ο‰ means that the Limited Principle of Omniscience (LPO) implies Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.)
(𝐴 ∈ Omni β†’ 𝐴 ∈ Markov)
 
18-Mar-2023ismkvmap 7155 The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 18-Mar-2023.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Markov ↔ βˆ€π‘“ ∈ (2o β†‘π‘š 𝐴)(Β¬ βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1o β†’ βˆƒπ‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = βˆ…)))
 
18-Mar-2023ismkv 7154 The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Markov ↔ βˆ€π‘“(𝑓:𝐴⟢2o β†’ (Β¬ βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1o β†’ βˆƒπ‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = βˆ…))))
 
18-Mar-2023df-markov 7153 A Markov set is one where if a predicate (here represented by a function 𝑓) on that set does not hold (where hold means is equal to 1o) for all elements, then there exists an element where it fails (is equal to βˆ…). Generalization of definition 2.5 of [Pierik], p. 9.

In particular, Ο‰ ∈ Markov is known as Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.)

Markov = {𝑦 ∣ βˆ€π‘“(𝑓:π‘¦βŸΆ2o β†’ (Β¬ βˆ€π‘₯ ∈ 𝑦 (π‘“β€˜π‘₯) = 1o β†’ βˆƒπ‘₯ ∈ 𝑦 (π‘“β€˜π‘₯) = βˆ…))}
 
17-Mar-2023finct 7118 A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.)
(𝐴 ∈ Fin β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐴 βŠ” 1o))
 
16-Mar-2023ctmlemr 7110 Lemma for ctm 7111. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.)
(βˆƒπ‘₯ π‘₯ ∈ 𝐴 β†’ (βˆƒπ‘“ 𝑓:ω–onto→𝐴 β†’ βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o)))
 
15-Mar-2023caseinl 7093 Applying the "case" construction to a left injection. (Contributed by Jim Kingdon, 15-Mar-2023.)
(πœ‘ β†’ 𝐹 Fn 𝐡)    &   (πœ‘ β†’ Fun 𝐺)    &   (πœ‘ β†’ 𝐴 ∈ 𝐡)    β‡’   (πœ‘ β†’ (case(𝐹, 𝐺)β€˜(inlβ€˜π΄)) = (πΉβ€˜π΄))
 
13-Mar-2023enumct 7117 A finitely enumerable set is countable. Lemma 8.1.14 of [AczelRathjen], p. 73 (except that our definition of countable does not require the set to be inhabited). "Finitely enumerable" is defined as βˆƒπ‘› ∈ Ο‰βˆƒπ‘“π‘“:𝑛–onto→𝐴 per Definition 8.1.4 of [AczelRathjen], p. 71 and "countable" is defined as βˆƒπ‘”π‘”:ω–ontoβ†’(𝐴 βŠ” 1o) per [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
(βˆƒπ‘› ∈ Ο‰ βˆƒπ‘“ 𝑓:𝑛–onto→𝐴 β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐴 βŠ” 1o))
 
13-Mar-2023enumctlemm 7116 Lemma for enumct 7117. The case where 𝑁 is greater than zero. (Contributed by Jim Kingdon, 13-Mar-2023.)
(πœ‘ β†’ 𝐹:𝑁–onto→𝐴)    &   (πœ‘ β†’ 𝑁 ∈ Ο‰)    &   (πœ‘ β†’ βˆ… ∈ 𝑁)    &   πΊ = (π‘˜ ∈ Ο‰ ↦ if(π‘˜ ∈ 𝑁, (πΉβ€˜π‘˜), (πΉβ€˜βˆ…)))    β‡’   (πœ‘ β†’ 𝐺:ω–onto→𝐴)
 
13-Mar-2023ctm 7111 Two equivalent definitions of countable for an inhabited set. Remark of [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
(βˆƒπ‘₯ π‘₯ ∈ 𝐴 β†’ (βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o) ↔ βˆƒπ‘“ 𝑓:ω–onto→𝐴))
 
13-Mar-20230ct 7109 The empty set is countable. Remark of [BauerSwan], p. 14:3 which also has the definition of countable used here. (Contributed by Jim Kingdon, 13-Mar-2023.)
βˆƒπ‘“ 𝑓:ω–ontoβ†’(βˆ… βŠ” 1o)
 
13-Mar-2023ctex 6756 A class dominated by Ο‰ is a set. See also ctfoex 7120 which says that a countable class is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof shortened by Jim Kingdon, 13-Mar-2023.)
(𝐴 β‰Ό Ο‰ β†’ 𝐴 ∈ V)
 
12-Mar-2023cls0 13794 The closure of the empty set. (Contributed by NM, 2-Oct-2007.) (Proof shortened by Jim Kingdon, 12-Mar-2023.)
(𝐽 ∈ Top β†’ ((clsβ€˜π½)β€˜βˆ…) = βˆ…)
 
12-Mar-2023algrp1 12049 The value of the algorithm iterator 𝑅 at (𝐾 + 1). (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Jim Kingdon, 12-Mar-2023.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   π‘… = seq𝑀((𝐹 ∘ 1st ), (𝑍 Γ— {𝐴}))    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   (πœ‘ β†’ 𝐹:π‘†βŸΆπ‘†)    β‡’   ((πœ‘ ∧ 𝐾 ∈ 𝑍) β†’ (π‘…β€˜(𝐾 + 1)) = (πΉβ€˜(π‘…β€˜πΎ)))
 
12-Mar-2023ialgr0 12047 The value of the algorithm iterator 𝑅 at 0 is the initial state 𝐴. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Jim Kingdon, 12-Mar-2023.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   π‘… = seq𝑀((𝐹 ∘ 1st ), (𝑍 Γ— {𝐴}))    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   (πœ‘ β†’ 𝐹:π‘†βŸΆπ‘†)    β‡’   (πœ‘ β†’ (π‘…β€˜π‘€) = 𝐴)
 
11-Mar-2023ntreq0 13793 Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜π‘†) = βˆ… ↔ βˆ€π‘₯ ∈ 𝐽 (π‘₯ βŠ† 𝑆 β†’ π‘₯ = βˆ…)))
 
11-Mar-2023clstop 13788 The closure of a topology's underlying set is the entire set. (Contributed by NM, 5-Oct-2007.) (Proof shortened by Jim Kingdon, 11-Mar-2023.)
𝑋 = βˆͺ 𝐽    β‡’   (𝐽 ∈ Top β†’ ((clsβ€˜π½)β€˜π‘‹) = 𝑋)
 
11-Mar-2023ntrss 13780 Subset relationship for interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ ((intβ€˜π½)β€˜π‘‡) βŠ† ((intβ€˜π½)β€˜π‘†))
 
10-Mar-2023iuncld 13776 A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) (Revised by Jim Kingdon, 10-Mar-2023.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ (Clsdβ€˜π½)) β†’ βˆͺ π‘₯ ∈ 𝐴 𝐡 ∈ (Clsdβ€˜π½))
 
5-Mar-20232basgeng 13743 Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Jim Kingdon, 5-Mar-2023.)
((𝐡 ∈ 𝑉 ∧ 𝐡 βŠ† 𝐢 ∧ 𝐢 βŠ† (topGenβ€˜π΅)) β†’ (topGenβ€˜π΅) = (topGenβ€˜πΆ))
 
5-Mar-2023exmidsssn 4204 Excluded middle is equivalent to the biconditionalized version of sssnr 3755 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.)
(EXMID ↔ βˆ€π‘₯βˆ€π‘¦(π‘₯ βŠ† {𝑦} ↔ (π‘₯ = βˆ… ∨ π‘₯ = {𝑦})))
 
5-Mar-2023exmidn0m 4203 Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.)
(EXMID ↔ βˆ€π‘₯(π‘₯ = βˆ… ∨ βˆƒπ‘¦ 𝑦 ∈ π‘₯))
 
4-Mar-2023eltg3 13718 Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.)
(𝐡 ∈ 𝑉 β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ βˆƒπ‘₯(π‘₯ βŠ† 𝐡 ∧ 𝐴 = βˆͺ π‘₯)))
 
4-Mar-2023tgvalex 12718 The topology generated by a basis is a set. (Contributed by Jim Kingdon, 4-Mar-2023.)
(𝐡 ∈ 𝑉 β†’ (topGenβ€˜π΅) ∈ V)
 
4-Mar-2023biadanii 613 Inference associated with biadani 612. Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.) (Proof shortened by BJ, 4-Mar-2023.)
(πœ‘ β†’ πœ“)    &   (πœ“ β†’ (πœ‘ ↔ πœ’))    β‡’   (πœ‘ ↔ (πœ“ ∧ πœ’))
 
4-Mar-2023biadani 612 An implication implies to the equivalence of some implied equivalence and some other equivalence involving a conjunction. (Contributed by BJ, 4-Mar-2023.)
(πœ‘ β†’ πœ“)    β‡’   ((πœ“ β†’ (πœ‘ ↔ πœ’)) ↔ (πœ‘ ↔ (πœ“ ∧ πœ’)))
 
16-Feb-2023ixp0 6734 The infinite Cartesian product of a family 𝐡(π‘₯) with an empty member is empty. (Contributed by NM, 1-Oct-2006.) (Revised by Jim Kingdon, 16-Feb-2023.)
(βˆƒπ‘₯ ∈ 𝐴 𝐡 = βˆ… β†’ Xπ‘₯ ∈ 𝐴 𝐡 = βˆ…)
 
16-Feb-2023ixpm 6733 If an infinite Cartesian product of a family 𝐡(π‘₯) is inhabited, every 𝐡(π‘₯) is inhabited. (Contributed by Mario Carneiro, 22-Jun-2016.) (Revised by Jim Kingdon, 16-Feb-2023.)
(βˆƒπ‘“ 𝑓 ∈ Xπ‘₯ ∈ 𝐴 𝐡 β†’ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘§ 𝑧 ∈ 𝐡)
 
16-Feb-2023exmidundifim 4209 Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4208 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.)
(EXMID ↔ βˆ€π‘₯βˆ€π‘¦(π‘₯ βŠ† 𝑦 β†’ (π‘₯ βˆͺ (𝑦 βˆ– π‘₯)) = 𝑦))
 
15-Feb-2023ixpintm 6728 The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.)
(βˆƒπ‘§ 𝑧 ∈ 𝐡 β†’ Xπ‘₯ ∈ 𝐴 ∩ 𝐡 = ∩ 𝑦 ∈ 𝐡 Xπ‘₯ ∈ 𝐴 𝑦)
 
15-Feb-2023ixpiinm 6727 The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.)
(βˆƒπ‘§ 𝑧 ∈ 𝐡 β†’ Xπ‘₯ ∈ 𝐴 ∩ 𝑦 ∈ 𝐡 𝐢 = ∩ 𝑦 ∈ 𝐡 Xπ‘₯ ∈ 𝐴 𝐢)
 
15-Feb-2023ixpexgg 6725 The existence of an infinite Cartesian product. π‘₯ is normally a free-variable parameter in 𝐡. Remark in Enderton p. 54. (Contributed by NM, 28-Sep-2006.) (Revised by Jim Kingdon, 15-Feb-2023.)
((𝐴 ∈ π‘Š ∧ βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ 𝑉) β†’ Xπ‘₯ ∈ 𝐴 𝐡 ∈ V)
 
15-Feb-2023nfixpxy 6720 Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Jim Kingdon, 15-Feb-2023.)
Ⅎ𝑦𝐴    &   β„²π‘¦π΅    β‡’   β„²π‘¦Xπ‘₯ ∈ 𝐴 𝐡
 
13-Feb-2023topnpropgd 12708 The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.)
(πœ‘ β†’ (Baseβ€˜πΎ) = (Baseβ€˜πΏ))    &   (πœ‘ β†’ (TopSetβ€˜πΎ) = (TopSetβ€˜πΏ))    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝐿 ∈ π‘Š)    β‡’   (πœ‘ β†’ (TopOpenβ€˜πΎ) = (TopOpenβ€˜πΏ))
 
12-Feb-2023slotex 12492 Existence of slot value. A corollary of slotslfn 12491. (Contributed by Jim Kingdon, 12-Feb-2023.)
(𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    β‡’   (𝐴 ∈ 𝑉 β†’ (πΈβ€˜π΄) ∈ V)
 
11-Feb-2023topnvalg 12706 Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.)
𝐡 = (Baseβ€˜π‘Š)    &   π½ = (TopSetβ€˜π‘Š)    β‡’   (π‘Š ∈ 𝑉 β†’ (𝐽 β†Ύt 𝐡) = (TopOpenβ€˜π‘Š))
 
10-Feb-2023slotslfn 12491 A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by Jim Kingdon, 10-Feb-2023.)
(𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    β‡’   πΈ Fn V
 
9-Feb-2023pleslid 12663 Slot property of le. (Contributed by Jim Kingdon, 9-Feb-2023.)
(le = Slot (leβ€˜ndx) ∧ (leβ€˜ndx) ∈ β„•)
 
9-Feb-2023topgrptsetd 12660 The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
π‘Š = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩}    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ π‘Š)    &   (πœ‘ β†’ 𝐽 ∈ 𝑋)    β‡’   (πœ‘ β†’ 𝐽 = (TopSetβ€˜π‘Š))
 
9-Feb-2023topgrpplusgd 12659 The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
π‘Š = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩}    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ π‘Š)    &   (πœ‘ β†’ 𝐽 ∈ 𝑋)    β‡’   (πœ‘ β†’ + = (+gβ€˜π‘Š))
 
9-Feb-2023topgrpbasd 12658 The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
π‘Š = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩}    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ π‘Š)    &   (πœ‘ β†’ 𝐽 ∈ 𝑋)    β‡’   (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘Š))
 
9-Feb-2023topgrpstrd 12657 A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
π‘Š = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩}    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ π‘Š)    &   (πœ‘ β†’ 𝐽 ∈ 𝑋)    β‡’   (πœ‘ β†’ π‘Š Struct ⟨1, 9⟩)
 
9-Feb-2023tsetslid 12649 Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.)
(TopSet = Slot (TopSetβ€˜ndx) ∧ (TopSetβ€˜ndx) ∈ β„•)
 
8-Feb-2023ipsipd 12643 The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
𝐴 = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Γ— ⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), Β· ⟩, ⟨(Β·π‘–β€˜ndx), 𝐼⟩})    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ π‘Š)    &   (πœ‘ β†’ Γ— ∈ 𝑋)    &   (πœ‘ β†’ 𝑆 ∈ π‘Œ)    &   (πœ‘ β†’ Β· ∈ 𝑄)    &   (πœ‘ β†’ 𝐼 ∈ 𝑍)    β‡’   (πœ‘ β†’ 𝐼 = (Β·π‘–β€˜π΄))
 
8-Feb-2023ipsvscad 12642 The scalar product operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
𝐴 = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Γ— ⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), Β· ⟩, ⟨(Β·π‘–β€˜ndx), 𝐼⟩})    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ π‘Š)    &   (πœ‘ β†’ Γ— ∈ 𝑋)    &   (πœ‘ β†’ 𝑆 ∈ π‘Œ)    &   (πœ‘ β†’ Β· ∈ 𝑄)    &   (πœ‘ β†’ 𝐼 ∈ 𝑍)    β‡’   (πœ‘ β†’ Β· = ( ·𝑠 β€˜π΄))
 
8-Feb-2023ipsscad 12641 The set of scalars of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
𝐴 = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Γ— ⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), Β· ⟩, ⟨(Β·π‘–β€˜ndx), 𝐼⟩})    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ π‘Š)    &   (πœ‘ β†’ Γ— ∈ 𝑋)    &   (πœ‘ β†’ 𝑆 ∈ π‘Œ)    &   (πœ‘ β†’ Β· ∈ 𝑄)    &   (πœ‘ β†’ 𝐼 ∈ 𝑍)    β‡’   (πœ‘ β†’ 𝑆 = (Scalarβ€˜π΄))
 
7-Feb-2023ipsmulrd 12640 The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Γ— ⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), Β· ⟩, ⟨(Β·π‘–β€˜ndx), 𝐼⟩})    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ π‘Š)    &   (πœ‘ β†’ Γ— ∈ 𝑋)    &   (πœ‘ β†’ 𝑆 ∈ π‘Œ)    &   (πœ‘ β†’ Β· ∈ 𝑄)    &   (πœ‘ β†’ 𝐼 ∈ 𝑍)    β‡’   (πœ‘ β†’ Γ— = (.rβ€˜π΄))
 
7-Feb-2023ipsaddgd 12639 The additive operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Γ— ⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), Β· ⟩, ⟨(Β·π‘–β€˜ndx), 𝐼⟩})    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ π‘Š)    &   (πœ‘ β†’ Γ— ∈ 𝑋)    &   (πœ‘ β†’ 𝑆 ∈ π‘Œ)    &   (πœ‘ β†’ Β· ∈ 𝑄)    &   (πœ‘ β†’ 𝐼 ∈ 𝑍)    β‡’   (πœ‘ β†’ + = (+gβ€˜π΄))
 
7-Feb-2023ipsbased 12638 The base set of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Γ— ⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), Β· ⟩, ⟨(Β·π‘–β€˜ndx), 𝐼⟩})    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ π‘Š)    &   (πœ‘ β†’ Γ— ∈ 𝑋)    &   (πœ‘ β†’ 𝑆 ∈ π‘Œ)    &   (πœ‘ β†’ Β· ∈ 𝑄)    &   (πœ‘ β†’ 𝐼 ∈ 𝑍)    β‡’   (πœ‘ β†’ 𝐡 = (Baseβ€˜π΄))
 
7-Feb-2023ipsstrd 12637 A constructed inner product space is a structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Γ— ⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), Β· ⟩, ⟨(Β·π‘–β€˜ndx), 𝐼⟩})    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ π‘Š)    &   (πœ‘ β†’ Γ— ∈ 𝑋)    &   (πœ‘ β†’ 𝑆 ∈ π‘Œ)    &   (πœ‘ β†’ Β· ∈ 𝑄)    &   (πœ‘ β†’ 𝐼 ∈ 𝑍)    β‡’   (πœ‘ β†’ 𝐴 Struct ⟨1, 8⟩)
 
7-Feb-2023ipslid 12632 Slot property of ·𝑖. (Contributed by Jim Kingdon, 7-Feb-2023.)
(·𝑖 = Slot (Β·π‘–β€˜ndx) ∧ (Β·π‘–β€˜ndx) ∈ β„•)
 
7-Feb-2023lmodvscad 12629 The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 7-Feb-2023.)
π‘Š = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(Scalarβ€˜ndx), 𝐹⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), Β· ⟩})    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ 𝑋)    &   (πœ‘ β†’ 𝐹 ∈ π‘Œ)    &   (πœ‘ β†’ Β· ∈ 𝑍)    β‡’   (πœ‘ β†’ Β· = ( ·𝑠 β€˜π‘Š))
 
6-Feb-2023lmodscad 12628 The set of scalars of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
π‘Š = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(Scalarβ€˜ndx), 𝐹⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), Β· ⟩})    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ 𝑋)    &   (πœ‘ β†’ 𝐹 ∈ π‘Œ)    &   (πœ‘ β†’ Β· ∈ 𝑍)    β‡’   (πœ‘ β†’ 𝐹 = (Scalarβ€˜π‘Š))
 
6-Feb-2023lmodplusgd 12627 The additive operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
π‘Š = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(Scalarβ€˜ndx), 𝐹⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), Β· ⟩})    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ 𝑋)    &   (πœ‘ β†’ 𝐹 ∈ π‘Œ)    &   (πœ‘ β†’ Β· ∈ 𝑍)    β‡’   (πœ‘ β†’ + = (+gβ€˜π‘Š))
 
6-Feb-2023lmodbased 12626 The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
π‘Š = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(Scalarβ€˜ndx), 𝐹⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), Β· ⟩})    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ 𝑋)    &   (πœ‘ β†’ 𝐹 ∈ π‘Œ)    &   (πœ‘ β†’ Β· ∈ 𝑍)    β‡’   (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘Š))
 
5-Feb-2023lmodstrd 12625 A constructed left module or left vector space is a structure. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
π‘Š = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(Scalarβ€˜ndx), 𝐹⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), Β· ⟩})    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ 𝑋)    &   (πœ‘ β†’ 𝐹 ∈ π‘Œ)    &   (πœ‘ β†’ Β· ∈ 𝑍)    β‡’   (πœ‘ β†’ π‘Š Struct ⟨1, 6⟩)
 
5-Feb-2023vscaslid 12624 Slot property of ·𝑠. (Contributed by Jim Kingdon, 5-Feb-2023.)
( ·𝑠 = Slot ( ·𝑠 β€˜ndx) ∧ ( ·𝑠 β€˜ndx) ∈ β„•)
 
5-Feb-2023scaslid 12614 Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.)
(Scalar = Slot (Scalarβ€˜ndx) ∧ (Scalarβ€˜ndx) ∈ β„•)
 
5-Feb-2023srngplusgd 12609 The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by Jim Kingdon, 5-Feb-2023.)
𝑅 = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Β· ⟩} βˆͺ {⟨(*π‘Ÿβ€˜ndx), βˆ— ⟩})    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ π‘Š)    &   (πœ‘ β†’ Β· ∈ 𝑋)    &   (πœ‘ β†’ βˆ— ∈ π‘Œ)    β‡’   (πœ‘ β†’ + = (+gβ€˜π‘…))
 
5-Feb-2023srngbased 12608 The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
𝑅 = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Β· ⟩} βˆͺ {⟨(*π‘Ÿβ€˜ndx), βˆ— ⟩})    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ π‘Š)    &   (πœ‘ β†’ Β· ∈ 𝑋)    &   (πœ‘ β†’ βˆ— ∈ π‘Œ)    β‡’   (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))
 
5-Feb-2023srngstrd 12607 A constructed star ring is a structure. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
𝑅 = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Β· ⟩} βˆͺ {⟨(*π‘Ÿβ€˜ndx), βˆ— ⟩})    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ + ∈ π‘Š)    &   (πœ‘ β†’ Β· ∈ 𝑋)    &   (πœ‘ β†’ βˆ— ∈ π‘Œ)    β‡’   (πœ‘ β†’ 𝑅 Struct ⟨1, 4⟩)
 
5-Feb-2023opelstrsl 12576 The slot of a structure which contains an ordered pair for that slot. (Contributed by Jim Kingdon, 5-Feb-2023.)
(𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   (πœ‘ β†’ 𝑆 Struct 𝑋)    &   (πœ‘ β†’ 𝑉 ∈ π‘Œ)    &   (πœ‘ β†’ ⟨(πΈβ€˜ndx), π‘‰βŸ© ∈ 𝑆)    β‡’   (πœ‘ β†’ 𝑉 = (πΈβ€˜π‘†))
 
4-Feb-2023starvslid 12602 Slot property of *π‘Ÿ. (Contributed by Jim Kingdon, 4-Feb-2023.)
(*π‘Ÿ = Slot (*π‘Ÿβ€˜ndx) ∧ (*π‘Ÿβ€˜ndx) ∈ β„•)
 
3-Feb-2023rngbaseg 12597 The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
𝑅 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Β· ⟩}    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š ∧ Β· ∈ 𝑋) β†’ 𝐡 = (Baseβ€˜π‘…))
 
3-Feb-2023rngstrg 12596 A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
𝑅 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Β· ⟩}    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š ∧ Β· ∈ 𝑋) β†’ 𝑅 Struct ⟨1, 3⟩)
 
3-Feb-2023mulrslid 12593 Slot property of .r. (Contributed by Jim Kingdon, 3-Feb-2023.)
(.r = Slot (.rβ€˜ndx) ∧ (.rβ€˜ndx) ∈ β„•)
 
3-Feb-2023plusgslid 12574 Slot property of +g. (Contributed by Jim Kingdon, 3-Feb-2023.)
(+g = Slot (+gβ€˜ndx) ∧ (+gβ€˜ndx) ∈ β„•)
 
2-Feb-20232strop1g 12585 The other slot of a constructed two-slot structure. Version of 2stropg 12582 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, βŸ¨π‘, + ⟩}    &   (Baseβ€˜ndx) < 𝑁    &   π‘ ∈ β„•    &   πΈ = Slot 𝑁    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ + = (πΈβ€˜πΊ))
 
2-Feb-20232strbas1g 12584 The base set of a constructed two-slot structure. Version of 2strbasg 12581 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, βŸ¨π‘, + ⟩}    &   (Baseβ€˜ndx) < 𝑁    &   π‘ ∈ β„•    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ 𝐡 = (Baseβ€˜πΊ))
 
2-Feb-20232strstr1g 12583 A constructed two-slot structure. Version of 2strstrg 12580 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, βŸ¨π‘, + ⟩}    &   (Baseβ€˜ndx) < 𝑁    &   π‘ ∈ β„•    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ 𝐺 Struct ⟨(Baseβ€˜ndx), π‘βŸ©)
 
31-Jan-2023baseslid 12522 The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.)
(Base = Slot (Baseβ€˜ndx) ∧ (Baseβ€˜ndx) ∈ β„•)
 
31-Jan-2023strsl0 12514 All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 31-Jan-2023.)
(𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    β‡’   βˆ… = (πΈβ€˜βˆ…)
 
31-Jan-2023strslss 12513 Propagate component extraction to a structure 𝑇 from a subset structure 𝑆. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Jim Kingdon, 31-Jan-2023.)
𝑇 ∈ V    &   Fun 𝑇    &   π‘† βŠ† 𝑇    &   (𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   βŸ¨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆    β‡’   (πΈβ€˜π‘‡) = (πΈβ€˜π‘†)
 
31-Jan-2023strslssd 12512 Deduction version of strslss 12513. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 31-Jan-2023.)
(𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   (πœ‘ β†’ 𝑇 ∈ 𝑉)    &   (πœ‘ β†’ Fun 𝑇)    &   (πœ‘ β†’ 𝑆 βŠ† 𝑇)    &   (πœ‘ β†’ ⟨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆)    β‡’   (πœ‘ β†’ (πΈβ€˜π‘‡) = (πΈβ€˜π‘†))
 
30-Jan-2023strslfv3 12511 Variant on strslfv 12510 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.)
(πœ‘ β†’ π‘ˆ = 𝑆)    &   π‘† Struct 𝑋    &   (𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   {⟨(πΈβ€˜ndx), 𝐢⟩} βŠ† 𝑆    &   (πœ‘ β†’ 𝐢 ∈ 𝑉)    &   π΄ = (πΈβ€˜π‘ˆ)    β‡’   (πœ‘ β†’ 𝐴 = 𝐢)
 
30-Jan-2023strslfv 12510 Extract a structure component 𝐢 (such as the base set) from a structure 𝑆 with a component extractor 𝐸 (such as the base set extractor df-base 12471). By virtue of ndxslid 12490, this can be done without having to refer to the hard-coded numeric index of 𝐸. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Jim Kingdon, 30-Jan-2023.)
𝑆 Struct 𝑋    &   (𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   {⟨(πΈβ€˜ndx), 𝐢⟩} βŠ† 𝑆    β‡’   (𝐢 ∈ 𝑉 β†’ 𝐢 = (πΈβ€˜π‘†))
 
30-Jan-2023strslfv2 12509 A variation on strslfv 12510 to avoid asserting that 𝑆 itself is a function, which involves sethood of all the ordered pair components of 𝑆. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
𝑆 ∈ V    &   Fun ◑◑𝑆    &   (𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   βŸ¨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆    β‡’   (𝐢 ∈ 𝑉 β†’ 𝐢 = (πΈβ€˜π‘†))
 
30-Jan-2023strslfv2d 12508 Deduction version of strslfv 12510. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
(𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ Fun ◑◑𝑆)    &   (πœ‘ β†’ ⟨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆)    &   (πœ‘ β†’ 𝐢 ∈ π‘Š)    β‡’   (πœ‘ β†’ 𝐢 = (πΈβ€˜π‘†))
 
30-Jan-2023strslfvd 12507 Deduction version of strslfv 12510. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 30-Jan-2023.)
(𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ Fun 𝑆)    &   (πœ‘ β†’ ⟨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆)    β‡’   (πœ‘ β†’ 𝐢 = (πΈβ€˜π‘†))
 
30-Jan-2023strsetsid 12498 Value of the structure replacement function. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 30-Jan-2023.)
𝐸 = Slot (πΈβ€˜ndx)    &   (πœ‘ β†’ 𝑆 Struct βŸ¨π‘€, π‘βŸ©)    &   (πœ‘ β†’ Fun 𝑆)    &   (πœ‘ β†’ (πΈβ€˜ndx) ∈ dom 𝑆)    β‡’   (πœ‘ β†’ 𝑆 = (𝑆 sSet ⟨(πΈβ€˜ndx), (πΈβ€˜π‘†)⟩))
 
30-Jan-2023funresdfunsndc 6510 Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself, where equality is decidable. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 30-Jan-2023.)
((βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ dom 𝐹DECID π‘₯ = 𝑦 ∧ Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) β†’ ((𝐹 β†Ύ (V βˆ– {𝑋})) βˆͺ {βŸ¨π‘‹, (πΉβ€˜π‘‹)⟩}) = 𝐹)
 
29-Jan-2023ndxslid 12490 A structure component extractor is defined by its own index. That the index is a natural number will also be needed in quite a few contexts so it is included in the conclusion of this theorem which can be used as a hypothesis of theorems like strslfv 12510. (Contributed by Jim Kingdon, 29-Jan-2023.)
𝐸 = Slot 𝑁    &   π‘ ∈ β„•    β‡’   (𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)
 
29-Jan-2023fnsnsplitdc 6509 Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 29-Jan-2023.)
((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) β†’ 𝐹 = ((𝐹 β†Ύ (𝐴 βˆ– {𝑋})) βˆͺ {βŸ¨π‘‹, (πΉβ€˜π‘‹)⟩}))
 
28-Jan-20232stropg 12582 The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(πΈβ€˜ndx), + ⟩}    &   πΈ = Slot 𝑁    &   1 < 𝑁    &   π‘ ∈ β„•    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ + = (πΈβ€˜πΊ))
 
28-Jan-20232strbasg 12581 The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(πΈβ€˜ndx), + ⟩}    &   πΈ = Slot 𝑁    &   1 < 𝑁    &   π‘ ∈ β„•    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ 𝐡 = (Baseβ€˜πΊ))
 
28-Jan-20232strstrg 12580 A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(πΈβ€˜ndx), + ⟩}    &   πΈ = Slot 𝑁    &   1 < 𝑁    &   π‘ ∈ β„•    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ 𝐺 Struct ⟨1, π‘βŸ©)
 
28-Jan-20231strstrg 12578 A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩}    β‡’   (𝐡 ∈ 𝑉 β†’ 𝐺 Struct ⟨1, 1⟩)
 
27-Jan-2023strle2g 12569 Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
𝐼 ∈ β„•    &   π΄ = 𝐼    &   πΌ < 𝐽    &   π½ ∈ β„•    &   π΅ = 𝐽    β‡’   ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ π‘Š) β†’ {⟨𝐴, π‘‹βŸ©, ⟨𝐡, π‘ŒβŸ©} Struct ⟨𝐼, 𝐽⟩)
 
27-Jan-2023strle1g 12568 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
𝐼 ∈ β„•    &   π΄ = 𝐼    β‡’   (𝑋 ∈ 𝑉 β†’ {⟨𝐴, π‘‹βŸ©} Struct ⟨𝐼, 𝐼⟩)
 
27-Jan-2023strleund 12565 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
(πœ‘ β†’ 𝐹 Struct ⟨𝐴, 𝐡⟩)    &   (πœ‘ β†’ 𝐺 Struct ⟨𝐢, 𝐷⟩)    &   (πœ‘ β†’ 𝐡 < 𝐢)    β‡’   (πœ‘ β†’ (𝐹 βˆͺ 𝐺) Struct ⟨𝐴, 𝐷⟩)
 
24-Jan-2023setsslnid 12517 Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
(𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   (πΈβ€˜ndx) β‰  𝐷    &   π· ∈ β„•    β‡’   ((π‘Š ∈ 𝐴 ∧ 𝐢 ∈ 𝑉) β†’ (πΈβ€˜π‘Š) = (πΈβ€˜(π‘Š sSet ⟨𝐷, 𝐢⟩)))
 
24-Jan-2023setsslid 12516 Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
(𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    β‡’   ((π‘Š ∈ 𝐴 ∧ 𝐢 ∈ 𝑉) β†’ 𝐢 = (πΈβ€˜(π‘Š sSet ⟨(πΈβ€˜ndx), 𝐢⟩)))
 
22-Jan-2023setsabsd 12504 Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
(πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝐴 ∈ π‘Š)    &   (πœ‘ β†’ 𝐡 ∈ 𝑋)    &   (πœ‘ β†’ 𝐢 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ ((𝑆 sSet ⟨𝐴, 𝐡⟩) sSet ⟨𝐴, 𝐢⟩) = (𝑆 sSet ⟨𝐴, 𝐢⟩))
 
22-Jan-2023setsresg 12503 The structure replacement function does not affect the value of 𝑆 away from 𝐴. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š ∧ 𝐡 ∈ 𝑋) β†’ ((𝑆 sSet ⟨𝐴, 𝐡⟩) β†Ύ (V βˆ– {𝐴})) = (𝑆 β†Ύ (V βˆ– {𝐴})))
 
22-Jan-2023setsex 12497 Applying the structure replacement function yields a set. (Contributed by Jim Kingdon, 22-Jan-2023.)
((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Š) β†’ (𝑆 sSet ⟨𝐴, 𝐡⟩) ∈ V)
 
22-Jan-20232zsupmax 11237 Two ways to express the maximum of two integers. Because order of integers is decidable, we have more flexibility than for real numbers. (Contributed by Jim Kingdon, 22-Jan-2023.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ sup({𝐴, 𝐡}, ℝ, < ) = if(𝐴 ≀ 𝐡, 𝐡, 𝐴))
 
22-Jan-2023elpwpwel 4477 A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.)
(𝐴 ∈ 𝒫 𝒫 𝐡 ↔ βˆͺ 𝐴 ∈ 𝒫 𝐡)
 
21-Jan-2023funresdfunsnss 5722 Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in a subset of the function itself. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 21-Jan-2023.)
((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) β†’ ((𝐹 β†Ύ (V βˆ– {𝑋})) βˆͺ {βŸ¨π‘‹, (πΉβ€˜π‘‹)⟩}) βŠ† 𝐹)
 
20-Jan-2023setsvala 12496 Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 20-Jan-2023.)
((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Š) β†’ (𝑆 sSet ⟨𝐴, 𝐡⟩) = ((𝑆 β†Ύ (V βˆ– {𝐴})) βˆͺ {⟨𝐴, 𝐡⟩}))
 
20-Jan-2023fnsnsplitss 5718 Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 20-Jan-2023.)
((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) β†’ ((𝐹 β†Ύ (𝐴 βˆ– {𝑋})) βˆͺ {βŸ¨π‘‹, (πΉβ€˜π‘‹)⟩}) βŠ† 𝐹)
 
19-Jan-2023strfvssn 12487 A structure component extractor produces a value which is contained in a set dependent on 𝑆, but not 𝐸. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Jim Kingdon, 19-Jan-2023.)
𝐸 = Slot 𝑁    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (πΈβ€˜π‘†) βŠ† βˆͺ ran 𝑆)
 
19-Jan-2023strnfvn 12486 Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 12471) and 𝑁 is a fixed integer such as 1. 𝑆 is a structure, i.e. a specific member of a class of structures.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strslfv 12510. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.)

𝑆 ∈ V    &   πΈ = Slot 𝑁    &   π‘ ∈ β„•    β‡’   (πΈβ€˜π‘†) = (π‘†β€˜π‘)
 
19-Jan-2023strnfvnd 12485 Deduction version of strnfvn 12486. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 19-Jan-2023.)
𝐸 = Slot 𝑁    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (πΈβ€˜π‘†) = (π‘†β€˜π‘))
 
18-Jan-2023isstructr 12480 The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(((𝑀 ∈ β„• ∧ 𝑁 ∈ β„• ∧ 𝑀 ≀ 𝑁) ∧ (Fun (𝐹 βˆ– {βˆ…}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 βŠ† (𝑀...𝑁))) β†’ 𝐹 Struct βŸ¨π‘€, π‘βŸ©)
 
18-Jan-2023isstructim 12479 The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(𝐹 Struct βŸ¨π‘€, π‘βŸ© β†’ ((𝑀 ∈ β„• ∧ 𝑁 ∈ β„• ∧ 𝑀 ≀ 𝑁) ∧ Fun (𝐹 βˆ– {βˆ…}) ∧ dom 𝐹 βŠ† (𝑀...𝑁)))
 
18-Jan-2023isstruct2r 12476 The property of being a structure with components in (1st β€˜π‘‹)...(2nd β€˜π‘‹). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(((𝑋 ∈ ( ≀ ∩ (β„• Γ— β„•)) ∧ Fun (𝐹 βˆ– {βˆ…})) ∧ (𝐹 ∈ 𝑉 ∧ dom 𝐹 βŠ† (...β€˜π‘‹))) β†’ 𝐹 Struct 𝑋)
 
18-Jan-2023isstruct2im 12475 The property of being a structure with components in (1st β€˜π‘‹)...(2nd β€˜π‘‹). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(𝐹 Struct 𝑋 β†’ (𝑋 ∈ ( ≀ ∩ (β„• Γ— β„•)) ∧ Fun (𝐹 βˆ– {βˆ…}) ∧ dom 𝐹 βŠ† (...β€˜π‘‹)))
 
18-Jan-2023sbiev 1792 Conversion of implicit substitution to explicit substitution. Version of sbie 1791 with a disjoint variable condition. (Contributed by Wolf Lammen, 18-Jan-2023.)
β„²π‘₯πœ“    &   (π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   ([𝑦 / π‘₯]πœ‘ ↔ πœ“)
 
16-Jan-2023toponsspwpwg 13683 The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)
(𝐴 ∈ 𝑉 β†’ (TopOnβ€˜π΄) βŠ† 𝒫 𝒫 𝐴)
 
14-Jan-2023istopfin 13661 Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 13660. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.)
(𝐽 ∈ Top β†’ (βˆ€π‘₯(π‘₯ βŠ† 𝐽 β†’ βˆͺ π‘₯ ∈ 𝐽) ∧ βˆ€π‘₯((π‘₯ βŠ† 𝐽 ∧ π‘₯ β‰  βˆ… ∧ π‘₯ ∈ Fin) β†’ ∩ π‘₯ ∈ 𝐽)))
 
14-Jan-2023fiintim 6931 If a class is closed under pairwise intersections, then it is closed under nonempty finite intersections. The converse would appear to require an additional condition, such as π‘₯ and 𝑦 not being equal, or 𝐴 having decidable equality.

This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use a pairwise intersection and some texts use a finite intersection, but most topology texts assume excluded middle (in which case the two intersection properties would be equivalent). (Contributed by NM, 22-Sep-2002.) (Revised by Jim Kingdon, 14-Jan-2023.)

(βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ∩ 𝑦) ∈ 𝐴 β†’ βˆ€π‘₯((π‘₯ βŠ† 𝐴 ∧ π‘₯ β‰  βˆ… ∧ π‘₯ ∈ Fin) β†’ ∩ π‘₯ ∈ 𝐴))
 
9-Jan-2023divccncfap 14238 Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Jim Kingdon, 9-Jan-2023.)
𝐹 = (π‘₯ ∈ β„‚ ↦ (π‘₯ / 𝐴))    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐴 # 0) β†’ 𝐹 ∈ (ℂ–cnβ†’β„‚))
 
7-Jan-2023eap1 11796 e is apart from 1. (Contributed by Jim Kingdon, 7-Jan-2023.)
e # 1
 
7-Jan-2023eap0 11794 e is apart from 0. (Contributed by Jim Kingdon, 7-Jan-2023.)
e # 0
 
7-Jan-2023egt2lt3 11790 Euler's constant e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 28-Nov-2008.) (Revised by Jim Kingdon, 7-Jan-2023.)
(2 < e ∧ e < 3)
 
6-Jan-2023eirr 11789 e is not rational. In the absence of excluded middle, we can distinguish between this and saying that e is irrational in the sense of being apart from any rational number, which is eirrap 11788. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.)
e βˆ‰ β„š
 
6-Jan-2023eirrap 11788 e is irrational. That is, for any rational number, e is apart from it. In the absence of excluded middle, we can distinguish between this and saying that e is not rational, which is eirr 11789. (Contributed by Jim Kingdon, 6-Jan-2023.)
(𝑄 ∈ β„š β†’ e # 𝑄)
 
6-Jan-2023btwnapz 9386 A number between an integer and its successor is apart from any integer. (Contributed by Jim Kingdon, 6-Jan-2023.)
(πœ‘ β†’ 𝐴 ∈ β„€)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ β„€)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐡 < (𝐴 + 1))    β‡’   (πœ‘ β†’ 𝐡 # 𝐢)
 
6-Jan-2023apmul2 8749 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 6-Jan-2023.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ (𝐢 ∈ β„‚ ∧ 𝐢 # 0)) β†’ (𝐴 # 𝐡 ↔ (𝐢 Β· 𝐴) # (𝐢 Β· 𝐡)))
 
1-Jan-2023nnap0i 8953 A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.)
𝐴 ∈ β„•    β‡’   π΄ # 0
 
31-Dec-20222logb9irrALT 14553 Alternate proof of 2logb9irr 14550: The logarithm of nine to base two is not rational. (Contributed by AV, 31-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(2 logb 9) ∈ (ℝ βˆ– β„š)
 
31-Dec-20222logb3irr 14552 Example for logbprmirr 14551. The logarithm of three to base two is not rational. (Contributed by AV, 31-Dec-2022.)
(2 logb 3) ∈ (ℝ βˆ– β„š)
 
31-Dec-2022logbprmirr 14551 The logarithm of a prime to a different prime base is not rational. For example, (2 logb 3) ∈ (ℝ βˆ– β„š) (see 2logb3irr 14552). (Contributed by AV, 31-Dec-2022.)
((𝑋 ∈ β„™ ∧ 𝐡 ∈ β„™ ∧ 𝑋 β‰  𝐡) β†’ (𝐡 logb 𝑋) ∈ (ℝ βˆ– β„š))
 
30-Dec-2022elpqb 9652 A class is a positive rational iff it is the quotient of two positive integers. (Contributed by AV, 30-Dec-2022.)
((𝐴 ∈ β„š ∧ 0 < 𝐴) ↔ βˆƒπ‘₯ ∈ β„• βˆƒπ‘¦ ∈ β„• 𝐴 = (π‘₯ / 𝑦))
 
29-Dec-2022sqrt2cxp2logb9e3 14554 The square root of two to the power of the logarithm of nine to base two is three. (βˆšβ€˜2) and (2 logb 9) are not rational (see sqrt2irr0 12167 resp. 2logb9irr 14550), satisfying the statement in 2irrexpq 14555. (Contributed by AV, 29-Dec-2022.)
((βˆšβ€˜2)↑𝑐(2 logb 9)) = 3
 
29-Dec-20222logb9irr 14550 Example for logbgcd1irr 14546. The logarithm of nine to base two is not rational. Also see 2logb9irrap 14556 which says that it is irrational (in the sense of being apart from any rational number). (Contributed by AV, 29-Dec-2022.)
(2 logb 9) ∈ (ℝ βˆ– β„š)
 
29-Dec-2022logbgcd1irrap 14549 The logarithm of an integer greater than 1 to an integer base greater than 1 is irrational (in the sense of being apart from any rational number) if the argument and the base are relatively prime. For example, (2 logb 9) # 𝑄 where 𝑄 is rational. (Contributed by AV, 29-Dec-2022.)
(((𝑋 ∈ (β„€β‰₯β€˜2) ∧ 𝐡 ∈ (β„€β‰₯β€˜2)) ∧ ((𝑋 gcd 𝐡) = 1 ∧ 𝑄 ∈ β„š)) β†’ (𝐡 logb 𝑋) # 𝑄)
 
29-Dec-2022logbgcd1irr 14546 The logarithm of an integer greater than 1 to an integer base greater than 1 is not rational if the argument and the base are relatively prime. For example, (2 logb 9) ∈ (ℝ βˆ– β„š). (Contributed by AV, 29-Dec-2022.)
((𝑋 ∈ (β„€β‰₯β€˜2) ∧ 𝐡 ∈ (β„€β‰₯β€˜2) ∧ (𝑋 gcd 𝐡) = 1) β†’ (𝐡 logb 𝑋) ∈ (ℝ βˆ– β„š))
 
29-Dec-2022logbgt0b 14545 The logarithm of a positive real number to a real base greater than 1 is positive iff the number is greater than 1. (Contributed by AV, 29-Dec-2022.)
((𝐴 ∈ ℝ+ ∧ (𝐡 ∈ ℝ+ ∧ 1 < 𝐡)) β†’ (0 < (𝐡 logb 𝐴) ↔ 1 < 𝐴))
 
29-Dec-2022cxpcom 14518 Commutative law for real exponentiation. (Contributed by AV, 29-Dec-2022.)
((𝐴 ∈ ℝ+ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ ((𝐴↑𝑐𝐡)↑𝑐𝐢) = ((𝐴↑𝑐𝐢)↑𝑐𝐡))
 
29-Dec-2022elpq 9651 A positive rational is the quotient of two positive integers. (Contributed by AV, 29-Dec-2022.)
((𝐴 ∈ β„š ∧ 0 < 𝐴) β†’ βˆƒπ‘₯ ∈ β„• βˆƒπ‘¦ ∈ β„• 𝐴 = (π‘₯ / 𝑦))
 
26-Dec-2022apdivmuld 8773 Relationship between division and multiplication. (Contributed by Jim Kingdon, 26-Dec-2022.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 # 0)    β‡’   (πœ‘ β†’ ((𝐴 / 𝐡) # 𝐢 ↔ (𝐡 Β· 𝐢) # 𝐴))
 
25-Dec-2022tanaddaplem 11749 A useful intermediate step in tanaddap 11750 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.) (Revised by Jim Kingdon, 25-Dec-2022.)
(((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) ∧ ((cosβ€˜π΄) # 0 ∧ (cosβ€˜π΅) # 0)) β†’ ((cosβ€˜(𝐴 + 𝐡)) # 0 ↔ ((tanβ€˜π΄) Β· (tanβ€˜π΅)) # 1))
 
25-Dec-2022subap0 8603 Two numbers being apart is equivalent to their difference being apart from zero. (Contributed by Jim Kingdon, 25-Dec-2022.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((𝐴 βˆ’ 𝐡) # 0 ↔ 𝐴 # 𝐡))
 
23-Dec-20222irrexpq 14555 There exist real numbers π‘Ž and 𝑏 which are not rational such that (π‘Žβ†‘π‘) is rational. Statement in the Metamath book, section 1.1.5, footnote 27 on page 17, and the "constructive proof" for theorem 1.2 of [Bauer], p. 483. This is a constructive proof because it is based on two explicitly named non-rational numbers (βˆšβ€˜2) and (2 logb 9), see sqrt2irr0 12167, 2logb9irr 14550 and sqrt2cxp2logb9e3 14554. Therefore, this proof is acceptable/usable in intuitionistic logic.

For a theorem which is the same but proves that π‘Ž and 𝑏 are irrational (in the sense of being apart from any rational number), see 2irrexpqap 14557. (Contributed by AV, 23-Dec-2022.)

βˆƒπ‘Ž ∈ (ℝ βˆ– β„š)βˆƒπ‘ ∈ (ℝ βˆ– β„š)(π‘Žβ†‘π‘π‘) ∈ β„š
 
23-Dec-2022rpcxpsqrtth 14511 Square root theorem over the complex numbers for the complex power function. Compare with resqrtth 11043. (Contributed by AV, 23-Dec-2022.)
(𝐴 ∈ ℝ+ β†’ ((βˆšβ€˜π΄)↑𝑐2) = 𝐴)
 
23-Dec-2022sqrt2irr0 12167 The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.)
(βˆšβ€˜2) ∈ (ℝ βˆ– β„š)
 
22-Dec-2022tanval3ap 11725 Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
((𝐴 ∈ β„‚ ∧ ((expβ€˜(2 Β· (i Β· 𝐴))) + 1) # 0) β†’ (tanβ€˜π΄) = (((expβ€˜(2 Β· (i Β· 𝐴))) βˆ’ 1) / (i Β· ((expβ€˜(2 Β· (i Β· 𝐴))) + 1))))
 
22-Dec-2022tanval2ap 11724 Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (tanβ€˜π΄) = (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (i Β· ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))))))
 
22-Dec-2022tanclapd 11723 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) (Revised by Jim Kingdon, 22-Dec-2022.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ (cosβ€˜π΄) # 0)    β‡’   (πœ‘ β†’ (tanβ€˜π΄) ∈ β„‚)
 
21-Dec-2022tanclap 11720 The closure of the tangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.)
((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (tanβ€˜π΄) ∈ β„‚)
 
21-Dec-2022tanvalap 11719 Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.)
((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (tanβ€˜π΄) = ((sinβ€˜π΄) / (cosβ€˜π΄)))
 
20-Dec-2022reef11 11710 The exponential function on real numbers is one-to-one. (Contributed by NM, 21-Aug-2008.) (Revised by Jim Kingdon, 20-Dec-2022.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ ((expβ€˜π΄) = (expβ€˜π΅) ↔ 𝐴 = 𝐡))
 
20-Dec-2022efltim 11709 The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 20-Dec-2022.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 < 𝐡 β†’ (expβ€˜π΄) < (expβ€˜π΅)))
 
20-Dec-2022eqord1 8443 A strictly increasing real function on a subset of ℝ is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by Jim Kingdon, 20-Dec-2022.)
(π‘₯ = 𝑦 β†’ 𝐴 = 𝐡)    &   (π‘₯ = 𝐢 β†’ 𝐴 = 𝑀)    &   (π‘₯ = 𝐷 β†’ 𝐴 = 𝑁)    &   π‘† βŠ† ℝ    &   ((πœ‘ ∧ π‘₯ ∈ 𝑆) β†’ 𝐴 ∈ ℝ)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ < 𝑦 β†’ 𝐴 < 𝐡))    β‡’   ((πœ‘ ∧ (𝐢 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) β†’ (𝐢 = 𝐷 ↔ 𝑀 = 𝑁))
 
14-Dec-2022iserabs 11486 Generalized triangle inequality: the absolute value of an infinite sum is less than or equal to the sum of absolute values. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Jim Kingdon, 14-Dec-2022.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ seq𝑀( + , 𝐹) ⇝ 𝐴)    &   (πœ‘ β†’ seq𝑀( + , 𝐺) ⇝ 𝐡)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΊβ€˜π‘˜) = (absβ€˜(πΉβ€˜π‘˜)))    β‡’   (πœ‘ β†’ (absβ€˜π΄) ≀ 𝐡)
 
12-Dec-2022efap0 11688 The exponential of a complex number is apart from zero. (Contributed by Jim Kingdon, 12-Dec-2022.)
(𝐴 ∈ β„‚ β†’ (expβ€˜π΄) # 0)
 
8-Dec-2022efcllem 11670 Lemma for efcl 11675. The series that defines the exponential function converges. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
𝐹 = (𝑛 ∈ β„•0 ↦ ((𝐴↑𝑛) / (!β€˜π‘›)))    β‡’   (𝐴 ∈ β„‚ β†’ seq0( + , 𝐹) ∈ dom ⇝ )
 
8-Dec-2022efcllemp 11669 Lemma for efcl 11675. The series that defines the exponential function converges. The ratio test cvgratgt0 11544 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
𝐹 = (𝑛 ∈ β„•0 ↦ ((𝐴↑𝑛) / (!β€˜π‘›)))    &   (πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐾 ∈ β„•)    &   (πœ‘ β†’ (2 Β· (absβ€˜π΄)) < 𝐾)    β‡’   (πœ‘ β†’ seq0( + , 𝐹) ∈ dom ⇝ )
 
8-Dec-2022eftvalcn 11668 The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 8-Dec-2022.)
𝐹 = (𝑛 ∈ β„•0 ↦ ((𝐴↑𝑛) / (!β€˜π‘›)))    β‡’   ((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„•0) β†’ (πΉβ€˜π‘) = ((𝐴↑𝑁) / (!β€˜π‘)))
 
8-Dec-2022mertensabs 11548 Mertens' theorem. If 𝐴(𝑗) is an absolutely convergent series and 𝐡(π‘˜) is convergent, then (Σ𝑗 ∈ β„•0𝐴(𝑗) Β· Ξ£π‘˜ ∈ β„•0𝐡(π‘˜)) = Ξ£π‘˜ ∈ β„•0Σ𝑗 ∈ (0...π‘˜)(𝐴(𝑗) Β· 𝐡(π‘˜ βˆ’ 𝑗)) (and this latter series is convergent). This latter sum is commonly known as the Cauchy product of the sequences. The proof follows the outline at http://en.wikipedia.org/wiki/Cauchy_product#Proof_of_Mertens.27_theorem. (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by Jim Kingdon, 8-Dec-2022.)
((πœ‘ ∧ 𝑗 ∈ β„•0) β†’ (πΉβ€˜π‘—) = 𝐴)    &   ((πœ‘ ∧ 𝑗 ∈ β„•0) β†’ (πΎβ€˜π‘—) = (absβ€˜π΄))    &   ((πœ‘ ∧ 𝑗 ∈ β„•0) β†’ 𝐴 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (πΊβ€˜π‘˜) = 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (π»β€˜π‘˜) = Σ𝑗 ∈ (0...π‘˜)(𝐴 Β· (πΊβ€˜(π‘˜ βˆ’ 𝑗))))    &   (πœ‘ β†’ seq0( + , 𝐾) ∈ dom ⇝ )    &   (πœ‘ β†’ seq0( + , 𝐺) ∈ dom ⇝ )    &   (πœ‘ β†’ seq0( + , 𝐹) ∈ dom ⇝ )    β‡’   (πœ‘ β†’ seq0( + , 𝐻) ⇝ (Σ𝑗 ∈ β„•0 𝐴 Β· Ξ£π‘˜ ∈ β„•0 𝐡))
 
3-Dec-2022mertenslemub 11545 Lemma for mertensabs 11548. An upper bound for 𝑇. (Contributed by Jim Kingdon, 3-Dec-2022.)
((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (πΊβ€˜π‘˜) = 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ seq0( + , 𝐺) ∈ dom ⇝ )    &   π‘‡ = {𝑧 ∣ βˆƒπ‘› ∈ (0...(𝑆 βˆ’ 1))𝑧 = (absβ€˜Ξ£π‘˜ ∈ (β„€β‰₯β€˜(𝑛 + 1))(πΊβ€˜π‘˜))}    &   (πœ‘ β†’ 𝑋 ∈ 𝑇)    &   (πœ‘ β†’ 𝑆 ∈ β„•)    β‡’   (πœ‘ β†’ 𝑋 ≀ Σ𝑛 ∈ (0...(𝑆 βˆ’ 1))(absβ€˜Ξ£π‘˜ ∈ (β„€β‰₯β€˜(𝑛 + 1))(πΊβ€˜π‘˜)))
 
2-Dec-2022mertenslemi1 11546 Lemma for mertensabs 11548. (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
((πœ‘ ∧ 𝑗 ∈ β„•0) β†’ (πΉβ€˜π‘—) = 𝐴)    &   ((πœ‘ ∧ 𝑗 ∈ β„•0) β†’ (πΎβ€˜π‘—) = (absβ€˜π΄))    &   ((πœ‘ ∧ 𝑗 ∈ β„•0) β†’ 𝐴 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (πΊβ€˜π‘˜) = 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ β„•0) β†’ (π»β€˜π‘˜) = Σ𝑗 ∈ (0...π‘˜)(𝐴 Β· (πΊβ€˜(π‘˜ βˆ’ 𝑗))))    &   (πœ‘ β†’ seq0( + , 𝐾) ∈ dom ⇝ )    &   (πœ‘ β†’ seq0( + , 𝐺) ∈ dom ⇝ )    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   π‘‡ = {𝑧 ∣ βˆƒπ‘› ∈ (0...(𝑠 βˆ’ 1))𝑧 = (absβ€˜Ξ£π‘˜ ∈ (β„€β‰₯β€˜(𝑛 + 1))(πΊβ€˜π‘˜))}    &   (πœ“ ↔ (𝑠 ∈ β„• ∧ βˆ€π‘› ∈ (β„€β‰₯β€˜π‘ )(absβ€˜Ξ£π‘˜ ∈ (β„€β‰₯β€˜(𝑛 + 1))(πΊβ€˜π‘˜)) < ((𝐸 / 2) / (Σ𝑗 ∈ β„•0 (πΎβ€˜π‘—) + 1))))    &   (πœ‘ β†’ 𝑃 ∈ ℝ)    &   (πœ‘ β†’ (πœ“ ∧ (𝑑 ∈ β„•0 ∧ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘‘)(πΎβ€˜π‘š) < (((𝐸 / 2) / 𝑠) / (𝑃 + 1)))))    &   (πœ‘ β†’ 0 ≀ 𝑃)    &   (πœ‘ β†’ βˆ€π‘€ ∈ 𝑇 𝑀 ≀ 𝑃)    β‡’   (πœ‘ β†’ βˆƒπ‘¦ ∈ β„•0 βˆ€π‘š ∈ (β„€β‰₯β€˜π‘¦)(absβ€˜Ξ£π‘— ∈ (0...π‘š)(𝐴 Β· Ξ£π‘˜ ∈ (β„€β‰₯β€˜((π‘š βˆ’ 𝑗) + 1))𝐡)) < 𝐸)
 
2-Dec-2022fsum3cvg3 11407 A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑍)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ DECID π‘˜ ∈ 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) = if(π‘˜ ∈ 𝐴, 𝐡, 0))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ seq𝑀( + , 𝐹) ∈ dom ⇝ )
 
2-Dec-2022fsum3cvg2 11405 The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) = if(π‘˜ ∈ 𝐴, 𝐡, 0))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ DECID π‘˜ ∈ 𝐴)    &   (πœ‘ β†’ 𝐴 βŠ† (𝑀...𝑁))    β‡’   (πœ‘ β†’ seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)β€˜π‘))
 
24-Nov-2022cvgratnnlembern 11534 Lemma for cvgratnn 11542. Upper bound for a geometric progression of positive ratio less than one. (Contributed by Jim Kingdon, 24-Nov-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 1)    &   (πœ‘ β†’ 0 < 𝐴)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    β‡’   (πœ‘ β†’ (𝐴↑𝑀) < ((1 / ((1 / 𝐴) βˆ’ 1)) / 𝑀))
 
23-Nov-2022cvgratnnlemfm 11540 Lemma for cvgratnn 11542. (Contributed by Jim Kingdon, 23-Nov-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 1)    &   (πœ‘ β†’ 0 < 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (absβ€˜(πΉβ€˜(π‘˜ + 1))) ≀ (𝐴 Β· (absβ€˜(πΉβ€˜π‘˜))))    &   (πœ‘ β†’ 𝑀 ∈ β„•)    β‡’   (πœ‘ β†’ (absβ€˜(πΉβ€˜π‘€)) < ((((1 / ((1 / 𝐴) βˆ’ 1)) / 𝐴) Β· ((absβ€˜(πΉβ€˜1)) + 1)) / 𝑀))
 
23-Nov-2022cvgratnnlemsumlt 11539 Lemma for cvgratnn 11542. (Contributed by Jim Kingdon, 23-Nov-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 1)    &   (πœ‘ β†’ 0 < 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (absβ€˜(πΉβ€˜(π‘˜ + 1))) ≀ (𝐴 Β· (absβ€˜(πΉβ€˜π‘˜))))    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    β‡’   (πœ‘ β†’ Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 βˆ’ 𝑀)) < (𝐴 / (1 βˆ’ 𝐴)))
 
21-Nov-2022cvgratnnlemrate 11541 Lemma for cvgratnn 11542. (Contributed by Jim Kingdon, 21-Nov-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 1)    &   (πœ‘ β†’ 0 < 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (absβ€˜(πΉβ€˜(π‘˜ + 1))) ≀ (𝐴 Β· (absβ€˜(πΉβ€˜π‘˜))))    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    β‡’   (πœ‘ β†’ (absβ€˜((seq1( + , 𝐹)β€˜π‘) βˆ’ (seq1( + , 𝐹)β€˜π‘€))) < (((((1 / ((1 / 𝐴) βˆ’ 1)) / 𝐴) Β· ((absβ€˜(πΉβ€˜1)) + 1)) Β· (𝐴 / (1 βˆ’ 𝐴))) / 𝑀))
 
21-Nov-2022cvgratnnlemabsle 11538 Lemma for cvgratnn 11542. (Contributed by Jim Kingdon, 21-Nov-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 1)    &   (πœ‘ β†’ 0 < 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (absβ€˜(πΉβ€˜(π‘˜ + 1))) ≀ (𝐴 Β· (absβ€˜(πΉβ€˜π‘˜))))    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    β‡’   (πœ‘ β†’ (absβ€˜Ξ£π‘– ∈ ((𝑀 + 1)...𝑁)(πΉβ€˜π‘–)) ≀ ((absβ€˜(πΉβ€˜π‘€)) Β· Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 βˆ’ 𝑀))))
 
21-Nov-2022cvgratnnlemseq 11537 Lemma for cvgratnn 11542. (Contributed by Jim Kingdon, 21-Nov-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 1)    &   (πœ‘ β†’ 0 < 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (absβ€˜(πΉβ€˜(π‘˜ + 1))) ≀ (𝐴 Β· (absβ€˜(πΉβ€˜π‘˜))))    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    β‡’   (πœ‘ β†’ ((seq1( + , 𝐹)β€˜π‘) βˆ’ (seq1( + , 𝐹)β€˜π‘€)) = Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(πΉβ€˜π‘–))
 
15-Nov-2022cvgratnnlemmn 11536 Lemma for cvgratnn 11542. (Contributed by Jim Kingdon, 15-Nov-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 1)    &   (πœ‘ β†’ 0 < 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (absβ€˜(πΉβ€˜(π‘˜ + 1))) ≀ (𝐴 Β· (absβ€˜(πΉβ€˜π‘˜))))    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    β‡’   (πœ‘ β†’ (absβ€˜(πΉβ€˜π‘)) ≀ ((absβ€˜(πΉβ€˜π‘€)) Β· (𝐴↑(𝑁 βˆ’ 𝑀))))
 
15-Nov-2022cvgratnnlemnexp 11535 Lemma for cvgratnn 11542. (Contributed by Jim Kingdon, 15-Nov-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 1)    &   (πœ‘ β†’ 0 < 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (absβ€˜(πΉβ€˜(π‘˜ + 1))) ≀ (𝐴 Β· (absβ€˜(πΉβ€˜π‘˜))))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (absβ€˜(πΉβ€˜π‘)) ≀ ((absβ€˜(πΉβ€˜1)) Β· (𝐴↑(𝑁 βˆ’ 1))))
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