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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.

See the MPE Most Recent Proofs page for news and some useful links.

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Last updated on 8-Jan-2026 at 7:15 AM ET.
Recent Additions to the Intuitionistic Logic Explorer
DateLabelDescription
Theorem
 
31-Dec-2025pw0ss 15723 There are no inhabited subsets of the empty set. (Contributed by Jim Kingdon, 31-Dec-2025.)
{𝑠 ∈ 𝒫 ∅ ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅
 
31-Dec-2025df-ushgrm 15710 Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a special (non-simple, multiple, multi-) hypergraph for which the edge function 𝑒 is an injective (one-to-one) function into subsets of the set of vertices 𝑣, representing the (one or more) vertices incident to the edge. This definition corresponds to the definition of hypergraphs in section I.1 of [Bollobas] p. 7 (except that the empty set seems to be allowed to be an "edge") or section 1.10 of [Diestel] p. 27, where "E is a subset of [...] the power set of V, that is the set of all subsets of V" resp. "the elements of E are nonempty subsets (of any cardinality) of V". (Contributed by AV, 19-Jan-2020.) (Revised by Jim Kingdon, 31-Dec-2025.)
USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}}
 
29-Dec-2025df-uhgrm 15709 Define the class of all undirected hypergraphs. An undirected hypergraph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into the set of inhabited subsets of this set. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by Jim Kingdon, 29-Dec-2025.)
UHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}}
 
29-Dec-2025iedgex 15662 Applying the indexed edge function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
(𝐺 ∈ 𝑉 → (iEdg‘𝐺) ∈ V)
 
29-Dec-2025vtxex 15661 Applying the vertex function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
(𝐺 ∈ 𝑉 → (Vtx‘𝐺) ∈ V)
 
29-Dec-2025snmb 3755 A singleton is inhabited iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.) (Revised by Jim Kingdon, 29-Dec-2025.)
(𝐴 ∈ V ↔ ∃𝑥 𝑥 ∈ {𝐴})
 
27-Dec-2025lswex 11052 Existence of the last symbol. The last symbol of a word is a set. See lsw0g 11049 or lswcl 11051 if you want more specific results for empty or nonempty words, respectively. (Contributed by Jim Kingdon, 27-Dec-2025.)
(𝑊 ∈ Word 𝑉 → (lastS‘𝑊) ∈ V)
 
23-Dec-2025fzowrddc 11108 Decidability of whether a range of integers is a subset of a word's domain. (Contributed by Jim Kingdon, 23-Dec-2025.)
((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → DECID (𝐹..^𝐿) ⊆ dom 𝑆)
 
19-Dec-2025ccatclab 11058 The concatenation of words over two sets is a word over the union of those sets. (Contributed by Jim Kingdon, 19-Dec-2025.)
((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → (𝑆 ++ 𝑇) ∈ Word (𝐴 ∪ 𝐵))
 
18-Dec-2025lswwrd 11047 Extract the last symbol of a word. (Contributed by Alexander van der Vekens, 18-Mar-2018.) (Revised by Jim Kingdon, 18-Dec-2025.)
(𝑊 ∈ Word 𝑉 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
 
14-Dec-20252strstrndx 12994 A constructed two-slot structure not depending on the hard-coded index value of the base set. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 14-Dec-2025.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   (Base‘ndx) < 𝑁    &   𝑁 ∈ ℕ    ⇒   ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩)
 
12-Dec-2025funiedgdm2vald 15675 The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 12-Dec-2025.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝜑 → 𝐺 ∈ 𝑋)    &   (𝜑 → Fun (𝐺 ∖ {∅}))    &   (𝜑 → 𝐴 ≠ 𝐵)    &   (𝜑 → {𝐴, 𝐵} ⊆ dom 𝐺)    ⇒   (𝜑 → (iEdg‘𝐺) = (.ef‘𝐺))
 
11-Dec-2025funvtxdm2vald 15674 The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 11-Dec-2025.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝜑 → 𝐺 ∈ 𝑋)    &   (𝜑 → Fun (𝐺 ∖ {∅}))    &   (𝜑 → 𝐴 ≠ 𝐵)    &   (𝜑 → {𝐴, 𝐵} ⊆ dom 𝐺)    ⇒   (𝜑 → (Vtx‘𝐺) = (Base‘𝐺))
 
11-Dec-2025funiedgdm2domval 15673 The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by Jim Kingdon, 11-Dec-2025.)
((𝐺 ∈ 𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ 2o ≼ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))
 
11-Dec-2025funvtxdm2domval 15672 The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by Jim Kingdon, 11-Dec-2025.)
((𝐺 ∈ 𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ 2o ≼ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))
 
4-Dec-2025hash2en 10995 Two equivalent ways to say a set has two elements. (Contributed by Jim Kingdon, 4-Dec-2025.)
(𝑉 ≈ 2o ↔ (𝑉 ∈ Fin ∧ (♯‘𝑉) = 2))
 
30-Nov-2025nninfnfiinf 16034 An element of ℕ∞ which is not finite is infinite. (Contributed by Jim Kingdon, 30-Nov-2025.)
((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → 𝐴 = (𝑖 ∈ ω ↦ 1o))
 
27-Nov-2025psrelbasfi 14482 Simpler form of psrelbas 14481 when the index set is finite. (Contributed by Jim Kingdon, 27-Nov-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐾 = (Base‘𝑅)    &   (𝜑 → 𝐼 ∈ Fin)    &   𝐵 = (Base‘𝑆)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → 𝑋:(ℕ0 ↑𝑚 𝐼)⟶𝐾)
 
26-Nov-2025mplsubgfileminv 14506 Lemma for mplsubgfi 14507. The additive inverse of a polynomial is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝑈)    &   𝑁 = (invg‘𝑆)    ⇒   (𝜑 → (𝑁‘𝑋) ∈ 𝑈)
 
26-Nov-2025mplsubgfilemcl 14505 Lemma for mplsubgfi 14507. The sum of two polynomials is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝑈)    &   (𝜑 → 𝑌 ∈ 𝑈)    &    + = (+g‘𝑆)    ⇒   (𝜑 → (𝑋 + 𝑌) ∈ 𝑈)
 
25-Nov-2025nninfinfwlpo 7289 The point at infinity in ℕ∞ being isolated is equivalent to the Weak Limited Principle of Omniscience (WLPO). By isolated, we mean that the equality of that point with every other element of ℕ∞ is decidable. From an online post by Martin Escardo. By contrast, elements of ℕ∞ corresponding to natural numbers are isolated (nninfisol 7242). (Contributed by Jim Kingdon, 25-Nov-2025.)
(∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ ω ∈ WOmni)
 
23-Nov-2025psrbagfi 14479 A finite index set gives a simpler expression for finite bags. (Contributed by Jim Kingdon, 23-Nov-2025.)
𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   (𝐼 ∈ Fin → 𝐷 = (ℕ0 ↑𝑚 𝐼))
 
22-Nov-2025df-acnm 7294 Define a local and length-limited version of the axiom of choice. The definition of the predicate 𝑋 ∈ AC 𝐴 is that for all families of inhabited subsets of 𝑋 indexed on 𝐴 (i.e. functions 𝐴⟶{𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗𝑗 ∈ 𝑧}), there is a function which selects an element from each set in the family. (Contributed by Mario Carneiro, 31-Aug-2015.) Change nonempty to inhabited. (Revised by Jim Kingdon, 22-Nov-2025.)
AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
 
21-Nov-2025mplsubgfilemm 14504 Lemma for mplsubgfi 14507. There exists a polynomial. (Contributed by Jim Kingdon, 21-Nov-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Grp)    ⇒   (𝜑 → ∃𝑗 𝑗 ∈ 𝑈)
 
14-Nov-20252omapen 16007 Equinumerosity of (2o ↑𝑚 𝐴) and the set of decidable subsets of 𝐴. (Contributed by Jim Kingdon, 14-Nov-2025.)
(𝐴 ∈ 𝑉 → (2o ↑𝑚 𝐴) ≈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑥})
 
12-Nov-20252omap 16006 Mapping between (2o ↑𝑚 𝐴) and decidable subsets of 𝐴. (Contributed by Jim Kingdon, 12-Nov-2025.)
𝐹 = (𝑠 ∈ (2o ↑𝑚 𝐴) ↦ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})    ⇒   (𝐴 ∈ 𝑉 → 𝐹:(2o ↑𝑚 𝐴)–1-1-onto→{𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑥})
 
11-Nov-2025domomsubct 16012 A set dominated by ω is subcountable. (Contributed by Jim Kingdon, 11-Nov-2025.)
(𝐴 ≼ ω → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴))
 
10-Nov-2025prdsbaslemss 13150 Lemma for prdsbas 13152 and similar theorems. (Contributed by Jim Kingdon, 10-Nov-2025.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐴 = (𝐸‘𝑃)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ∈ ℕ    &   (𝜑 → 𝑇 ∈ 𝑋)    &   (𝜑 → {⟨(𝐸‘ndx), 𝑇⟩} ⊆ 𝑃)    ⇒   (𝜑 → 𝐴 = 𝑇)
 
5-Nov-2025fnmpl 14499 mPoly has universal domain. (Contributed by Jim Kingdon, 5-Nov-2025.)
mPoly Fn (V × V)
 
4-Nov-2025mplelbascoe 14498 Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    0 = (0g‘𝑅)    &   𝑈 = (Base‘𝑃)    ⇒   ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑋‘𝑏) = 0 ))))
 
4-Nov-2025mplbascoe 14497 Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    0 = (0g‘𝑅)    &   𝑈 = (Base‘𝑃)    ⇒   ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑈 = {𝑓 ∈ 𝐵 ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = 0 )})
 
4-Nov-2025mplvalcoe 14496 Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    0 = (0g‘𝑅)    &   𝑈 = {𝑓 ∈ 𝐵 ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = 0 )}    ⇒   ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑃 = (𝑆 ↾s 𝑈))
 
1-Nov-2025ficardon 7303 The cardinal number of a finite set is an ordinal. (Contributed by Jim Kingdon, 1-Nov-2025.)
(𝐴 ∈ Fin → (card‘𝐴) ∈ On)
 
31-Oct-2025bitsdc 12302 Whether a bit is set is decidable. (Contributed by Jim Kingdon, 31-Oct-2025.)
((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → DECID 𝑀 ∈ (bits‘𝑁))
 
28-Oct-2025nn0maxcl 11580 The maximum of two nonnegative integers is a nonnegative integer. (Contributed by Jim Kingdon, 28-Oct-2025.)
((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → sup({𝐴, 𝐵}, ℝ, < ) ∈ ℕ0)
 
28-Oct-2025qdcle 10396 Rational ≤ is decidable. (Contributed by Jim Kingdon, 28-Oct-2025.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 ≤ 𝐵)
 
17-Oct-2025plycoeid3 15273 Reconstruct a polynomial as an explicit sum of the coefficient function up to an index no smaller than the degree of the polynomial. (Contributed by Jim Kingdon, 17-Oct-2025.)
(𝜑 → 𝐷 ∈ ℕ0)    &   (𝜑 → 𝐴:ℕ0⟶ℂ)    &   (𝜑 → (𝐴 “ (ℤ≥‘(𝐷 + 1))) = {0})    &   (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝐷)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   (𝜑 → 𝑀 ∈ (ℤ≥‘𝐷))    &   (𝜑 → 𝑋 ∈ ℂ)    ⇒   (𝜑 → (𝐹‘𝑋) = Σ𝑗 ∈ (0...𝑀)((𝐴‘𝑗) · (𝑋↑𝑗)))
 
13-Oct-2025tpfidceq 7034 A triple is finite if it consists of elements of a class with decidable equality. (Contributed by Jim Kingdon, 13-Oct-2025.)
(𝜑 → 𝐴 ∈ 𝐷)    &   (𝜑 → 𝐵 ∈ 𝐷)    &   (𝜑 → 𝐶 ∈ 𝐷)    &   (𝜑 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 DECID 𝑥 = 𝑦)    ⇒   (𝜑 → {𝐴, 𝐵, 𝐶} ∈ Fin)
 
13-Oct-2025prfidceq 7032 A pair is finite if it consists of elements of a class with decidable equality. (Contributed by Jim Kingdon, 13-Oct-2025.)
(𝜑 → 𝐴 ∈ 𝐶)    &   (𝜑 → 𝐵 ∈ 𝐶)    &   (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 DECID 𝑥 = 𝑦)    ⇒   (𝜑 → {𝐴, 𝐵} ∈ Fin)
 
13-Oct-2025dcun 3571 The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.) (Revised by Jim Kingdon, 13-Oct-2025.)
(𝜑 → DECID 𝐶 ∈ 𝐴)    &   (𝜑 → DECID 𝐶 ∈ 𝐵)    ⇒   (𝜑 → DECID 𝐶 ∈ (𝐴 ∪ 𝐵))
 
9-Oct-2025dvdsfi 12605 A natural number has finitely many divisors. (Contributed by Jim Kingdon, 9-Oct-2025.)
(𝑁 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ∈ Fin)
 
7-Oct-2025df-mplcoe 14470 Define the subalgebra of the power series algebra generated by the variables; this is the polynomial algebra (the set of power series with finite degree).

The index set (which has an element for each variable) is 𝑖, the coefficients are in ring 𝑟, and for each variable there is a "degree" such that the coefficient is zero for a term where the powers are all greater than those degrees. (Degree is in quotes because there is no guarantee that coefficients below that degree are nonzero, as we do not assume decidable equality for 𝑟). (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 7-Oct-2025.)

mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑤⦌(𝑤 ↾s {𝑓 ∈ (Base‘𝑤) ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))}))
 
6-Oct-2025dvconstss 15214 Derivative of a constant function defined on an open set. (Contributed by Jim Kingdon, 6-Oct-2025.)
(𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   𝐽 = (𝐾 ↾t 𝑆)    &   𝐾 = (MetOpen‘(abs ∘ − ))    &   (𝜑 → 𝑋 ∈ 𝐽)    &   (𝜑 → 𝐴 ∈ ℂ)    ⇒   (𝜑 → (𝑆 D (𝑋 × {𝐴})) = (𝑋 × {0}))
 
6-Oct-2025dcfrompeirce 1470 The decidability of a proposition 𝜒 follows from a suitable instance of Peirce's law. Therefore, if we were to introduce Peirce's law as a general principle (without the decidability condition in peircedc 916), then we could prove that every proposition is decidable, giving us the classical system of propositional calculus (since Perice's law is itself classically valid). (Contributed by Adrian Ducourtial, 6-Oct-2025.)
(𝜑 ↔ (𝜒 ∨ ¬ 𝜒))    &   (𝜓 ↔ ⊥)    &   (((𝜑 → 𝜓) → 𝜑) → 𝜑)    ⇒   DECID 𝜒
 
6-Oct-2025dcfromcon 1469 The decidability of a proposition 𝜒 follows from a suitable instance of the principle of contraposition. Therefore, if we were to introduce contraposition as a general principle (without the decidability condition in condc 855), then we could prove that every proposition is decidable, giving us the classical system of propositional calculus (since the principle of contraposition is itself classically valid). (Contributed by Adrian Ducourtial, 6-Oct-2025.)
(𝜑 ↔ (𝜒 ∨ ¬ 𝜒))    &   (𝜓 ↔ ⊤)    &   ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))    ⇒   DECID 𝜒
 
6-Oct-2025dcfromnotnotr 1468 The decidability of a proposition 𝜓 follows from a suitable instance of double negation elimination (DNE). Therefore, if we were to introduce DNE as a general principle (without the decidability condition in notnotrdc 845), then we could prove that every proposition is decidable, giving us the classical system of propositional calculus (since DNE itself is classically valid). (Contributed by Adrian Ducourtial, 6-Oct-2025.)
(𝜑 ↔ (𝜓 ∨ ¬ 𝜓))    &   (¬ ¬ 𝜑 → 𝜑)    ⇒   DECID 𝜓
 
3-Oct-2025dvidre 15213 Real derivative of the identity function. (Contributed by Jim Kingdon, 3-Oct-2025.)
(ℝ D ( I ↾ ℝ)) = (ℝ × {1})
 
3-Oct-2025dvconstre 15212 Real derivative of a constant function. (Contributed by Jim Kingdon, 3-Oct-2025.)
(𝐴 ∈ ℂ → (ℝ D (ℝ × {𝐴})) = (ℝ × {0}))
 
3-Oct-2025dvidsslem 15209 Lemma for dvconstss 15214. Analogue of dvidlemap 15207 where 𝐹 is defined on an open subset of the real or complex numbers. (Contributed by Jim Kingdon, 3-Oct-2025.)
(𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   𝐽 = (𝐾 ↾t 𝑆)    &   𝐾 = (MetOpen‘(abs ∘ − ))    &   (𝜑 → 𝐹:𝑋⟶ℂ)    &   (𝜑 → 𝑋 ∈ 𝐽)    &   ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)) = 𝐵)    &   𝐵 ∈ ℂ    ⇒   (𝜑 → (𝑆 D 𝐹) = (𝑋 × {𝐵}))
 
3-Oct-2025dvidrelem 15208 Lemma for dvidre 15213 and dvconstre 15212. Analogue of dvidlemap 15207 for real numbers rather than complex numbers. (Contributed by Jim Kingdon, 3-Oct-2025.)
(𝜑 → 𝐹:ℝ⟶ℂ)    &   ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ 𝑧 # 𝑥)) → (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)) = 𝐵)    &   𝐵 ∈ ℂ    ⇒   (𝜑 → (ℝ D 𝐹) = (ℝ × {𝐵}))
 
28-Sep-2025metuex 14361 Applying metUnif yields a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
(𝐴 ∈ 𝑉 → (metUnif‘𝐴) ∈ V)
 
28-Sep-2025cndsex 14359 The standard distance function on the complex numbers is a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
(abs ∘ − ) ∈ V
 
25-Sep-2025cntopex 14360 The standard topology on the complex numbers is a set. (Contributed by Jim Kingdon, 25-Sep-2025.)
(MetOpen‘(abs ∘ − )) ∈ V
 
24-Sep-2025mopnset 14358 Getting a set by applying MetOpen. (Contributed by Jim Kingdon, 24-Sep-2025.)
(𝐷 ∈ 𝑉 → (MetOpen‘𝐷) ∈ V)
 
24-Sep-2025blfn 14357 The ball function has universal domain. (Contributed by Jim Kingdon, 24-Sep-2025.)
ball Fn V
 
23-Sep-2025elfzoext 10328 Membership of an integer in an extended open range of integers, extension added to the right. (Contributed by AV, 30-Apr-2020.) (Proof shortened by AV, 23-Sep-2025.)
((𝑍 ∈ (𝑀..^𝑁) ∧ 𝐼 ∈ ℕ0) → 𝑍 ∈ (𝑀..^(𝑁 + 𝐼)))
 
22-Sep-2025plycjlemc 15276 Lemma for plycj 15277. (Contributed by Mario Carneiro, 24-Jul-2014.) (Revised by Jim Kingdon, 22-Sep-2025.)
(𝜑 → 𝑁 ∈ ℕ0)    &   𝐺 = ((∗ ∘ 𝐹) ∘ ∗)    &   (𝜑 → 𝐴:ℕ0⟶(𝑆 ∪ {0}))    &   (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   (𝜑 → 𝐹 ∈ (Poly‘𝑆))    ⇒   (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((∗ ∘ 𝐴)‘𝑘) · (𝑧↑𝑘))))
 
20-Sep-2025plycolemc 15274 Lemma for plyco 15275. The result expressed as a sum, with a degree and coefficients for 𝐹 specified as hypotheses. (Contributed by Jim Kingdon, 20-Sep-2025.)
(𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐴:ℕ0⟶(𝑆 ∪ {0}))    &   (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   (𝜑 → 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑥↑𝑘))))    ⇒   (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))
 
18-Sep-2025elfzoextl 10327 Membership of an integer in an extended open range of integers, extension added to the left. (Contributed by AV, 31-Aug-2025.) Generalized by replacing the left border of the ranges. (Revised by SN, 18-Sep-2025.)
((𝑍 ∈ (𝑀..^𝑁) ∧ 𝐼 ∈ ℕ0) → 𝑍 ∈ (𝑀..^(𝐼 + 𝑁)))
 
16-Sep-2025lgsquadlemofi 15597 Lemma for lgsquad 15601. There are finitely many members of 𝑆 with odd first part. (Contributed by Jim Kingdon, 16-Sep-2025.)
(𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))    &   (𝜑 → 𝑃 ≠ 𝑄)    &   𝑀 = ((𝑃 − 1) / 2)    &   𝑁 = ((𝑄 − 1) / 2)    &   𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}    ⇒   (𝜑 → {𝑧 ∈ 𝑆 ∣ ¬ 2 ∥ (1st ‘𝑧)} ∈ Fin)
 
16-Sep-2025lgsquadlemsfi 15596 Lemma for lgsquad 15601. 𝑆 is finite. (Contributed by Jim Kingdon, 16-Sep-2025.)
(𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))    &   (𝜑 → 𝑃 ≠ 𝑄)    &   𝑀 = ((𝑃 − 1) / 2)    &   𝑁 = ((𝑄 − 1) / 2)    &   𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}    ⇒   (𝜑 → 𝑆 ∈ Fin)
 
16-Sep-2025opabfi 7042 Finiteness of an ordered pair abstraction which is a decidable subset of finite sets. (Contributed by Jim Kingdon, 16-Sep-2025.)
𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐵 ∈ Fin)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 DECID 𝜓)    ⇒   (𝜑 → 𝑆 ∈ Fin)
 
13-Sep-2025uchoice 6230 Principle of unique choice. This is also called non-choice. The name choice results in its similarity to something like acfun 7326 (with the key difference being the change of ∃ to ∃!) but unique choice in fact follows from the axiom of collection and our other axioms. This is somewhat similar to Corollary 3.9.2 of [HoTT], p. (varies) but is better described by the paragraph at the end of Section 3.9 which starts "A similar issue arises in set-theoretic mathematics". (Contributed by Jim Kingdon, 13-Sep-2025.)
((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))
 
11-Sep-2025expghmap 14413 Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) (Revised by AV, 10-Jun-2019.) (Revised by Jim Kingdon, 11-Sep-2025.)
𝑀 = (mulGrp‘ℂfld)    &   𝑈 = (𝑀 ↾s {𝑧 ∈ ℂ ∣ 𝑧 # 0})    ⇒   ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈))
 
11-Sep-2025cnfldui 14395 The invertible complex numbers are exactly those apart from zero. This is recapb 8751 but expressed in terms of ℂfld. (Contributed by Jim Kingdon, 11-Sep-2025.)
{𝑧 ∈ ℂ ∣ 𝑧 # 0} = (Unit‘ℂfld)
 
9-Sep-2025gsumfzfsumlemm 14393 Lemma for gsumfzfsum 14394. The case where the sum is inhabited. (Contributed by Jim Kingdon, 9-Sep-2025.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐵 ∈ ℂ)    ⇒   (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑁)𝐵)
 
9-Sep-2025gsumfzfsumlem0 14392 Lemma for gsumfzfsum 14394. The case where the sum is empty. (Contributed by Jim Kingdon, 9-Sep-2025.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝑁 < 𝑀)    ⇒   (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑁)𝐵)
 
9-Sep-2025gsumfzmhm2 13724 Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 9-Sep-2025.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝐻 ∈ Mnd)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ (𝐺 MndHom 𝐻))    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑋 ∈ 𝐵)    &   (𝑥 = 𝑋 → 𝐶 = 𝐷)    &   (𝑥 = (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) → 𝐶 = 𝐸)    ⇒   (𝜑 → (𝐻 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐷)) = 𝐸)
 
8-Sep-2025gsumfzmhm 13723 Apply a monoid homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 8-Sep-2025.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝐻 ∈ Mnd)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻))    &   (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    ⇒   (𝜑 → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
 
8-Sep-20255ndvds6 12290 5 does not divide 6. (Contributed by AV, 8-Sep-2025.)
¬ 5 ∥ 6
 
8-Sep-20255ndvds3 12289 5 does not divide 3. (Contributed by AV, 8-Sep-2025.)
¬ 5 ∥ 3
 
6-Sep-2025gsumfzconst 13721 Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Jim Kingdon, 6-Sep-2025.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    ⇒   ((𝐺 ∈ Mnd ∧ 𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) = (((𝑁 − 𝑀) + 1) · 𝑋))
 
31-Aug-2025gsumfzmptfidmadd 13719 The sum of two group sums expressed as mappings with finite domain. (Contributed by AV, 23-Jul-2019.) (Revised by Jim Kingdon, 31-Aug-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐶 ∈ 𝐵)    &   ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐷 ∈ 𝐵)    &   𝐹 = (𝑥 ∈ (𝑀...𝑁) ↦ 𝐶)    &   𝐻 = (𝑥 ∈ (𝑀...𝑁) ↦ 𝐷)    ⇒   (𝜑 → (𝐺 Σg (𝑥 ∈ (𝑀...𝑁) ↦ (𝐶 + 𝐷))) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
 
30-Aug-2025gsumfzsubmcl 13718 Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 30-Aug-2025.)
(𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝑆 ∈ (SubMnd‘𝐺))    &   (𝜑 → 𝐹:(𝑀...𝑁)⟶𝑆)    ⇒   (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆)
 
30-Aug-2025seqm1g 10626 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 30-Aug-2025.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))    &   (𝜑 → + ∈ 𝑉)    &   (𝜑 → 𝐹 ∈ 𝑊)    ⇒   (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘𝑁)))
 
29-Aug-2025seqf1og 10673 Rearrange a sum via an arbitrary bijection on (𝑀...𝑁). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Jim Kingdon, 29-Aug-2025.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐶 ⊆ 𝑆)    &   (𝜑 → + ∈ 𝑉)    &   (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘𝑥) ∈ 𝐶)    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = (𝐺‘(𝐹‘𝑘)))    &   (𝜑 → 𝐺 ∈ 𝑊)    &   (𝜑 → 𝐻 ∈ 𝑋)    ⇒   (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
 
25-Aug-2025irrmulap 9776 The product of an irrational with a nonzero rational is irrational. By irrational we mean apart from any rational number. For a similar theorem with not rational in place of irrational, see irrmul 9775. (Contributed by Jim Kingdon, 25-Aug-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → ∀𝑞 ∈ ℚ 𝐴 # 𝑞)    &   (𝜑 → 𝐵 ∈ ℚ)    &   (𝜑 → 𝐵 ≠ 0)    &   (𝜑 → 𝑄 ∈ ℚ)    ⇒   (𝜑 → (𝐴 · 𝐵) # 𝑄)
 
19-Aug-2025seqp1g 10618 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 19-Aug-2025.)
((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐹 ∈ 𝑉 ∧ + ∈ 𝑊) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))))
 
19-Aug-2025seq1g 10615 Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 19-Aug-2025.)
((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉 ∧ + ∈ 𝑊) → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
 
18-Aug-2025iswrdiz 11008 A zero-based sequence is a word. In iswrdinn0 11006 we can specify a length as an nonnegative integer. However, it will occasionally be helpful to allow a negative length, as well as zero, to specify an empty sequence. (Contributed by Jim Kingdon, 18-Aug-2025.)
((𝑊:(0..^𝐿)⟶𝑆 ∧ 𝐿 ∈ ℤ) → 𝑊 ∈ Word 𝑆)
 
16-Aug-2025gsumfzcl 13375 Closure of a finite group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 16-Aug-2025.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    ⇒   (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵)
 
16-Aug-2025iswrdinn0 11006 A zero-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 16-Aug-2025.)
((𝑊:(0..^𝐿)⟶𝑆 ∧ 𝐿 ∈ ℕ0) → 𝑊 ∈ Word 𝑆)
 
15-Aug-2025gsumfzz 13371 Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 15-Aug-2025.)
0 = (0g‘𝐺)    ⇒   ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = 0 )
 
14-Aug-2025gsumfzval 13267 An expression for Σg when summing over a finite set of sequential integers. (Contributed by Jim Kingdon, 14-Aug-2025.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    ⇒   (𝜑 → (𝐺 Σg 𝐹) = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)))
 
13-Aug-2025znidom 14463 The ℤ/nℤ structure is an integral domain when 𝑛 is prime. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by Jim Kingdon, 13-Aug-2025.)
𝑌 = (ℤ/nℤ‘𝑁)    ⇒   (𝑁 ∈ ℙ → 𝑌 ∈ IDomn)
 
12-Aug-2025rrgmex 14067 A structure whose set of left-regular elements is inhabited is a set. (Contributed by Jim Kingdon, 12-Aug-2025.)
𝐸 = (RLReg‘𝑅)    ⇒   (𝐴 ∈ 𝐸 → 𝑅 ∈ V)
 
10-Aug-2025gausslemma2dlem1cl 15580 Lemma for gausslemma2dlem1 15582. Closure of the body of the definition of 𝑅. (Contributed by Jim Kingdon, 10-Aug-2025.)
(𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   𝐻 = ((𝑃 − 1) / 2)    &   𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    &   (𝜑 → 𝐴 ∈ (1...𝐻))    ⇒   (𝜑 → if((𝐴 · 2) < (𝑃 / 2), (𝐴 · 2), (𝑃 − (𝐴 · 2))) ∈ ℤ)
 
9-Aug-2025gausslemma2dlem1f1o 15581 Lemma for gausslemma2dlem1 15582. (Contributed by Jim Kingdon, 9-Aug-2025.)
(𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   𝐻 = ((𝑃 − 1) / 2)    &   𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    ⇒   (𝜑 → 𝑅:(1...𝐻)–1-1-onto→(1...𝐻))
 
7-Aug-2025qdclt 10395 Rational < is decidable. (Contributed by Jim Kingdon, 7-Aug-2025.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 < 𝐵)
 
22-Jul-2025ivthdich 15169 The intermediate value theorem implies real number dichotomy. Because real number dichotomy (also known as analytic LLPO) is a constructive taboo, this means we will be unable to prove the intermediate value theorem as stated here (although versions with additional conditions, such as ivthinc 15159 for strictly monotonic functions, can be proved).

The proof is via a function which we call the hover function and which is also described in Section 5.1 of [Bauer], p. 493. Consider any real number 𝑧. We want to show that 𝑧 ≤ 0 ∨ 0 ≤ 𝑧. Because of hovercncf 15162, hovera 15163, and hoverb 15164, we are able to apply the intermediate value theorem to get a value 𝑐 such that the hover function at 𝑐 equals 𝑧. By axltwlin 8147, 𝑐 < 1 or 0 < 𝑐, and that leads to 𝑧 ≤ 0 by hoverlt1 15165 or 0 ≤ 𝑧 by hovergt0 15166. (Contributed by Jim Kingdon and Mario Carneiro, 22-Jul-2025.)

(∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))) → ∀𝑟 ∈ ℝ ∀𝑠 ∈ ℝ (𝑟 ≤ 𝑠 ∨ 𝑠 ≤ 𝑟))
 
22-Jul-2025dich0 15168 Real number dichotomy stated in terms of two real numbers or a real number and zero. (Contributed by Jim Kingdon, 22-Jul-2025.)
(∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ↔ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
 
22-Jul-2025ivthdichlem 15167 Lemma for ivthdich 15169. The result, with a few notational conveniences. (Contributed by Jim Kingdon, 22-Jul-2025.)
𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))    &   (𝜑 → 𝑍 ∈ ℝ)    &   (𝜑 → ∀𝑓(𝑓 ∈ (ℝ–cn→ℝ) → ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((𝑎 < 𝑏 ∧ (𝑓‘𝑎) < 0 ∧ 0 < (𝑓‘𝑏)) → ∃𝑥 ∈ ℝ (𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ∧ (𝑓‘𝑥) = 0))))    ⇒   (𝜑 → (𝑍 ≤ 0 ∨ 0 ≤ 𝑍))
 
22-Jul-2025hovergt0 15166 The hover function evaluated at a point greater than zero. (Contributed by Jim Kingdon, 22-Jul-2025.)
𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))    ⇒   ((𝐶 ∈ ℝ ∧ 0 < 𝐶) → 0 ≤ (𝐹‘𝐶))
 
22-Jul-2025hoverlt1 15165 The hover function evaluated at a point less than one. (Contributed by Jim Kingdon, 22-Jul-2025.)
𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))    ⇒   ((𝐶 ∈ ℝ ∧ 𝐶 < 1) → (𝐹‘𝐶) ≤ 0)
 
21-Jul-2025hoverb 15164 A point at which the hover function is greater than a given value. (Contributed by Jim Kingdon, 21-Jul-2025.)
𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))    ⇒   (𝑍 ∈ ℝ → 𝑍 < (𝐹‘(𝑍 + 2)))
 
21-Jul-2025hovera 15163 A point at which the hover function is less than a given value. (Contributed by Jim Kingdon, 21-Jul-2025.)
𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))    ⇒   (𝑍 ∈ ℝ → (𝐹‘(𝑍 − 1)) < 𝑍)
 
21-Jul-2025rexeqtrrdv 2714 Substitution of equal classes into a restricted existential quantifier. (Contributed by Matthew House, 21-Jul-2025.)
(𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    &   (𝜑 → 𝐵 = 𝐴)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
 
21-Jul-2025raleqtrrdv 2713 Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.)
(𝜑 → ∀𝑥 ∈ 𝐴 𝜓)    &   (𝜑 → 𝐵 = 𝐴)    ⇒   (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
 
21-Jul-2025rexeqtrdv 2712 Substitution of equal classes into a restricted existential quantifier. (Contributed by Matthew House, 21-Jul-2025.)
(𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    &   (𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
 
21-Jul-2025raleqtrdv 2711 Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.)
(𝜑 → ∀𝑥 ∈ 𝐴 𝜓)    &   (𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
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