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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 11-May-2026 at 7:16 AM ET.
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DateLabelDescription
Theorem
 
5-May-2026fmelpw1o 7470 With a formula 𝜑 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 ⊥, by nndc 858, which translate to 1o and ∅ respectively by iftrue 3611 and iffalse 3614, giving pwtrufal 16658).

As proved in if0ab 3607, the associated element of 𝒫 1o is the extension, in 𝒫 1o, of the formula 𝜑. (Contributed by BJ, 15-Aug-2024.) (Proof shortened by BJ, 5-May-2026.)

if(𝜑, 1o, ∅) ∈ 𝒫 1o
 
5-May-2026if0elpw 4250 A conditional class with the False alternative being sent to the empty class is an element of the powerset of the class corresponding to the True alternative when that class is a set. This statement requires fewer axioms than the general case ifelpwung 4580. (Contributed by BJ, 5-May-2026.)
(𝐴 ∈ 𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴)
 
5-May-2026if0ss 3608 A conditional class with the False alternative being sent to the empty class is included in the class corresponding to the True alternative. (Contributed by BJ, 5-May-2026.)
if(𝜑, 𝐴, ∅) ⊆ 𝐴
 
27-Apr-2026repiecef 16699 Piecewise definition on the reals yields a function. The function agrees with 𝐹 and 𝐺 on their respective parts of the real line; see repiecele0 16697 and repiecege0 16698. From an online post by James E Hanson. The construction was published in Martín Hötzel Escardó, "Effective and sequential definition by cases on the reals via infinite signed-digit numerals", Electronic Notes in Theoretical Computer Science 10 (1998), page 2, https://martinescardo.github.io/papers/lexnew.pdf. 16698 (Contributed by Jim Kingdon, 27-Apr-2026.)
(𝜑 → 𝐹:(-∞(,]0)⟶ℝ)    &   (𝜑 → 𝐺:(0[,)+∞)⟶ℝ)    &   (𝜑 → (𝐹‘0) = (𝐺‘0))    &   𝐻 = (𝑥 ∈ ℝ ↦ (((𝐹‘inf({𝑥, 0}, ℝ, < )) + (𝐺‘sup({𝑥, 0}, ℝ, < ))) − (𝐹‘0)))    ⇒   (𝜑 → 𝐻:ℝ⟶ℝ)
 
27-Apr-2026repiecege0 16698 Piecewise definition on the reals agrees with the nonnegative part of the definition. See repiecef 16699 for more on this construction. (Contributed by Jim Kingdon, 27-Apr-2026.)
(𝜑 → 𝐹:(-∞(,]0)⟶ℝ)    &   (𝜑 → 𝐺:(0[,)+∞)⟶ℝ)    &   (𝜑 → (𝐹‘0) = (𝐺‘0))    &   𝐻 = (𝑥 ∈ ℝ ↦ (((𝐹‘inf({𝑥, 0}, ℝ, < )) + (𝐺‘sup({𝑥, 0}, ℝ, < ))) − (𝐹‘0)))    ⇒   ((𝜑 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐻‘𝐴) = (𝐺‘𝐴))
 
27-Apr-2026repiecele0 16697 Piecewise definition on the reals agrees with the nonpositive part of the definition. See repiecef 16699 for more on this construction. (Contributed by Jim Kingdon, 27-Apr-2026.)
(𝜑 → 𝐹:(-∞(,]0)⟶ℝ)    &   (𝜑 → 𝐺:(0[,)+∞)⟶ℝ)    &   (𝜑 → (𝐹‘0) = (𝐺‘0))    &   𝐻 = (𝑥 ∈ ℝ ↦ (((𝐹‘inf({𝑥, 0}, ℝ, < )) + (𝐺‘sup({𝑥, 0}, ℝ, < ))) − (𝐹‘0)))    ⇒   ((𝜑 ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → (𝐻‘𝐴) = (𝐹‘𝐴))
 
27-Apr-2026repiecelem 16696 Lemma for repiecele0 16697, repiecege0 16698, and repiecef 16699. The function 𝐻 is defined everywhere. (Contributed by Jim Kingdon, 27-Apr-2026.)
(𝜑 → 𝐹:(-∞(,]0)⟶ℝ)    &   (𝜑 → 𝐺:(0[,)+∞)⟶ℝ)    &   (𝜑 → (𝐹‘0) = (𝐺‘0))    &   𝐻 = (𝑥 ∈ ℝ ↦ (((𝐹‘inf({𝑥, 0}, ℝ, < )) + (𝐺‘sup({𝑥, 0}, ℝ, < ))) − (𝐹‘0)))    ⇒   ((𝜑 ∧ 𝐴 ∈ ℝ) → (((𝐹‘inf({𝐴, 0}, ℝ, < )) + (𝐺‘sup({𝐴, 0}, ℝ, < ))) − (𝐹‘0)) ∈ ℝ)
 
24-Apr-2026qdiff 16720 The rationals are exactly those reals for which there exist two distinct rationals that are the same distance from the original number. Similar to apdiff 16719 but by stating the result positively we can completely sidestep the issue of not equal versus apart in the statement of the result. From an online post by Ingo Blechschmidt. (Contributed by Jim Kingdon, 24-Apr-2026.)
(𝐴 ∈ ℝ → (𝐴 ∈ ℚ ↔ ∃𝑞 ∈ ℚ ∃𝑟 ∈ ℚ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))))
 
23-Apr-2026exmidpeirce 16668 Excluded middle is equivalent to Peirce's law. Read an element of 𝒫 1o as being a truth value and 𝑥 = 1o being that 𝑥 is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1o, see dcfrompeirce 1494. (Contributed by Jim Kingdon, 23-Apr-2026.)
(EXMID ↔ ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o))
 
22-Apr-2026exmidcon 16667 Excluded middle is equivalent to the form of contraposition which removes negation. Read an element of 𝒫 1o as being a truth value and 𝑥 = 1o being that 𝑥 is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1o, see dcfromcon 1493. (Contributed by Jim Kingdon, 22-Apr-2026.)
(EXMID ↔ ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o((¬ 𝑦 = 1o → ¬ 𝑥 = 1o) → (𝑥 = 1o → 𝑦 = 1o)))
 
22-Apr-2026exmidnotnotr 16666 Excluded middle is equivalent to double negation elimination. Read an element of 𝒫 1o as being a truth value and 𝑥 = 1o being that 𝑥 is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1o, see dcfromnotnotr 1492. (Contributed by Jim Kingdon, 22-Apr-2026.)
(EXMID ↔ ∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
 
18-Apr-2026hashtpglem 11116 Lemma for hashtpg 11117. This is one of the three not-equal conclusions required for the reverse direction. (Contributed by Jim Kingdon, 18-Apr-2026.)
(𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑊)    &   (𝜑 → (♯‘{𝐴, 𝐵, 𝐶}) = 3)    ⇒   (𝜑 → 𝐵 ≠ 𝐶)
 
17-Apr-2026hashtpgim 11115 The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 18-Sep-2021.) (Revised by Jim Kingdon, 17-Apr-2026.)
((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) → (♯‘{𝐴, 𝐵, 𝐶}) = 3))
 
14-Apr-2026depind 16389 Theorem related to a dependently typed induction principle in type theory. (Contributed by Matthew House, 14-Apr-2026.)
(𝜑 → 𝑃:ℕ0⟶V)    &   (𝜑 → 𝐴 ∈ (𝑃‘0))    &   (𝜑 → ∀𝑛 ∈ ℕ0 (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)))    ⇒   (𝜑 → ∃!𝑓 ∈ X 𝑛 ∈ ℕ0 (𝑃‘𝑛)((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛))))
 
14-Apr-2026depindlem3 16388 Lemma for depind 16389. (Contributed by Matthew House, 14-Apr-2026.)
(𝜑 → 𝑃:ℕ0⟶V)    &   (𝜑 → 𝐴 ∈ (𝑃‘0))    &   (𝜑 → ∀𝑛 ∈ ℕ0 (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)))    &   𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))    ⇒   (𝜑 → ∀𝑓 ∈ X 𝑛 ∈ ℕ0 (𝑃‘𝑛)(((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛))) → 𝑓 = 𝐹))
 
14-Apr-2026depindlem2 16387 Lemma for depind 16389. (Contributed by Matthew House, 14-Apr-2026.)
(𝜑 → 𝑃:ℕ0⟶V)    &   (𝜑 → 𝐴 ∈ (𝑃‘0))    &   (𝜑 → ∀𝑛 ∈ ℕ0 (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)))    &   𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))    ⇒   (𝜑 → 𝐹 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛))
 
14-Apr-2026depindlem1 16386 Lemma for depind 16389. (Contributed by Matthew House, 14-Apr-2026.)
(𝜑 → 𝑃:ℕ0⟶V)    &   (𝜑 → 𝐴 ∈ (𝑃‘0))    &   (𝜑 → ∀𝑛 ∈ ℕ0 (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)))    &   𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))    ⇒   (𝜑 → (𝐹:ℕ0⟶V ∧ (𝐹‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛))))
 
8-Apr-2026gfsumcl 16755 Closure of a finite group sum. (Contributed by Jim Kingdon, 8-Apr-2026.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   (𝜑 → (𝐺 Σgf 𝐹) ∈ 𝐵)
 
4-Apr-2026gsumsplit0 13956 Splitting off the rightmost summand of a group sum (even if it is the only summand). Similar to gsumsplit1r 13504 except that 𝑁 can equal 𝑀 − 1. (Contributed by Jim Kingdon, 4-Apr-2026.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 − 1)))    &   (𝜑 → 𝐹:(𝑀...(𝑁 + 1))⟶𝐵)    ⇒   (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ (𝑀...𝑁))) + (𝐹‘(𝑁 + 1))))
 
4-Apr-2026fzf1o 11959 A finite set can be enumerated by integers starting at one. (Contributed by Jim Kingdon, 4-Apr-2026.)
(𝐴 ∈ Fin → ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
 
3-Apr-2026gfsump1 16754 Splitting off one element from a finite group sum. This would typically used in a proof by induction. (Contributed by Jim Kingdon, 3-Apr-2026.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝐹:(𝑌 ∪ {𝑍})⟶𝐵)    &   (𝜑 → 𝑌 ∈ Fin)    &   (𝜑 → 𝑍 ∈ 𝑉)    &   (𝜑 → ¬ 𝑍 ∈ 𝑌)    ⇒   (𝜑 → (𝐺 Σgf 𝐹) = ((𝐺 Σgf (𝐹 ↾ 𝑌)) + (𝐹‘𝑍)))
 
2-Apr-2026gfsumsn 16753 Group sum of a singleton. (Contributed by Jim Kingdon, 2-Apr-2026.)
𝐵 = (Base‘𝐺)    &   (𝑘 = 𝑀 → 𝐴 = 𝐶)    ⇒   ((𝐺 ∈ CMnd ∧ 𝑀 ∈ 𝑉 ∧ 𝐶 ∈ 𝐵) → (𝐺 Σgf (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)
 
31-Mar-2026sspw1or2 7408 The set of subsets of a given set with one or two elements can be expressed as elements of the power set or as inhabited elements of the power set. (Contributed by Jim Kingdon, 31-Mar-2026.)
{𝑥 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} = {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
 
28-Mar-2026imaf1fi 7130 The image of a finite set under a one-to-one mapping is finite. (Contributed by Jim Kingdon, 28-Mar-2026.)
((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin)
 
26-Mar-2026gsumgfsumlem 16751 Shifting the indexes of a group sum indexed by consecutive integers. (Contributed by Jim Kingdon, 26-Mar-2026.)
𝐵 = (Base‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    &   𝑆 = (𝑗 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀)))    ⇒   (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝑆)))
 
26-Mar-2026gfsum0 16750 An empty finite group sum is the identity. (Contributed by Jim Kingdon, 26-Mar-2026.)
(𝐺 ∈ CMnd → (𝐺 Σgf ∅) = (0g‘𝐺))
 
25-Mar-2026gsumgfsum 16752 On an integer range, Σg and Σgf agree. (Contributed by Jim Kingdon, 25-Mar-2026.)
𝐵 = (Base‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    ⇒   (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σgf 𝐹))
 
25-Mar-2026gsumgfsum1 16749 On an integer range starting at one, Σg and Σgf agree. (Contributed by Jim Kingdon, 25-Mar-2026.)
𝐵 = (Base‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐹:(1...𝑁)⟶𝐵)    ⇒   (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σgf 𝐹))
 
24-Mar-2026gfsumval 16748 Value of the finite group sum over an unordered finite set. (Contributed by Jim Kingdon, 24-Mar-2026.)
𝐵 = (Base‘𝑊)    &   (𝜑 → 𝑊 ∈ CMnd)    &   (𝜑 → 𝐹:𝐴⟶𝐵)    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐺:(1...(♯‘𝐴))–1-1-onto→𝐴)    ⇒   (𝜑 → (𝑊 Σgf 𝐹) = (𝑊 Σg (𝐹 ∘ 𝐺)))
 
23-Mar-2026df-gfsum 16747 Define the finite group sum (iterated sum) over an unordered finite set. As currently defined, df-igsum 13365 is indexed by consecutive integers, but in the case of a commutative monoid, the order of the sum doesn't matter and we can define a sum indexed by any finite set without needing to specify an order. (Contributed by Jim Kingdon, 23-Mar-2026.)
Σgf = (𝑤 ∈ CMnd, 𝑓 ∈ V ↦ (℩𝑥(dom 𝑓 ∈ Fin ∧ ∃𝑔(𝑔:(1...(♯‘dom 𝑓))–1-1-onto→dom 𝑓 ∧ 𝑥 = (𝑤 Σg (𝑓 ∘ 𝑔))))))
 
20-Mar-2026exmidssfi 7136 Excluded middle is equivalent to any subset of a finite set being finite. Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 20-Mar-2026.)
(EXMID ↔ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin))
 
18-Mar-2026umgr1een 16005 A graph with one non-loop edge is a multigraph. (Contributed by Jim Kingdon, 18-Mar-2026.)
(𝜑 → 𝐾 ∈ 𝑋)    &   (𝜑 → 𝑉 ∈ 𝑌)    &   (𝜑 → 𝐸 ∈ 𝒫 𝑉)    &   (𝜑 → 𝐸 ≈ 2o)    ⇒   (𝜑 → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UMGraph)
 
18-Mar-2026upgr1een 16004 A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 16001 for a different way of specifying a graph with one edge. (Contributed by Jim Kingdon, 18-Mar-2026.)
(𝜑 → 𝐾 ∈ 𝑋)    &   (𝜑 → 𝑉 ∈ 𝑌)    &   (𝜑 → 𝐸 ∈ 𝒫 𝑉)    &   (𝜑 → 𝐸 ≈ 2o)    ⇒   (𝜑 → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UPGraph)
 
14-Mar-2026trlsex 16267 The class of trails on a graph is a set. (Contributed by Jim Kingdon, 14-Mar-2026.)
(𝐺 ∈ 𝑉 → (Trails‘𝐺) ∈ V)
 
13-Mar-2026eupthv 16326 The classes involved in a Eulerian path are sets. (Contributed by Jim Kingdon, 13-Mar-2026.)
(𝐹(EulerPaths‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
 
13-Mar-20261hevtxdg0fi 16187 The vertex degree of vertex 𝐷 in a finite pseudograph 𝐺 with only one edge 𝐸 is 0 if 𝐷 is not incident with the edge 𝐸. (Contributed by AV, 2-Mar-2021.) (Revised by Jim Kingdon, 13-Mar-2026.)
(𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   (𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑 → 𝐴 ∈ 𝑋)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑉 ∈ Fin)    &   (𝜑 → 𝐺 ∈ UPGraph)    &   (𝜑 → 𝐸 ∈ 𝑌)    &   (𝜑 → 𝐷 ∉ 𝐸)    ⇒   (𝜑 → ((VtxDeg‘𝐺)‘𝐷) = 0)
 
11-Mar-2026en1hash 11068 A set equinumerous to the ordinal one has size 1 . (Contributed by Jim Kingdon, 11-Mar-2026.)
(𝐴 ≈ 1o → (♯‘𝐴) = 1)
 
4-Mar-2026elmpom 6408 If a maps-to operation is inhabited, the first class it is defined with is inhabited. (Contributed by Jim Kingdon, 4-Mar-2026.)
𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   (𝐷 ∈ 𝐹 → ∃𝑧 𝑧 ∈ 𝐴)
 
22-Feb-2026isclwwlkni 16287 A word over the set of vertices representing a closed walk of a fixed length. (Contributed by Jim Kingdon, 22-Feb-2026.)
(𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁))
 
21-Feb-2026clwwlkex 16278 Existence of the set of closed walks (represented by words). (Contributed by Jim Kingdon, 21-Feb-2026.)
(𝐺 ∈ 𝑉 → (ClWWalks‘𝐺) ∈ V)
 
17-Feb-2026vtxdgfif 16173 In a finite graph, the vertex degree function is a function from vertices to nonnegative integers. (Contributed by Jim Kingdon, 17-Feb-2026.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝑉 ∈ Fin)    &   (𝜑 → 𝐺 ∈ UPGraph)    ⇒   (𝜑 → (VtxDeg‘𝐺):𝑉⟶ℕ0)
 
16-Feb-2026vtxlpfi 16170 In a finite graph, the number of loops from a given vertex is finite. (Contributed by Jim Kingdon, 16-Feb-2026.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝑉 ∈ Fin)    &   (𝜑 → 𝑈 ∈ 𝑉)    &   (𝜑 → 𝐺 ∈ UPGraph)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}} ∈ Fin)
 
16-Feb-2026vtxedgfi 16169 In a finite graph, the number of edges from a given vertex is finite. (Contributed by Jim Kingdon, 16-Feb-2026.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝑉 ∈ Fin)    &   (𝜑 → 𝑈 ∈ 𝑉)    &   (𝜑 → 𝐺 ∈ UPGraph)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)} ∈ Fin)
 
15-Feb-2026eqsndc 7100 Decidability of equality between a finite subset of a set with decidable equality, and a singleton whose element is an element of the larger set. (Contributed by Jim Kingdon, 15-Feb-2026.)
(𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → 𝐴 ∈ Fin)    ⇒   (𝜑 → DECID 𝐴 = {𝑋})
 
14-Feb-2026pw1ninf 16650 The powerset of 1o is not infinite. Since we cannot prove it is finite (see pw1fin 7107), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7103. (Contributed by Jim Kingdon, 14-Feb-2026.)
¬ ω ≼ 𝒫 1o
 
14-Feb-2026pw1ndom3 16649 The powerset of 1o does not dominate 3o. This is another way of saying that 𝒫 1o does not have three elements (like pwntru 4291). (Contributed by Steven Nguyen and Jim Kingdon, 14-Feb-2026.)
¬ 3o ≼ 𝒫 1o
 
14-Feb-2026pw1ndom3lem 16648 Lemma for pw1ndom3 16649. (Contributed by Jim Kingdon, 14-Feb-2026.)
(𝜑 → 𝑋 ∈ 𝒫 1o)    &   (𝜑 → 𝑌 ∈ 𝒫 1o)    &   (𝜑 → 𝑍 ∈ 𝒫 1o)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝑋 ≠ 𝑍)    &   (𝜑 → 𝑌 ≠ 𝑍)    ⇒   (𝜑 → 𝑋 = ∅)
 
12-Feb-2026pw1dceq 16665 The powerset of 1o having decidable equality is equivalent to excluded middle. (Contributed by Jim Kingdon, 12-Feb-2026.)
(EXMID ↔ ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦)
 
12-Feb-20263dom 16647 A set that dominates ordinal 3 has at least 3 different members. (Contributed by Jim Kingdon, 12-Feb-2026.)
(3o ≼ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧))
 
11-Feb-2026elssdc 7099 Membership in a finite subset of a set with decidable equality is decidable. (Contributed by Jim Kingdon, 11-Feb-2026.)
(𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → 𝐴 ∈ Fin)    ⇒   (𝜑 → DECID 𝑋 ∈ 𝐴)
 
10-Feb-2026vtxdgfifival 16171 The degree of a vertex for graphs with finite vertex and edge sets. (Contributed by Jim Kingdon, 10-Feb-2026.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝑉 ∈ Fin)    &   (𝜑 → 𝑈 ∈ 𝑉)    &   (𝜑 → 𝐺 ∈ UPGraph)    ⇒   (𝜑 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))
 
10-Feb-2026fidcen 7093 Equinumerosity of finite sets is decidable. (Contributed by Jim Kingdon, 10-Feb-2026.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → DECID 𝐴 ≈ 𝐵)
 
8-Feb-2026wlkvtxm 16220 A graph with a walk has at least one vertex. (Contributed by Jim Kingdon, 8-Feb-2026.)
𝑉 = (Vtx‘𝐺)    ⇒   (𝐹(Walks‘𝐺)𝑃 → ∃𝑥 𝑥 ∈ 𝑉)
 
7-Feb-2026trlsv 16264 The classes involved in a trail are sets. (Contributed by Jim Kingdon, 7-Feb-2026.)
(𝐹(Trails‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
 
7-Feb-2026wlkex 16205 The class of walks on a graph is a set. (Contributed by Jim Kingdon, 7-Feb-2026.)
(𝐺 ∈ 𝑉 → (Walks‘𝐺) ∈ V)
 
3-Feb-2026dom1oi 7008 A set with an element dominates one. (Contributed by Jim Kingdon, 3-Feb-2026.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → 1o ≼ 𝐴)
 
2-Feb-2026edginwlkd 16235 The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 9-Dec-2021.) (Revised by Jim Kingdon, 2-Feb-2026.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑 → 𝐹 ∈ Word dom 𝐼)    &   (𝜑 → 𝐾 ∈ (0..^(♯‘𝐹)))    &   (𝜑 → 𝐺 ∈ 𝑉)    ⇒   (𝜑 → (𝐼‘(𝐹‘𝐾)) ∈ 𝐸)
 
2-Feb-2026wlkelvv 16229 A walk is an ordered pair. (Contributed by Jim Kingdon, 2-Feb-2026.)
(𝑊 ∈ (Walks‘𝐺) → 𝑊 ∈ (V × V))
 
1-Feb-2026wlkcprim 16230 A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Revised by Jim Kingdon, 1-Feb-2026.)
(𝑊 ∈ (Walks‘𝐺) → (1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊))
 
1-Feb-2026wlkmex 16199 If there are walks on a graph, the graph is a set. (Contributed by Jim Kingdon, 1-Feb-2026.)
(𝑊 ∈ (Walks‘𝐺) → 𝐺 ∈ V)
 
31-Jan-2026fvmbr 5677 If a function value is inhabited, the argument is related to the function value. (Contributed by Jim Kingdon, 31-Jan-2026.)
(𝐴 ∈ (𝐹‘𝑋) → 𝑋𝐹(𝐹‘𝑋))
 
30-Jan-2026elfvfvex 5676 If a function value is inhabited, the function value is a set. (Contributed by Jim Kingdon, 30-Jan-2026.)
(𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ∈ V)
 
30-Jan-2026reldmm 4952 A relation is inhabited iff its domain is inhabited. (Contributed by Jim Kingdon, 30-Jan-2026.)
(Rel 𝐴 → (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ dom 𝐴))
 
25-Jan-2026ifp2 988 Forward direction of dfifp2dc 989. This direction does not require decidability. (Contributed by Jim Kingdon, 25-Jan-2026.)
(if-(𝜑, 𝜓, 𝜒) → ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
 
25-Jan-2026ifpdc 987 The conditional operator for propositions implies decidability. (Contributed by Jim Kingdon, 25-Jan-2026.)
(if-(𝜑, 𝜓, 𝜒) → DECID 𝜑)
 
20-Jan-2026cats1fvd 11356 A symbol other than the last in a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 20-Jan-2026.)
𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   (𝜑 → 𝑆 ∈ Word V)    &   (𝜑 → (♯‘𝑆) = 𝑀)    &   (𝜑 → 𝑌 ∈ 𝑉)    &   (𝜑 → 𝑋 ∈ 𝑊)    &   (𝜑 → (𝑆‘𝑁) = 𝑌)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑁 < 𝑀)    ⇒   (𝜑 → (𝑇‘𝑁) = 𝑌)
 
20-Jan-2026cats1fvnd 11355 The last symbol of a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 20-Jan-2026.)
𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   (𝜑 → 𝑆 ∈ Word V)    &   (𝜑 → 𝑋 ∈ 𝑉)    &   (𝜑 → (♯‘𝑆) = 𝑀)    ⇒   (𝜑 → (𝑇‘𝑀) = 𝑋)
 
19-Jan-2026cats2catd 11359 Closure of concatenation of concatenations with singleton words. (Contributed by AV, 1-Mar-2021.) (Revised by Jim Kingdon, 19-Jan-2026.)
(𝜑 → 𝐵 ∈ Word V)    &   (𝜑 → 𝐷 ∈ Word V)    &   (𝜑 → 𝑋 ∈ 𝑉)    &   (𝜑 → 𝑌 ∈ 𝑊)    &   (𝜑 → 𝐴 = (𝐵 ++ ⟨“𝑋”⟩))    &   (𝜑 → 𝐶 = (⟨“𝑌”⟩ ++ 𝐷))    ⇒   (𝜑 → (𝐴 ++ 𝐶) = ((𝐵 ++ ⟨“𝑋𝑌”⟩) ++ 𝐷))
 
19-Jan-2026cats1catd 11358 Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 19-Jan-2026.)
𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   (𝜑 → 𝐴 ∈ Word V)    &   (𝜑 → 𝑆 ∈ Word V)    &   (𝜑 → 𝑋 ∈ 𝑊)    &   (𝜑 → 𝐶 = (𝐵 ++ ⟨“𝑋”⟩))    &   (𝜑 → 𝐵 = (𝐴 ++ 𝑆))    ⇒   (𝜑 → 𝐶 = (𝐴 ++ 𝑇))
 
19-Jan-2026cats1lend 11357 The length of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 19-Jan-2026.)
𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   (𝜑 → 𝑆 ∈ Word V)    &   (𝜑 → 𝑋 ∈ 𝑊)    &   (♯‘𝑆) = 𝑀    &   (𝑀 + 1) = 𝑁    ⇒   (𝜑 → (♯‘𝑇) = 𝑁)
 
18-Jan-2026rexanaliim 2637 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) (Revised by Jim Kingdon, 18-Jan-2026.)
(∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))
 
15-Jan-2026df-uspgren 16035 Define the class of all undirected simple pseudographs (which could have loops). An undirected simple pseudograph is a special undirected pseudograph or a special undirected simple hypergraph, consisting of a set 𝑣 (of "vertices") and an injective (one-to-one) function 𝑒 (representing (indexed) "edges") into subsets of 𝑣 of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. In contrast to a pseudograph, there is at most one edge between two vertices resp. at most one loop for a vertex. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by Jim Kingdon, 15-Jan-2026.)
USPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}}
 
11-Jan-2026en2prde 7403 A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.) (Revised by Jim Kingdon, 11-Jan-2026.)
(𝑉 ≈ 2o → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))
 
10-Jan-2026pw1mapen 16657 Equinumerosity of (𝒫 1o ↑𝑚 𝐴) and the set of subsets of 𝐴. (Contributed by Jim Kingdon, 10-Jan-2026.)
(𝐴 ∈ 𝑉 → (𝒫 1o ↑𝑚 𝐴) ≈ 𝒫 𝐴)
 
10-Jan-2026pw1if 7448 Expressing a truth value in terms of an if expression. (Contributed by Jim Kingdon, 10-Jan-2026.)
(𝐴 ∈ 𝒫 1o → if(𝐴 = 1o, 1o, ∅) = 𝐴)
 
10-Jan-2026pw1m 7447 A truth value which is inhabited is equal to true. This is a variation of pwntru 4291 and pwtrufal 16658. (Contributed by Jim Kingdon, 10-Jan-2026.)
((𝐴 ∈ 𝒫 1o ∧ ∃𝑥 𝑥 ∈ 𝐴) → 𝐴 = 1o)
 
10-Jan-20261ndom2 7056 Two is not dominated by one. (Contributed by Jim Kingdon, 10-Jan-2026.)
¬ 2o ≼ 1o
 
9-Jan-2026pw1map 16656 Mapping between (𝒫 1o ↑𝑚 𝐴) and subsets of 𝐴. (Contributed by Jim Kingdon, 9-Jan-2026.)
𝐹 = (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ↦ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})    ⇒   (𝐴 ∈ 𝑉 → 𝐹:(𝒫 1o ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴)
 
9-Jan-2026iftrueb01 7446 Using an if expression to represent a truth value by ∅ or 1o. Unlike some theorems using if, 𝜑 does not need to be decidable. (Contributed by Jim Kingdon, 9-Jan-2026.)
(if(𝜑, 1o, ∅) = 1o ↔ 𝜑)
 
8-Jan-2026pfxclz 11269 Closure of the prefix extractor. This extends pfxclg 11268 from ℕ0 to ℤ (negative lengths are trivial, resulting in the empty word). (Contributed by Jim Kingdon, 8-Jan-2026.)
((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ℤ) → (𝑆 prefix 𝐿) ∈ Word 𝐴)
 
8-Jan-2026fnpfx 11267 The domain of the prefix extractor. (Contributed by Jim Kingdon, 8-Jan-2026.)
prefix Fn (V × ℕ0)
 
7-Jan-2026pr1or2 7404 An unordered pair, with decidable equality for the specified elements, has either one or two elements. (Contributed by Jim Kingdon, 7-Jan-2026.)
((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ DECID 𝐴 = 𝐵) → ({𝐴, 𝐵} ≈ 1o ∨ {𝐴, 𝐵} ≈ 2o))
 
6-Jan-2026upgr1elem1 16000 Lemma for upgr1edc 16001. (Contributed by AV, 16-Oct-2020.) (Revised by Jim Kingdon, 6-Jan-2026.)
(𝜑 → {𝐵, 𝐶} ∈ 𝑆)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐶 ∈ 𝑋)    &   (𝜑 → DECID 𝐵 = 𝐶)    ⇒   (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ 𝑆 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
 
3-Jan-2026df-umgren 15974 Define the class of all undirected multigraphs. An (undirected) multigraph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into subsets of 𝑣 of cardinality two, representing the two vertices incident to the edge. In contrast to a pseudograph, a multigraph has no loop. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "A multigraph M consists of a finite nonempty set V of vertices and a set E of edges, where every two vertices of M are joined by a finite number of edges (possibly zero). If two or more edges join the same pair of (distinct) vertices, then these edges are called parallel edges." (Contributed by AV, 24-Nov-2020.) (Revised by Jim Kingdon, 3-Jan-2026.)
UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o}}
 
3-Jan-2026df-upgren 15973 Define the class of all undirected pseudographs. An (undirected) pseudograph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into subsets of 𝑣 of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "In a pseudograph, not only are parallel edges permitted but an edge is also permitted to join a vertex to itself. Such an edge is called a loop." (in contrast to a multigraph, see df-umgren 15974). (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 24-Nov-2020.) (Revised by Jim Kingdon, 3-Jan-2026.)
UPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}}
 
3-Jan-2026dom1o 7007 Two ways of saying that a set is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
(𝐴 ∈ 𝑉 → (1o ≼ 𝐴 ↔ ∃𝑗 𝑗 ∈ 𝐴))
 
3-Jan-2026en2m 7004 A set with two elements is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
(𝐴 ≈ 2o → ∃𝑥 𝑥 ∈ 𝐴)
 
3-Jan-2026en1m 6984 A set with one element is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
(𝐴 ≈ 1o → ∃𝑥 𝑥 ∈ 𝐴)
 
31-Dec-2025pw0ss 15963 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 15950 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 15949 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 15899 Applying the indexed edge function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
(𝐺 ∈ 𝑉 → (iEdg‘𝐺) ∈ V)
 
29-Dec-2025vtxex 15898 Applying the vertex function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
(𝐺 ∈ 𝑉 → (Vtx‘𝐺) ∈ V)
 
29-Dec-2025snmb 3794 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 11174 Existence of the last symbol. The last symbol of a word is a set. See lsw0g 11171 or lswcl 11173 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 11237 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 11180 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 11169 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 13224 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 15912 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 15911 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 15910 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 15909 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 11113 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 16688 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))
 
30-Nov-2025eluz3nn 9806 An integer greater than or equal to 3 is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.) (Proof shortened by AV, 30-Nov-2025.)
(𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
 
27-Nov-2025psrelbasfi 14719 Simpler form of psrelbas 14718 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 14743 Lemma for mplsubgfi 14744. 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 14742 Lemma for mplsubgfi 14744. 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 7384 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 7337). (Contributed by Jim Kingdon, 25-Nov-2025.)
(∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ ω ∈ WOmni)
 
23-Nov-2025psrbagfi 14712 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 7389 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 14741 Lemma for mplsubgfi 14744. 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)
 
15-Nov-2025uzuzle35 9804 An integer greater than or equal to 5 is an integer greater than or equal to 3. (Contributed by AV, 15-Nov-2025.)
(𝐴 ∈ (ℤ≥‘5) → 𝐴 ∈ (ℤ≥‘3))
 
14-Nov-20252omapen 16655 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 16654 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 16662 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 13380 Lemma for prdsbas 13382 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 14736 mPoly has universal domain. (Contributed by Jim Kingdon, 5-Nov-2025.)
mPoly Fn (V × V)
 
4-Nov-2025mplelbascoe 14735 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 14734 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 14733 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 7398 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 12531 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 11808 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 10512 Rational ≤ is decidable. (Contributed by Jim Kingdon, 28-Oct-2025.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 ≤ 𝐵)
 
17-Oct-2025plycoeid3 15510 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 7127 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 7125 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 3603 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 12834 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 14702 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 15451 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 1494 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 921), 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 1493 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 860), 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 1492 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 850), 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 15450 Real derivative of the identity function. (Contributed by Jim Kingdon, 3-Oct-2025.)
(ℝ D ( I ↾ ℝ)) = (ℝ × {1})
 
3-Oct-2025dvconstre 15449 Real derivative of a constant function. (Contributed by Jim Kingdon, 3-Oct-2025.)
(𝐴 ∈ ℂ → (ℝ D (ℝ × {𝐴})) = (ℝ × {0}))
 
3-Oct-2025dvidsslem 15446 Lemma for dvconstss 15451. Analogue of dvidlemap 15444 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 15445 Lemma for dvidre 15450 and dvconstre 15449. Analogue of dvidlemap 15444 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 14593 Applying metUnif yields a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
(𝐴 ∈ 𝑉 → (metUnif‘𝐴) ∈ V)
 
28-Sep-2025cndsex 14591 The standard distance function on the complex numbers is a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
(abs ∘ − ) ∈ V
 
25-Sep-2025cntopex 14592 The standard topology on the complex numbers is a set. (Contributed by Jim Kingdon, 25-Sep-2025.)
(MetOpen‘(abs ∘ − )) ∈ V
 
24-Sep-2025mopnset 14590 Getting a set by applying MetOpen. (Contributed by Jim Kingdon, 24-Sep-2025.)
(𝐷 ∈ 𝑉 → (MetOpen‘𝐷) ∈ V)
 
24-Sep-2025blfn 14589 The ball function has universal domain. (Contributed by Jim Kingdon, 24-Sep-2025.)
ball Fn V
 
23-Sep-2025elfzoext 10443 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 15513 Lemma for plycj 15514. (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 15511 Lemma for plyco 15512. 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 10442 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 15834 Lemma for lgsquad 15838. 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 15833 Lemma for lgsquad 15838. 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 7137 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 6305 Principle of unique choice. This is also called non-choice. The name choice results in its similarity to something like acfun 7427 (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 14645 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 14627 The invertible complex numbers are exactly those apart from zero. This is recapb 8856 but expressed in terms of ℂfld. (Contributed by Jim Kingdon, 11-Sep-2025.)
{𝑧 ∈ ℂ ∣ 𝑧 # 0} = (Unit‘ℂfld)
 
9-Sep-2025gsumfzfsumlemm 14625 Lemma for gsumfzfsum 14626. 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 14624 Lemma for gsumfzfsum 14626. 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 13954 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 13953 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 12519 5 does not divide 6. (Contributed by AV, 8-Sep-2025.)
¬ 5 ∥ 6
 
8-Sep-20255ndvds3 12518 5 does not divide 3. (Contributed by AV, 8-Sep-2025.)
¬ 5 ∥ 3
 
7-Sep-20255eluz3 9800 5 is an integer greater than or equal to 3. (Contributed by AV, 7-Sep-2025.)
5 ∈ (ℤ≥‘3)
 
6-Sep-2025gsumfzconst 13951 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))
 
5-Sep-2025uzuzle34 9803 An integer greater than or equal to 4 is an integer greater than or equal to 3. (Contributed by AV, 5-Sep-2025.)
(𝑋 ∈ (ℤ≥‘4) → 𝑋 ∈ (ℤ≥‘3))
 
31-Aug-2025gsumfzmptfidmadd 13949 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 13948 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 10742 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 10789 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 9887 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 9886. (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 10734 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 10731 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 11129 A zero-based sequence is a word. In iswrdinn0 11127 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 13605 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 11127 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 13601 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 13497 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 14695 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 14299 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 15817 Lemma for gausslemma2dlem1 15819. 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 15818 Lemma for gausslemma2dlem1 15819. (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 10511 Rational < is decidable. (Contributed by Jim Kingdon, 7-Aug-2025.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 < 𝐵)
 
22-Jul-2025ivthdich 15406 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 15396 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 15399, hovera 15400, and hoverb 15401, 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 8252, 𝑐 < 1 or 0 < 𝑐, and that leads to 𝑧 ≤ 0 by hoverlt1 15402 or 0 ≤ 𝑧 by hovergt0 15403. (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 15405 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 15404 Lemma for ivthdich 15406. 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 15403 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 15402 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 15401 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 15400 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 2740 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 2739 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 2738 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 2737 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)
 
20-Jul-2025hovercncf 15399 The hover function is continuous. By hover function, we mean a a function which starts out as a line of slope one, is constant at zero from zero to one, and then resumes as a slope of one. (Contributed by Jim Kingdon, 20-Jul-2025.)
𝐹 = (𝑥 ∈ ℝ ↦ sup({inf({𝑥, 0}, ℝ, < ), (𝑥 − 1)}, ℝ, < ))    ⇒   𝐹 ∈ (ℝ–cn→ℝ)
 
19-Jul-2025mincncf 15369 The minimum of two continuous real functions is continuous. (Contributed by Jim Kingdon, 19-Jul-2025.)
(𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℝ))    &   (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℝ))    ⇒   (𝜑 → (𝑥 ∈ 𝑋 ↦ inf({𝐴, 𝐵}, ℝ, < )) ∈ (𝑋–cn→ℝ))
 
18-Jul-2025maxcncf 15368 The maximum of two continuous real functions is continuous. (Contributed by Jim Kingdon, 18-Jul-2025.)
(𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℝ))    &   (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℝ))    ⇒   (𝜑 → (𝑥 ∈ 𝑋 ↦ sup({𝐴, 𝐵}, ℝ, < )) ∈ (𝑋–cn→ℝ))
 
14-Jul-2025xnn0nnen 10705 The set of extended nonnegative integers is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 14-Jul-2025.)
ℕ0* ≈ ℕ
 
12-Jul-2025nninfninc 7327 All values beyond a zero in an ℕ∞ sequence are zero. This is another way of stating that elements of ℕ∞ are nonincreasing. (Contributed by Jim Kingdon, 12-Jul-2025.)
(𝜑 → 𝐴 ∈ ℕ∞)    &   (𝜑 → 𝑋 ∈ ω)    &   (𝜑 → 𝑌 ∈ ω)    &   (𝜑 → 𝑋 ⊆ 𝑌)    &   (𝜑 → (𝐴‘𝑋) = ∅)    ⇒   (𝜑 → (𝐴‘𝑌) = ∅)
 
10-Jul-2025nninfctlemfo 12634 Lemma for nninfct 12635. (Contributed by Jim Kingdon, 10-Jul-2025.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))    &   𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})    ⇒   (ω ∈ Omni → 𝐼:ℕ0*–onto→ℕ∞)
 
8-Jul-2025nnnninfen 16686 Equinumerosity of the natural numbers and ℕ∞ is equivalent to the Limited Principle of Omniscience (LPO). Remark in Section 1.1 of [Pradic2025], p. 2. (Contributed by Jim Kingdon, 8-Jul-2025.)
(ω ≈ ℕ∞ ↔ ω ∈ Omni)
 
8-Jul-2025nninfct 12635 The limited principle of omniscience (LPO) implies that ℕ∞ is countable. (Contributed by Jim Kingdon, 8-Jul-2025.)
(ω ∈ Omni → ∃𝑓 𝑓:ω–onto→(ℕ∞ ⊔ 1o))
 
8-Jul-2025nninfinf 10711 ℕ∞ is infinte. (Contributed by Jim Kingdon, 8-Jul-2025.)
ω ≼ ℕ∞
 
7-Jul-2025ivthreinc 15398 Restating the intermediate value theorem. Given a hypothesis stating the intermediate value theorem (in a strong form which is not provable given our axioms alone), provide a conclusion similar to the theorem as stated in the Metamath Proof Explorer (which is also similar to how we state the theorem for a strictly monotonic function at ivthinc 15396). Being able to have a hypothesis stating the intermediate value theorem will be helpful when it comes time to show that it implies a constructive taboo. This version of the theorem requires that the function 𝐹 is continuous on the entire real line, not just (𝐴[,]𝐵) which may be an unnecessary condition but which is sufficient for the way we want to use it. (Contributed by Jim Kingdon, 7-Jul-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → 𝐹 ∈ (ℝ–cn→ℝ))    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (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 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)
 
28-Jun-2025fngsum 13494 Iterated sum has a universal domain. (Contributed by Jim Kingdon, 28-Jun-2025.)
Σg Fn (V × V)
 
28-Jun-2025iotaexel 5981 Set existence of an iota expression in which all values are contained within a set. (Contributed by Jim Kingdon, 28-Jun-2025.)
((𝐴 ∈ 𝑉 ∧ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) → (℩𝑥𝜑) ∈ V)
 
27-Jun-2025df-igsum 13365 Define a finite group sum (also called "iterated sum") of a structure.

Given 𝐺 Σg 𝐹 where 𝐹:𝐴⟶(Base‘𝐺), the set of indices is 𝐴 and the values are given by 𝐹 at each index. A group sum over a multiplicative group may be viewed as a product. The definition is meaningful in different contexts, depending on the size of the index set 𝐴 and each demanding different properties of 𝐺.

1. If 𝐴 = ∅ and 𝐺 has an identity element, then the sum equals this identity.

2. If 𝐴 = (𝑀...𝑁) and 𝐺 is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e., ((𝐹‘1) + (𝐹‘2)) + (𝐹‘3), etc.

3. This definition does not handle other cases.

(Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 27-Jun-2025.)

Σg = (𝑤 ∈ V, 𝑓 ∈ V ↦ (℩𝑥((dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))))
 
20-Jun-2025opprnzrbg 14223 The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr 14224. (Contributed by SN, 20-Jun-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing))
 
16-Jun-2025fnpsr 14705 The multivariate power series constructor has a universal domain. (Contributed by Jim Kingdon, 16-Jun-2025.)
mPwSer Fn (V × V)
 
14-Jun-2025basm 13167 A structure whose base is inhabited is inhabited. (Contributed by Jim Kingdon, 14-Jun-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐴 ∈ 𝐵 → ∃𝑗 𝑗 ∈ 𝐺)
 
14-Jun-2025elfvm 5675 If a function value has a member, the function is inhabited. (Contributed by Jim Kingdon, 14-Jun-2025.)
(𝐴 ∈ (𝐹‘𝐵) → ∃𝑗 𝑗 ∈ 𝐹)
 
6-Jun-2025pcxqcl 12908 The general prime count function is an integer or infinite. (Contributed by Jim Kingdon, 6-Jun-2025.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℚ) → ((𝑃 pCnt 𝑁) ∈ ℤ ∨ (𝑃 pCnt 𝑁) = +∞))
 
5-Jun-2025xqltnle 10533 "Less than" expressed in terms of "less than or equal to", for extended numbers which are rational or +∞. We have not yet had enough usage of such numbers to warrant fully developing the concept, as in ℕ0* or ℝ*, so for now we just have a handful of theorems for what we need. (Contributed by Jim Kingdon, 5-Jun-2025.)
(((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
 
5-Jun-2025ceqsexv2d 2842 Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.) Shorten, reduce dv conditions. (Revised by Wolf Lammen, 5-Jun-2025.) (Proof shortened by SN, 5-Jun-2025.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   𝜓    ⇒   ∃𝑥𝜑
 
31-May-2025vtocl4ga 2875 Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.) (Proof shortened by Wolf Lammen, 31-May-2025.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   (𝑧 = 𝐶 → (𝜒 ↔ 𝜌))    &   (𝑤 = 𝐷 → (𝜌 ↔ 𝜃))    &   (((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) → 𝜑)    ⇒   (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)) → 𝜃)
 
30-May-20254sqexercise2 12995 Exercise which may help in understanding the proof of 4sqlemsdc 12996. (Contributed by Jim Kingdon, 30-May-2025.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑛 = ((𝑥↑2) + (𝑦↑2))}    ⇒   (𝐴 ∈ ℕ0 → DECID 𝐴 ∈ 𝑆)
 
27-May-2025iotaexab 5307 Existence of the ℩ class when all the possible values are contained in a set. (Contributed by Jim Kingdon, 27-May-2025.)
({𝑥 ∣ 𝜑} ∈ 𝑉 → (℩𝑥𝜑) ∈ V)
 
25-May-20254sqlemsdc 12996 Lemma for 4sq 13006. The property of being the sum of four squares is decidable.

The proof involves showing that (for a particular 𝐴) there are only a finite number of possible ways that it could be the sum of four squares, so checking each of those possibilities in turn decides whether the number is the sum of four squares. If this proof is hard to follow, especially because of its length, the simplified versions at 4sqexercise1 12994 and 4sqexercise2 12995 may help clarify, as they are using very much the same techniques on simplified versions of this lemma. (Contributed by Jim Kingdon, 25-May-2025.)

𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}    ⇒   (𝐴 ∈ ℕ0 → DECID 𝐴 ∈ 𝑆)
 
25-May-20254sqexercise1 12994 Exercise which may help in understanding the proof of 4sqlemsdc 12996. (Contributed by Jim Kingdon, 25-May-2025.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ 𝑛 = (𝑥↑2)}    ⇒   (𝐴 ∈ ℕ0 → DECID 𝐴 ∈ 𝑆)
 
24-May-20254sqleminfi 12993 Lemma for 4sq 13006. 𝐴 ∩ ran 𝐹 is finite. (Contributed by Jim Kingdon, 24-May-2025.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∈ ℕ)    &   𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}    &   𝐹 = (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣))    ⇒   (𝜑 → (𝐴 ∩ ran 𝐹) ∈ Fin)
 
24-May-20254sqlemffi 12992 Lemma for 4sq 13006. ran 𝐹 is finite. (Contributed by Jim Kingdon, 24-May-2025.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∈ ℕ)    &   𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}    &   𝐹 = (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣))    ⇒   (𝜑 → ran 𝐹 ∈ Fin)
 
24-May-20254sqlemafi 12991 Lemma for 4sq 13006. 𝐴 is finite. (Contributed by Jim Kingdon, 24-May-2025.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∈ ℕ)    &   𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}    ⇒   (𝜑 → 𝐴 ∈ Fin)
 
24-May-2025infidc 7138 The intersection of two sets is finite if one of them is and the other is decidable. (Contributed by Jim Kingdon, 24-May-2025.)
((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → (𝐴 ∩ 𝐵) ∈ Fin)
 
19-May-2025zrhex 14659 Set existence for ℤRHom. (Contributed by Jim Kingdon, 19-May-2025.)
𝐿 = (ℤRHom‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → 𝐿 ∈ V)
 
16-May-2025rhmex 14195 Set existence for ring homomorphism. (Contributed by Jim Kingdon, 16-May-2025.)
((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 RingHom 𝑆) ∈ V)
 
15-May-2025ghmex 13865 The set of group homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)
((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) ∈ V)
 
15-May-2025mhmex 13568 The set of monoid homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)
((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 MndHom 𝑇) ∈ V)
 
14-May-2025idomcringd 14316 An integral domain is a commutative ring with unity. (Contributed by Thierry Arnoux, 4-May-2025.) (Proof shortened by SN, 14-May-2025.)
(𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → 𝑅 ∈ CRing)
 
6-May-2025rrgnz 14306 In a nonzero ring, the zero is a left zero divisor (that is, not a left-regular element). (Contributed by Thierry Arnoux, 6-May-2025.)
𝐸 = (RLReg‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ NzRing → ¬ 0 ∈ 𝐸)
 
5-May-2025rngressid 13991 A non-unital ring restricted to its base set is a non-unital ring. It will usually be the original non-unital ring exactly, of course, but to show that needs additional conditions such as those in strressid 13177. (Contributed by Jim Kingdon, 5-May-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Rng → (𝐺 ↾s 𝐵) ∈ Rng)
 
5-May-2025ablressid 13945 A commutative group restricted to its base set is a commutative group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 13177. (Contributed by Jim Kingdon, 5-May-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Abel → (𝐺 ↾s 𝐵) ∈ Abel)
 
30-Apr-2025dvply2g 15519 The derivative of a polynomial with coefficients in a subring is a polynomial with coefficients in the same ring. (Contributed by Mario Carneiro, 1-Jan-2017.) (Revised by GG, 30-Apr-2025.)
((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℂ D 𝐹) ∈ (Poly‘𝑆))
 
29-Apr-2025rlmscabas 14498 Scalars in the ring module have the same base set. (Contributed by Jim Kingdon, 29-Apr-2025.)
(𝑅 ∈ 𝑋 → (Base‘𝑅) = (Base‘(Scalar‘(ringLMod‘𝑅))))
 
29-Apr-2025ressbasid 13176 The trivial structure restriction leaves the base set unchanged. (Contributed by Jim Kingdon, 29-Apr-2025.)
𝐵 = (Base‘𝑊)    ⇒   (𝑊 ∈ 𝑉 → (Base‘(𝑊 ↾s 𝐵)) = 𝐵)
 
28-Apr-2025lssmex 14393 If a linear subspace is inhabited, the class it is built from is a set. (Contributed by Jim Kingdon, 28-Apr-2025.)
𝑆 = (LSubSp‘𝑊)    ⇒   (𝑈 ∈ 𝑆 → 𝑊 ∈ V)
 
27-Apr-2025cnfldmul 14602 The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 27-Apr-2025.)
· = (.r‘ℂfld)
 
27-Apr-2025cnfldadd 14600 The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 27-Apr-2025.)
+ = (+g‘ℂfld)
 
27-Apr-2025lidlex 14511 Existence of the set of left ideals. (Contributed by Jim Kingdon, 27-Apr-2025.)
(𝑊 ∈ 𝑉 → (LIdeal‘𝑊) ∈ V)
 
27-Apr-2025lssex 14392 Existence of a linear subspace. (Contributed by Jim Kingdon, 27-Apr-2025.)
(𝑊 ∈ 𝑉 → (LSubSp‘𝑊) ∈ V)
 
25-Apr-2025rspex 14512 Existence of the ring span. (Contributed by Jim Kingdon, 25-Apr-2025.)
(𝑊 ∈ 𝑉 → (RSpan‘𝑊) ∈ V)
 
25-Apr-2025lspex 14433 Existence of the span of a set of vectors. (Contributed by Jim Kingdon, 25-Apr-2025.)
(𝑊 ∈ 𝑋 → (LSpan‘𝑊) ∈ V)
 
25-Apr-2025eqgex 13831 The left coset equivalence relation exists. (Contributed by Jim Kingdon, 25-Apr-2025.)
((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐺 ~QG 𝑆) ∈ V)
 
25-Apr-2025qusex 13431 Existence of a quotient structure. (Contributed by Jim Kingdon, 25-Apr-2025.)
((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) ∈ V)
 
23-Apr-20251dom1el 6998 If a set is dominated by one, then any two of its elements are equal. (Contributed by Jim Kingdon, 23-Apr-2025.)
((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → 𝐵 = 𝐶)
 
22-Apr-2025mulgex 13733 Existence of the group multiple operation. (Contributed by Jim Kingdon, 22-Apr-2025.)
(𝐺 ∈ 𝑉 → (.g‘𝐺) ∈ V)
 
21-Apr-2025uspgruhgr 16067 An undirected simple pseudograph is an undirected hypergraph. (Contributed by AV, 21-Apr-2025.)
(𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
 
20-Apr-2025uspgriedgedg 16059 In a simple pseudograph, for each indexed edge there is exactly one edge. (Contributed by AV, 20-Apr-2025.)
𝐸 = (Edg‘𝐺)    &   𝐼 = (iEdg‘𝐺)    ⇒   ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ dom 𝐼) → ∃!𝑘 ∈ 𝐸 𝑘 = (𝐼‘𝑋))
 
20-Apr-2025uspgredgiedg 16058 In a simple pseudograph, for each edge there is exactly one indexed edge. (Contributed by AV, 20-Apr-2025.)
𝐸 = (Edg‘𝐺)    &   𝐼 = (iEdg‘𝐺)    ⇒   ((𝐺 ∈ USPGraph ∧ 𝐾 ∈ 𝐸) → ∃!𝑥 ∈ dom 𝐼 𝐾 = (𝐼‘𝑥))
 
20-Apr-2025elovmpod 6225 Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 6226 in deduction form. (Revised by AV, 20-Apr-2025.)
𝑂 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → 𝐶 = 𝐷)    ⇒   (𝜑 → (𝐸 ∈ (𝑋𝑂𝑌) ↔ 𝐸 ∈ 𝐷))
 
20-Apr-2025fdmeu 5692 There is exactly one codomain element for each element of the domain of a function. (Contributed by AV, 20-Apr-2025.)
((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 (𝐹‘𝑋) = 𝑦)
 
18-Apr-2025fsumdvdsmul 15744 Product of two divisor sums. (This is also the main part of the proof that "Σ𝑘 ∥ 𝑁𝐹(𝑘) is a multiplicative function if 𝐹 is".) (Contributed by Mario Carneiro, 2-Jul-2015.) Avoid ax-mulf 8160. (Revised by GG, 18-Apr-2025.)
(𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   𝑋 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}    &   𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}    &   𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}    &   ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → (𝐴 · 𝐵) = 𝐷)    &   (𝑖 = (𝑗 · 𝑘) → 𝐶 = 𝐷)    ⇒   (𝜑 → (Σ𝑗 ∈ 𝑋 𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑖 ∈ 𝑍 𝐶)
 
18-Apr-2025mpodvdsmulf1o 15743 If 𝑀 and 𝑁 are two coprime integers, multiplication forms a bijection from the set of pairs ⟨𝑗, 𝑘⟩ where 𝑗 ∥ 𝑀 and 𝑘 ∥ 𝑁, to the set of divisors of 𝑀 · 𝑁. (Contributed by GG, 18-Apr-2025.)
(𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   𝑋 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}    &   𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}    &   𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}    ⇒   (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍)
 
18-Apr-2025df2idl2 14547 Alternate (the usual textbook) definition of a two-sided ideal of a ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
𝑈 = (2Ideal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 ((𝑥 · 𝑦) ∈ 𝐼 ∧ (𝑦 · 𝑥) ∈ 𝐼))))
 
18-Apr-20252idlmex 14539 Existence of the set a two-sided ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
𝑇 = (2Ideal‘𝑊)    ⇒   (𝑈 ∈ 𝑇 → 𝑊 ∈ V)
 
18-Apr-2025dflidl2 14526 Alternate (the usual textbook) definition of a (left) ideal of a ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥 · 𝑦) ∈ 𝐼)))
 
18-Apr-2025lidlmex 14513 Existence of the set a left ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
𝐼 = (LIdeal‘𝑊)    ⇒   (𝑈 ∈ 𝐼 → 𝑊 ∈ V)
 
18-Apr-2025lsslsp 14467 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 = (𝑊 ↾s 𝑈)    &   𝑀 = (LSpan‘𝑊)    &   𝑁 = (LSpan‘𝑋)    &   𝐿 = (LSubSp‘𝑊)    ⇒   ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → (𝑁‘𝐺) = (𝑀‘𝐺))
 
16-Apr-2025sraex 14484 Existence of a subring algebra. (Contributed by Jim Kingdon, 16-Apr-2025.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → 𝐴 ∈ V)
 
14-Apr-2025grpmgmd 13632 A group is a magma, deduction form. (Contributed by SN, 14-Apr-2025.)
(𝜑 → 𝐺 ∈ Grp)    ⇒   (𝜑 → 𝐺 ∈ Mgm)
 
12-Apr-2025psraddcl 14723 Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.) Generalize to magmas. (Revised by SN, 12-Apr-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    + = (+g‘𝑆)    &   (𝜑 → 𝑅 ∈ Mgm)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
 
10-Apr-2025cndcap 16731 Real number trichotomy is equivalent to decidability of complex number apartness. (Contributed by Jim Kingdon, 10-Apr-2025.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑧 ∈ ℂ ∀𝑤 ∈ ℂ DECID 𝑧 # 𝑤)
 
4-Apr-2025ghmf1 13883 Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 4-Apr-2025.)
𝐴 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑁 = (0g‘𝑅)    &    0 = (0g‘𝑆)    ⇒   (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)))
 
3-Apr-2025quscrng 14571 The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) (Proof shortened by AV, 3-Apr-2025.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (LIdeal‘𝑅)    ⇒   ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ CRing)
 
31-Mar-2025cnfldds 14606 The metric of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 14595. (Revised by GG, 31-Mar-2025.)
(abs ∘ − ) = (dist‘ℂfld)
 
31-Mar-2025cnfldle 14605 The ordering of the field of complex numbers. Note that this is not actually an ordering on ℂ, but we put it in the structure anyway because restricting to ℝ does not affect this component, so that (ℂfld ↾s ℝ) is an ordered field even though ℂfld itself is not. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 14595. (Revised by GG, 31-Mar-2025.)
≤ = (le‘ℂfld)
 
31-Mar-2025cnfldtset 14604 The topology component of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 31-Mar-2025.)
(MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld)
 
31-Mar-2025mpocnfldmul 14601 The multiplication operation of the field of complex numbers. Version of cnfldmul 14602 using maps-to notation, which does not require ax-mulf 8160. (Contributed by GG, 31-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (.r‘ℂfld)
 
31-Mar-2025mpocnfldadd 14599 The addition operation of the field of complex numbers. Version of cnfldadd 14600 using maps-to notation, which does not require ax-addf 8159. (Contributed by GG, 31-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (+g‘ℂfld)
 
31-Mar-2025df-cnfld 14595 The field of complex numbers. Other number fields and rings can be constructed by applying the ↾s restriction operator.

The contract of this set is defined entirely by cnfldex 14597, cnfldadd 14600, cnfldmul 14602, cnfldcj 14603, cnfldtset 14604, cnfldle 14605, cnfldds 14606, and cnfldbas 14598. We may add additional members to this in the future. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) Use maps-to notation for addition and multiplication. (Revised by GG, 31-Mar-2025.) (New usage is discouraged.)

ℂfld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
 
31-Mar-20252idlcpbl 14562 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) (Proof shortened by AV, 31-Mar-2025.)
𝑋 = (Base‘𝑅)    &   𝐸 = (𝑅 ~QG 𝑆)    &   𝐼 = (2Ideal‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
 
22-Mar-2025idomringd 14317 An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
(𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → 𝑅 ∈ Ring)
 
22-Mar-2025idomdomd 14315 An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
(𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → 𝑅 ∈ Domn)
 
21-Mar-2025df2idl2rng 14546 Alternate (the usual textbook) definition of a two-sided ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
𝑈 = (2Ideal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 ((𝑥 · 𝑦) ∈ 𝐼 ∧ (𝑦 · 𝑥) ∈ 𝐼)))
 
21-Mar-2025isridlrng 14520 A right ideal is a left ideal of the opposite non-unital ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
𝑈 = (LIdeal‘(oppr‘𝑅))    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼))
 
21-Mar-2025dflidl2rng 14519 Alternate (the usual textbook) definition of a (left) ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥 · 𝑦) ∈ 𝐼))
 
20-Mar-2025ccoslid 13344 Slot property of comp. (Contributed by Jim Kingdon, 20-Mar-2025.)
(comp = Slot (comp‘ndx) ∧ (comp‘ndx) ∈ ℕ)
 
20-Mar-2025homslid 13341 Slot property of Hom. (Contributed by Jim Kingdon, 20-Mar-2025.)
(Hom = Slot (Hom ‘ndx) ∧ (Hom ‘ndx) ∈ ℕ)
 
19-Mar-2025ptex 13370 Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.)
(𝐹 ∈ 𝑉 → (∏t‘𝐹) ∈ V)
 
18-Mar-2025prdsex 13375 Existence of the structure product. (Contributed by Jim Kingdon, 18-Mar-2025.)
((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑆Xs𝑅) ∈ V)
 
16-Mar-2025plycn 15515 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8160. (Revised by GG, 16-Mar-2025.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))
 
16-Mar-2025expcn 15322 The power function on complex numbers, for fixed exponent 𝑁, is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) Avoid ax-mulf 8160. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpen‘ℂfld)    ⇒   (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽))
 
16-Mar-2025mpomulcn 15319 Complex number multiplication is a continuous function. (Contributed by GG, 16-Mar-2025.)
𝐽 = (TopOpen‘ℂfld)    ⇒   (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
 
16-Mar-2025mpomulf 8174 Multiplication is an operation on complex numbers. Version of ax-mulf 8160 using maps-to notation, proved from the axioms of set theory and ax-mulcl 8135. (Contributed by GG, 16-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
 
13-Mar-20252idlss 14552 A two-sided ideal is a subset of the base set. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.) (Proof shortened by AV, 13-Mar-2025.)
𝐵 = (Base‘𝑊)    &   𝐼 = (2Ideal‘𝑊)    ⇒   (𝑈 ∈ 𝐼 → 𝑈 ⊆ 𝐵)
 
13-Mar-2025imasex 13411 Existence of the image structure. (Contributed by Jim Kingdon, 13-Mar-2025.)
((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐹 “s 𝑅) ∈ V)
 
11-Mar-2025rng2idlsubgsubrng 14558 A two-sided ideal of a non-unital ring which is a subgroup of the ring is a subring of the ring. (Contributed by AV, 11-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   (𝜑 → 𝐼 ∈ (SubRng‘𝑅))
 
11-Mar-2025rng2idlsubrng 14555 A two-sided ideal of a non-unital ring which is a non-unital ring is a subring of the ring. (Contributed by AV, 20-Feb-2025.) (Revised by AV, 11-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   (𝜑 → 𝐼 ∈ (SubRng‘𝑅))
 
11-Mar-2025rnglidlrng 14536 A (left) ideal of a non-unital ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 𝑈 ∈ (SubGrp‘𝑅) is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅 ↾s 𝑈)    ⇒   ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Rng)
 
11-Mar-2025rnglidlmsgrp 14535 The multiplicative group of a (left) ideal of a non-unital ring is a semigroup. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 0 ∈ 𝑈 is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅 ↾s 𝑈)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Smgrp)
 
11-Mar-2025rnglidlmmgm 14534 The multiplicative group of a (left) ideal of a non-unital ring is a magma. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 0 ∈ 𝑈 is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅 ↾s 𝑈)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Mgm)
 
11-Mar-2025imasival 13412 Value of an image structure. The is a lemma for the theorems imasbas 13413, imasplusg 13414, and imasmulr 13415 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.)
(𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   (𝜑 → 𝑉 = (Base‘𝑅))    &    + = (+g‘𝑅)    &    × = (.r‘𝑅)    &    · = ( ·𝑠 ‘𝑅)    &   (𝜑 → ✚ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩})    &   (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩})    &   (𝜑 → 𝐹:𝑉–onto→𝐵)    &   (𝜑 → 𝑅 ∈ 𝑍)    ⇒   (𝜑 → 𝑈 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), ∙ ⟩})
 
9-Mar-20252idlridld 14545 A two-sided ideal is a right ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
(𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝑂 = (oppr‘𝑅)    ⇒   (𝜑 → 𝐼 ∈ (LIdeal‘𝑂))
 
9-Mar-20252idllidld 14544 A two-sided ideal is a left ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
(𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    ⇒   (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
 
9-Mar-2025quseccl 13843 Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) (Proof shortened by AV, 9-Mar-2025.)
𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    &   𝑉 = (Base‘𝐺)    &   𝐵 = (Base‘𝐻)    ⇒   ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [𝑋](𝐺 ~QG 𝑆) ∈ 𝐵)
 
9-Mar-2025fovcl 6132 Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Proof shortened by AV, 9-Mar-2025.)
𝐹:(𝑅 × 𝑆)⟶𝐶    ⇒   ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
 
8-Mar-2025subgex 13786 The class of subgroups of a group is a set. (Contributed by Jim Kingdon, 8-Mar-2025.)
(𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ V)
 
7-Mar-2025ringrzd 14083 The zero of a unital ring is a right-absorbing element. (Contributed by SN, 7-Mar-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 · 0 ) = 0 )
 
7-Mar-2025ringlzd 14082 The zero of a unital ring is a left-absorbing element. (Contributed by SN, 7-Mar-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → ( 0 · 𝑋) = 0 )
 
7-Mar-2025qusecsub 13941 Two subgroup cosets are equal if and only if the difference of their representatives is a member of the subgroup. (Contributed by AV, 7-Mar-2025.)
𝐵 = (Base‘𝐺)    &    − = (-g‘𝐺)    &    ∼ = (𝐺 ~QG 𝑆)    ⇒   (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ([𝑋] ∼ = [𝑌] ∼ ↔ (𝑌 − 𝑋) ∈ 𝑆))
 
1-Mar-2025quselbasg 13840 Membership in the base set of a quotient group. (Contributed by AV, 1-Mar-2025.)
∼ = (𝐺 ~QG 𝑆)    &   𝑈 = (𝐺 /s ∼ )    &   𝐵 = (Base‘𝐺)    ⇒   ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑆 ∈ 𝑍) → (𝑋 ∈ (Base‘𝑈) ↔ ∃𝑥 ∈ 𝐵 𝑋 = [𝑥] ∼ ))
 
28-Feb-2025qusmulrng 14570 Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 14571. Similar to qusmul2 14567. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 28-Feb-2025.)
∼ = (𝑅 ~QG 𝑆)    &   𝐻 = (𝑅 /s ∼ )    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    ∙ = (.r‘𝐻)    ⇒   (((𝑅 ∈ Rng ∧ 𝑆 ∈ (2Ideal‘𝑅) ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ )
 
28-Feb-2025ringressid 14100 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 13177. (Contributed by Jim Kingdon, 28-Feb-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Ring)
 
28-Feb-2025grpressid 13667 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 13177. (Contributed by Jim Kingdon, 28-Feb-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp)
 
27-Feb-2025imasringf1 14102 The image of a ring under an injection is a ring. (Contributed by AV, 27-Feb-2025.)
𝑈 = (𝐹 “s 𝑅)    &   𝑉 = (Base‘𝑅)    ⇒   ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Ring) → 𝑈 ∈ Ring)
 
26-Feb-2025strext 13211 Extending the upper range of a structure. This works because when we say that a structure has components in 𝐴...𝐶 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.)
(𝜑 → 𝐹 Struct ⟨𝐴, 𝐵⟩)    &   (𝜑 → 𝐶 ∈ (ℤ≥‘𝐵))    ⇒   (𝜑 → 𝐹 Struct ⟨𝐴, 𝐶⟩)
 
25-Feb-2025subrngringnsg 14243 A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.)
(𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅))
 
25-Feb-2025rngansg 13987 Every additive subgroup of a non-unital ring is normal. (Contributed by AV, 25-Feb-2025.)
(𝑅 ∈ Rng → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))
 
25-Feb-2025ecqusaddd 13848 Addition of equivalence classes in a quotient group. (Contributed by AV, 25-Feb-2025.)
(𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))    &   𝐵 = (Base‘𝑅)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    ⇒   ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → [(𝐴(+g‘𝑅)𝐶)] ∼ = ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ ))
 
24-Feb-2025ecqusaddcl 13849 Closure of the addition in a quotient group. (Contributed by AV, 24-Feb-2025.)
(𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))    &   𝐵 = (Base‘𝑅)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    ⇒   ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ ) ∈ (Base‘𝑄))
 
24-Feb-2025quseccl0g 13841 Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) Generalization of quseccl 13843 for arbitrary sets 𝐺. (Revised by AV, 24-Feb-2025.)
∼ = (𝐺 ~QG 𝑆)    &   𝐻 = (𝐺 /s ∼ )    &   𝐶 = (Base‘𝐺)    &   𝐵 = (Base‘𝐻)    ⇒   ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝐶 ∧ 𝑆 ∈ 𝑍) → [𝑋] ∼ ∈ 𝐵)
 
23-Feb-2025ltlenmkv 16742 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 16740). (Contributed by Jim Kingdon, 23-Feb-2025.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥)) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≠ 𝑦 → 𝑥 # 𝑦))
 
23-Feb-2025neap0mkv 16741 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.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≠ 𝑦 → 𝑥 # 𝑦) ↔ ∀𝑥 ∈ ℝ (𝑥 ≠ 0 → 𝑥 # 0))
 
23-Feb-2025qus2idrng 14563 The quotient of a non-unital ring modulo a two-sided ideal, which is a subgroup of the additive group of the non-unital ring, is a non-unital ring (qusring 14565 analog). (Contributed by AV, 23-Feb-2025.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (2Ideal‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑈 ∈ Rng)
 
23-Feb-20252idlcpblrng 14561 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) Generalization for non-unital rings and two-sided ideals which are subgroups of the additive group of the non-unital ring. (Revised by AV, 23-Feb-2025.)
𝑋 = (Base‘𝑅)    &   𝐸 = (𝑅 ~QG 𝑆)    &   𝐼 = (2Ideal‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
 
23-Feb-2025lringuplu 14234 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.)
(𝜑 → 𝐵 = (Base‘𝑅))    &   (𝜑 → 𝑈 = (Unit‘𝑅))    &   (𝜑 → + = (+g‘𝑅))    &   (𝜑 → 𝑅 ∈ LRing)    &   (𝜑 → (𝑋 + 𝑌) ∈ 𝑈)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))
 
23-Feb-2025lringnz 14233 A local ring is a nonzero ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
1 = (1r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ LRing → 1 ≠ 0 )
 
23-Feb-2025lringring 14232 A local ring is a ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
(𝑅 ∈ LRing → 𝑅 ∈ Ring)
 
23-Feb-2025lringnzr 14231 A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
(𝑅 ∈ LRing → 𝑅 ∈ NzRing)
 
23-Feb-2025islring 14230 The predicate "is a local ring". (Contributed by SN, 23-Feb-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    1 = (1r‘𝑅)    &   𝑈 = (Unit‘𝑅)    ⇒   (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 1 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
 
23-Feb-2025df-lring 14229 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 = {𝑟 ∈ NzRing ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑥(+g‘𝑟)𝑦) = (1r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)))}
 
23-Feb-202501eq0ring 14227 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.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 })
 
23-Feb-2025nzrring 14221 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof shortened by SN, 23-Feb-2025.)
(𝑅 ∈ NzRing → 𝑅 ∈ Ring)
 
23-Feb-2025qusrng 13995 The quotient structure of a non-unital ring is a non-unital ring (qusring2 14103 analog). (Contributed by AV, 23-Feb-2025.)
(𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   (𝜑 → 𝑉 = (Base‘𝑅))    &    + = (+g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → ∼ Er 𝑉)    &   (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞)))    &   (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   (𝜑 → 𝑅 ∈ Rng)    ⇒   (𝜑 → 𝑈 ∈ Rng)
 
23-Feb-2025rngsubdir 13989 Ring multiplication distributes over subtraction. (subdir 8570 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 14094. (Revised by AV, 23-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    − = (-g‘𝑅)    &   (𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 − 𝑌) · 𝑍) = ((𝑋 · 𝑍) − (𝑌 · 𝑍)))
 
23-Feb-2025rngsubdi 13988 Ring multiplication distributes over subtraction. (subdi 8569 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdi 14093. (Revised by AV, 23-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    − = (-g‘𝑅)    &   (𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 · (𝑌 − 𝑍)) = ((𝑋 · 𝑌) − (𝑋 · 𝑍)))
 
22-Feb-2025imasrngf1 13994 The image of a non-unital ring under an injection is a non-unital ring. (Contributed by AV, 22-Feb-2025.)
𝑈 = (𝐹 “s 𝑅)    &   𝑉 = (Base‘𝑅)    ⇒   ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Rng) → 𝑈 ∈ Rng)
 
22-Feb-2025imasrng 13993 The image structure of a non-unital ring is a non-unital ring (imasring 14101 analog). (Contributed by AV, 22-Feb-2025.)
(𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   (𝜑 → 𝑉 = (Base‘𝑅))    &    + = (+g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑 → 𝑅 ∈ Rng)    ⇒   (𝜑 → 𝑈 ∈ Rng)
 
22-Feb-2025rngmgpf 13974 Restricted functionality of the multiplicative group on non-unital rings (mgpf 14048 analog). (Contributed by AV, 22-Feb-2025.)
(mulGrp ↾ Rng):Rng⟶Smgrp
 
22-Feb-2025imasabl 13946 The image structure of an abelian group is an abelian group (imasgrp 13721 analog). (Contributed by AV, 22-Feb-2025.)
(𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   (𝜑 → 𝑉 = (Base‘𝑅))    &   (𝜑 → + = (+g‘𝑅))    &   (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑 → 𝑅 ∈ Abel)    &    0 = (0g‘𝑅)    ⇒   (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘ 0 ) = (0g‘𝑈)))
 
21-Feb-2025prdssgrpd 13521 The product of a family of semigroups is a semigroup. (Contributed by AV, 21-Feb-2025.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑 → 𝐼 ∈ 𝑊)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅:𝐼⟶Smgrp)    ⇒   (𝜑 → 𝑌 ∈ Smgrp)
 
21-Feb-2025prdsplusgsgrpcl 13520 Structure product pointwise sums are closed when the factors are semigroups. (Contributed by AV, 21-Feb-2025.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    + = (+g‘𝑌)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐼 ∈ 𝑊)    &   (𝜑 → 𝑅:𝐼⟶Smgrp)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ 𝐵)    ⇒   (𝜑 → (𝐹 + 𝐺) ∈ 𝐵)
 
21-Feb-2025dftap2 7475 Tight apartness with the apartness properties from df-pap 7472 expanded. (Contributed by Jim Kingdon, 21-Feb-2025.)
(𝑅 TAp 𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦))))
 
20-Feb-2025rng2idlsubg0 14560 The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   (𝜑 → (0g‘𝑅) ∈ 𝐼)
 
20-Feb-2025rng2idlsubgnsg 14559 A two-sided ideal of a non-unital ring which is a subgroup of the ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
 
20-Feb-2025rng2idl0 14557 The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a non-unital ring. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   (𝜑 → (0g‘𝑅) ∈ 𝐼)
 
20-Feb-2025rng2idlnsg 14556 A two-sided ideal of a non-unital ring which is a non-unital ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
 
20-Feb-20252idlelbas 14554 The base set of a two-sided ideal as structure is a left and right ideal. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   𝐵 = (Base‘𝐽)    ⇒   (𝜑 → (𝐵 ∈ (LIdeal‘𝑅) ∧ 𝐵 ∈ (LIdeal‘(oppr‘𝑅))))
 
20-Feb-20252idlbas 14553 The base set of a two-sided ideal as structure. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   𝐵 = (Base‘𝐽)    ⇒   (𝜑 → 𝐵 = 𝐼)
 
20-Feb-20252idlelb 14543 Membership in a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.)
𝐼 = (LIdeal‘𝑅)    &   𝑂 = (oppr‘𝑅)    &   𝐽 = (LIdeal‘𝑂)    &   𝑇 = (2Ideal‘𝑅)    ⇒   (𝑈 ∈ 𝑇 ↔ (𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽))
 
20-Feb-2025aprap 14324 The relation given by df-apr 14319 for a local ring is an apartness relation. (Contributed by Jim Kingdon, 20-Feb-2025.)
(𝑅 ∈ LRing → (#r‘𝑅) Ap (Base‘𝑅))
 
20-Feb-2025setscomd 13146 Different components can be set in any order. (Contributed by Jim Kingdon, 20-Feb-2025.)
(𝜑 → 𝐴 ∈ 𝑌)    &   (𝜑 → 𝐵 ∈ 𝑍)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐴 ≠ 𝐵)    &   (𝜑 → 𝐶 ∈ 𝑊)    &   (𝜑 → 𝐷 ∈ 𝑋)    ⇒   (𝜑 → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
 
20-Feb-2025ifnebibdc 3652 The converse of ifbi 3627 holds if the two values are not equal. (Contributed by Thierry Arnoux, 20-Feb-2025.)
((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ (𝜑 ↔ 𝜓)))
 
20-Feb-2025ifnefals 3651 Deduce falsehood from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑)
 
20-Feb-2025ifnetruedc 3650 Deduce truth from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)
 
18-Feb-2025rnglidlmcl 14518 A (left) ideal containing the zero element is closed under left-multiplication by elements of the full non-unital ring. If the ring is not a unital ring, and the ideal does not contain the zero element of the ring, then the closure cannot be proven. (Contributed by AV, 18-Feb-2025.)
0 = (0g‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝑈 = (LIdeal‘𝑅)    ⇒   (((𝑅 ∈ Rng ∧ 𝐼 ∈ 𝑈 ∧ 0 ∈ 𝐼) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → (𝑋 · 𝑌) ∈ 𝐼)
 
17-Feb-2025aprcotr 14323 The apartness relation given by df-apr 14319 for a local ring is cotransitive. (Contributed by Jim Kingdon, 17-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝑅))    &   (𝜑 → # = (#r‘𝑅))    &   (𝜑 → 𝑅 ∈ LRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 # 𝑌 → (𝑋 # 𝑍 ∨ 𝑌 # 𝑍)))
 
17-Feb-2025aprsym 14322 The apartness relation given by df-apr 14319 for a ring is symmetric. (Contributed by Jim Kingdon, 17-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝑅))    &   (𝜑 → # = (#r‘𝑅))    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 # 𝑌 → 𝑌 # 𝑋))
 
17-Feb-2025aprval 14320 Expand Definition df-apr 14319. (Contributed by Jim Kingdon, 17-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝑅))    &   (𝜑 → # = (#r‘𝑅))    &   (𝜑 → − = (-g‘𝑅))    &   (𝜑 → 𝑈 = (Unit‘𝑅))    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 # 𝑌 ↔ (𝑋 − 𝑌) ∈ 𝑈))
 
17-Feb-2025subrngpropd 14254 If two structures have the same ring components (properties), they have the same set of subrings. (Contributed by AV, 17-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    ⇒   (𝜑 → (SubRng‘𝐾) = (SubRng‘𝐿))
 
17-Feb-2025rngm2neg 13986 Double negation of a product in a non-unital ring (mul2neg 8582 analog). (Contributed by Mario Carneiro, 4-Dec-2014.) Generalization of ringm2neg 14092. (Revised by AV, 17-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝑁 = (invg‘𝑅)    &   (𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ((𝑁‘𝑋) · (𝑁‘𝑌)) = (𝑋 · 𝑌))
 
17-Feb-2025rngmneg2 13985 Negation of a product in a non-unital ring (mulneg2 8580 analog). In contrast to ringmneg2 14091, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝑁 = (invg‘𝑅)    &   (𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 · (𝑁‘𝑌)) = (𝑁‘(𝑋 · 𝑌)))
 
17-Feb-2025rngmneg1 13984 Negation of a product in a non-unital ring (mulneg1 8579 analog). In contrast to ringmneg1 14090, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝑁 = (invg‘𝑅)    &   (𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ((𝑁‘𝑋) · 𝑌) = (𝑁‘(𝑋 · 𝑌)))
 
16-Feb-2025aprirr 14321 The apartness relation given by df-apr 14319 for a nonzero ring is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝑅))    &   (𝜑 → # = (#r‘𝑅))    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → (1r‘𝑅) ≠ (0g‘𝑅))    ⇒   (𝜑 → ¬ 𝑋 # 𝑋)
 
16-Feb-2025rngrz 13983 The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 14081. (Revised by AV, 16-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )
 
16-Feb-2025rng0cl 13980 The zero element of a non-unital ring belongs to its base set. (Contributed by AV, 16-Feb-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ Rng → 0 ∈ 𝐵)
 
16-Feb-2025rngacl 13979 Closure of the addition operation of a non-unital ring. (Contributed by AV, 16-Feb-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
 
16-Feb-2025rnggrp 13975 A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)
(𝑅 ∈ Rng → 𝑅 ∈ Grp)
 
16-Feb-2025aptap 8835 Complex apartness (as defined at df-ap 8767) is a tight apartness (as defined at df-tap 7474). (Contributed by Jim Kingdon, 16-Feb-2025.)
# TAp ℂ
 
15-Feb-2025subsubrng2 14253 The set of subrings of a subring are the smaller subrings. (Contributed by AV, 15-Feb-2025.)
𝑆 = (𝑅 ↾s 𝐴)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → (SubRng‘𝑆) = ((SubRng‘𝑅) ∩ 𝒫 𝐴))
 
15-Feb-2025subsubrng 14252 A subring of a subring is a subring. (Contributed by AV, 15-Feb-2025.)
𝑆 = (𝑅 ↾s 𝐴)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → (𝐵 ∈ (SubRng‘𝑆) ↔ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)))
 
15-Feb-2025subrngin 14251 The intersection of two subrings is a subring. (Contributed by AV, 15-Feb-2025.)
((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑅)) → (𝐴 ∩ 𝐵) ∈ (SubRng‘𝑅))
 
15-Feb-2025subrngintm 14250 The intersection of a nonempty collection of subrings is a subring. (Contributed by AV, 15-Feb-2025.)
((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → ∩ 𝑆 ∈ (SubRng‘𝑅))
 
15-Feb-2025opprsubrngg 14249 Being a subring is a symmetric property. (Contributed by AV, 15-Feb-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → (SubRng‘𝑅) = (SubRng‘𝑂))
 
15-Feb-2025issubrng2 14248 Characterize the subrings of a ring by closure properties. (Contributed by AV, 15-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
 
15-Feb-2025opprrngbg 14115 A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 14114. (Contributed by AV, 15-Feb-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → (𝑅 ∈ Rng ↔ 𝑂 ∈ Rng))
 
15-Feb-2025opprrng 14114 An opposite non-unital ring is a non-unital ring. (Contributed by AV, 15-Feb-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ Rng → 𝑂 ∈ Rng)
 
15-Feb-2025rngpropd 13992 If two structures have the same base set, and the values of their group (addition) and ring (multiplication) operations are equal for all pairs of elements of the base set, one is a non-unital ring iff the other one is. (Contributed by AV, 15-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    ⇒   (𝜑 → (𝐾 ∈ Rng ↔ 𝐿 ∈ Rng))
 
15-Feb-2025sgrppropd 13519 If two structures are sets, have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a semigroup iff the other one is. (Contributed by AV, 15-Feb-2025.)
(𝜑 → 𝐾 ∈ 𝑉)    &   (𝜑 → 𝐿 ∈ 𝑊)    &   (𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    ⇒   (𝜑 → (𝐾 ∈ Smgrp ↔ 𝐿 ∈ Smgrp))
 
15-Feb-2025sgrpcl 13515 Closure of the operation of a semigroup. (Contributed by AV, 15-Feb-2025.)
𝐵 = (Base‘𝐺)    &    ⚬ = (+g‘𝐺)    ⇒   ((𝐺 ∈ Smgrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)
 
15-Feb-2025tapeq2 7477 Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.)
(𝐴 = 𝐵 → (𝑅 TAp 𝐴 ↔ 𝑅 TAp 𝐵))
 
14-Feb-2025subrngmcl 14247 A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 14271. (Revised by AV, 14-Feb-2025.)
· = (.r‘𝑅)    ⇒   ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴)
 
14-Feb-2025subrngacl 14246 A subring is closed under addition. (Contributed by AV, 14-Feb-2025.)
+ = (+g‘𝑅)    ⇒   ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 + 𝑌) ∈ 𝐴)
 
14-Feb-2025subrng0 14245 A subring always has the same additive identity. (Contributed by AV, 14-Feb-2025.)
𝑆 = (𝑅 ↾s 𝐴)    &    0 = (0g‘𝑅)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → 0 = (0g‘𝑆))
 
14-Feb-2025subrngbas 14244 Base set of a subring structure. (Contributed by AV, 14-Feb-2025.)
𝑆 = (𝑅 ↾s 𝐴)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → 𝐴 = (Base‘𝑆))
 
14-Feb-2025subrngsubg 14242 A subring is a subgroup. (Contributed by AV, 14-Feb-2025.)
(𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
 
14-Feb-2025subrngrcl 14241 Reverse closure for a subring predicate. (Contributed by AV, 14-Feb-2025.)
(𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
 
14-Feb-2025subrngrng 14240 A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025.)
𝑆 = (𝑅 ↾s 𝐴)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng)
 
14-Feb-2025subrngid 14239 Every non-unital ring is a subring of itself. (Contributed by AV, 14-Feb-2025.)
𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ Rng → 𝐵 ∈ (SubRng‘𝑅))
 
14-Feb-2025subrngss 14238 A subring is a subset. (Contributed by AV, 14-Feb-2025.)
𝐵 = (Base‘𝑅)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵)
 
14-Feb-2025issubrng 14237 The subring of non-unital ring predicate. (Contributed by AV, 14-Feb-2025.)
𝐵 = (Base‘𝑅)    ⇒   (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))
 
14-Feb-2025df-subrng 14236 Define a subring of a non-unital ring as a set of elements that is a non-unital ring in its own right. In this section, a subring of a non-unital ring is simply called "subring", unless it causes any ambiguity with SubRing. (Contributed by AV, 14-Feb-2025.)
SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Rng})
 
14-Feb-2025isrngd 13990 Properties that determine a non-unital ring. (Contributed by AV, 14-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝑅))    &   (𝜑 → + = (+g‘𝑅))    &   (𝜑 → · = (.r‘𝑅))    &   (𝜑 → 𝑅 ∈ Abel)    &   ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))    ⇒   (𝜑 → 𝑅 ∈ Rng)
 
14-Feb-2025rngdi 13977 Distributive law for the multiplication operation of a non-unital ring (left-distributivity). (Contributed by AV, 14-Feb-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
 
14-Feb-2025exmidmotap 7485 The proposition that every class has at most one tight apartness is equivalent to excluded middle. (Contributed by Jim Kingdon, 14-Feb-2025.)
(EXMID ↔ ∀𝑥∃*𝑟 𝑟 TAp 𝑥)
 
14-Feb-2025exmidapne 7484 Excluded middle implies there is only one tight apartness on any class, namely negated equality. (Contributed by Jim Kingdon, 14-Feb-2025.)
(EXMID → (𝑅 TAp 𝐴 ↔ 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}))
 
14-Feb-2025df-pap 7472 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 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)))))
 
13-Feb-20252idl1 14551 Every ring contains a unit two-sided ideal. (Contributed by AV, 13-Feb-2025.)
𝐼 = (2Ideal‘𝑅)    &   𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ Ring → 𝐵 ∈ 𝐼)
 
13-Feb-20252idl0 14550 Every ring contains a zero two-sided ideal. (Contributed by AV, 13-Feb-2025.)
𝐼 = (2Ideal‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ Ring → { 0 } ∈ 𝐼)
 
13-Feb-2025ridl1 14549 Every ring contains a unit right ideal. (Contributed by AV, 13-Feb-2025.)
𝑈 = (LIdeal‘(oppr‘𝑅))    &   𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ Ring → 𝐵 ∈ 𝑈)
 
13-Feb-2025ridl0 14548 Every ring contains a zero right ideal. (Contributed by AV, 13-Feb-2025.)
𝑈 = (LIdeal‘(oppr‘𝑅))    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ Ring → { 0 } ∈ 𝑈)
 
13-Feb-2025isridl 14542 A right ideal is a left ideal of the opposite ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.)
𝑈 = (LIdeal‘(oppr‘𝑅))    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼)))
 
13-Feb-2025df-apr 14319 The relation between elements whose difference is invertible, which for a local ring is an apartness relation by aprap 14324. (Contributed by Jim Kingdon, 13-Feb-2025.)
#r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ (𝑥(-g‘𝑤)𝑦) ∈ (Unit‘𝑤))})
 
13-Feb-2025rngass 13976 Associative law for the multiplication operation of a non-unital ring. (Contributed by NM, 27-Aug-2011.) (Revised by AV, 13-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍)))
 
13-Feb-2025issgrpd 13518 Deduce a semigroup from its properties. (Contributed by AV, 13-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝐺))    &   (𝜑 → + = (+g‘𝐺))    &   ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝐺 ∈ 𝑉)    ⇒   (𝜑 → 𝐺 ∈ Smgrp)
 
8-Feb-20252oneel 7480 ∅ and 1o are two unequal elements of 2o. (Contributed by Jim Kingdon, 8-Feb-2025.)
⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)}
 
8-Feb-2025tapeq1 7476 Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.)
(𝑅 = 𝑆 → (𝑅 TAp 𝐴 ↔ 𝑆 TAp 𝐴))
 
7-Feb-2025psrgrp 14728 The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof shortened by SN, 7-Feb-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Grp)    ⇒   (𝜑 → 𝑆 ∈ Grp)
 
7-Feb-2025resrhm2b 14287 Restriction of the codomain of a (ring) homomorphism. resghm2b 13872 analog. (Contributed by SN, 7-Feb-2025.)
𝑈 = (𝑇 ↾s 𝑋)    ⇒   ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ 𝐹 ∈ (𝑆 RingHom 𝑈)))
 
6-Feb-2025zzlesq 10976 An integer is less than or equal to its square. (Contributed by BJ, 6-Feb-2025.)
(𝑁 ∈ ℤ → 𝑁 ≤ (𝑁↑2))
 
6-Feb-20252omotap 7483 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.)
(∃*𝑟 𝑟 TAp 2o → EXMID)
 
6-Feb-20252omotaplemst 7482 Lemma for 2omotap 7483. (Contributed by Jim Kingdon, 6-Feb-2025.)
((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → 𝜑)
 
6-Feb-20252omotaplemap 7481 Lemma for 2omotap 7483. (Contributed by Jim Kingdon, 6-Feb-2025.)
(¬ ¬ 𝜑 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} TAp 2o)
 
6-Feb-20252onetap 7479 Negated equality is a tight apartness on 2o. (Contributed by Jim Kingdon, 6-Feb-2025.)
{⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} TAp 2o
 
5-Feb-2025netap 7478 Negated equality on a set with decidable equality is a tight apartness. (Contributed by Jim Kingdon, 5-Feb-2025.)
(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} TAp 𝐴)
 
5-Feb-2025df-tap 7474 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 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦)))
 
1-Feb-2025mulgnn0cld 13753 Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 13748. (Contributed by SN, 1-Feb-2025.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    &   (𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑁 · 𝑋) ∈ 𝐵)
 
31-Jan-20250subg 13809 The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.) (Proof shortened by SN, 31-Jan-2025.)
0 = (0g‘𝐺)    ⇒   (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))
 
29-Jan-2025grprinvd 13662 The right inverse of a group element. Deduction associated with grprinv 13657. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   𝑁 = (invg‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 + (𝑁‘𝑋)) = 0 )
 
29-Jan-2025grplinvd 13661 The left inverse of a group element. Deduction associated with grplinv 13656. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   𝑁 = (invg‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → ((𝑁‘𝑋) + 𝑋) = 0 )
 
29-Jan-2025grpinvcld 13655 A group element's inverse is a group element. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑁‘𝑋) ∈ 𝐵)
 
29-Jan-2025grpridd 13640 The identity element of a group is a right identity. Deduction associated with grprid 13638. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 + 0 ) = 𝑋)
 
29-Jan-2025grplidd 13639 The identity element of a group is a left identity. Deduction associated with grplid 13637. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → ( 0 + 𝑋) = 𝑋)
 
29-Jan-2025grpassd 13618 A group operation is associative. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
 
28-Jan-2025dvdsrex 14136 Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)
(𝑅 ∈ SRing → (∥r‘𝑅) ∈ V)
 
24-Jan-2025reldvdsrsrg 14130 The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2025.)
(𝑅 ∈ SRing → Rel (∥r‘𝑅))
 
18-Jan-2025rerecapb 9028 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 → (𝐴 # 0 ↔ ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1))
 
18-Jan-2025recapb 8856 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 ∈ ℂ → (𝐴 # 0 ↔ ∃𝑥 ∈ ℂ (𝐴 · 𝑥) = 1))
 
17-Jan-2025ressval3d 13178 Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)
𝑅 = (𝑆 ↾s 𝐴)    &   𝐵 = (Base‘𝑆)    &   𝐸 = (Base‘ndx)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → 𝐸 ∈ dom 𝑆)    &   (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
 
17-Jan-2025strressid 13177 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 17-Jan-2025.)
(𝜑 → 𝐵 = (Base‘𝑊))    &   (𝜑 → 𝑊 Struct ⟨𝑀, 𝑁⟩)    &   (𝜑 → Fun 𝑊)    &   (𝜑 → (Base‘ndx) ∈ dom 𝑊)    ⇒   (𝜑 → (𝑊 ↾s 𝐵) = 𝑊)
 
17-Jan-2025snelpwg 4304 A singleton of a set is a member of the powerclass of a class if and only if that set is a member of that class. (Contributed by NM, 1-Apr-1998.) Put in closed form and avoid ax-nul 4216. (Revised by BJ, 17-Jan-2025.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))
 
16-Jan-2025ressex 13171 Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) ∈ V)
 
16-Jan-2025ressvalsets 13170 Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
 
12-Jan-2025isrim 14207 An isomorphism of rings is a bijective homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 12-Jan-2025.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    ⇒   (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))
 
10-Jan-2025rimrhm 14209 A ring isomorphism is a homomorphism. (Contributed by AV, 22-Oct-2019.) Remove hypotheses. (Revised by SN, 10-Jan-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RingHom 𝑆))
 
10-Jan-2025isrim0 14199 A ring isomorphism is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 10-Jan-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)))
 
10-Jan-2025opprex 14110 Existence of the opposite ring. If you know that 𝑅 is a ring, see opprring 14116. (Contributed by Jim Kingdon, 10-Jan-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
 
10-Jan-2025mgpex 13962 Existence of the multiplication group. If 𝑅 is known to be a semiring, see srgmgp 14005. (Contributed by Jim Kingdon, 10-Jan-2025.)
𝑀 = (mulGrp‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → 𝑀 ∈ V)
 
6-Jan-2025ord3 6599 Ordinal 3 is an ordinal class. (Contributed by BTernaryTau, 6-Jan-2025.)
Ord 3o
 
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.)
(((𝜑 → 𝜓) ↔ 𝜒) → (𝜑 → (𝜓 ↔ 𝜒)))
 
1-Jan-2025snss 3809 The singleton of an element of a class is a subset of the class (inference form of snssg 3808). Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.) (Proof shortened by BJ, 1-Jan-2025.)
𝐴 ∈ V    ⇒   (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)
 
1-Jan-2025snssg 3808 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 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))
 
1-Jan-2025snssb 3807 Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.)
({𝐴} ⊆ 𝐵 ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵))
 
30-Dec-2024rex2dom 7001 A set that has at least 2 different members dominates ordinal 2. (Contributed by BTernaryTau, 30-Dec-2024.)
((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦) → 2o ≼ 𝐴)
 
23-Dec-2024en2prd 6997 Two proper unordered pairs are equinumerous. (Contributed by BTernaryTau, 23-Dec-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐶 ∈ 𝑋)    &   (𝜑 → 𝐷 ∈ 𝑌)    &   (𝜑 → 𝐴 ≠ 𝐵)    &   (𝜑 → 𝐶 ≠ 𝐷)    ⇒   (𝜑 → {𝐴, 𝐵} ≈ {𝐶, 𝐷})
 
11-Dec-2024elopabr 4379 Membership in an ordered-pair class abstraction defined by a binary relation. (Contributed by AV, 16-Feb-2021.) (Proof shortened by SN, 11-Dec-2024.)
(𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅)
 
10-Dec-2024cbvreuw 2761 Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 2764 with a disjoint variable condition. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by GG, 10-Jan-2024.) (Revised by Wolf Lammen, 10-Dec-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
 
9-Dec-2024nninfwlpoim 7383 Decidable equality for ℕ∞ implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.)
(∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 → ω ∈ WOmni)
 
8-Dec-2024nninfinfwlpolem 7382 Lemma for nninfinfwlpo 7384. (Contributed by Jim Kingdon, 8-Dec-2024.)
(𝜑 → 𝐹:ω⟶2o)    &   𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o))    &   (𝜑 → ∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o))    ⇒   (𝜑 → DECID ∀𝑛 ∈ ω (𝐹‘𝑛) = 1o)
 
8-Dec-2024nninfwlpoimlemdc 7381 Lemma for nninfwlpoim 7383. (Contributed by Jim Kingdon, 8-Dec-2024.)
(𝜑 → 𝐹:ω⟶2o)    &   𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o))    &   (𝜑 → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦)    ⇒   (𝜑 → DECID ∀𝑛 ∈ ω (𝐹‘𝑛) = 1o)
 
8-Dec-2024nninfwlpoimlemginf 7380 Lemma for nninfwlpoim 7383. (Contributed by Jim Kingdon, 8-Dec-2024.)
(𝜑 → 𝐹:ω⟶2o)    &   𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o))    ⇒   (𝜑 → (𝐺 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑛 ∈ ω (𝐹‘𝑛) = 1o))
 
8-Dec-2024nninfwlpoimlemg 7379 Lemma for nninfwlpoim 7383. (Contributed by Jim Kingdon, 8-Dec-2024.)
(𝜑 → 𝐹:ω⟶2o)    &   𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o))    ⇒   (𝜑 → 𝐺 ∈ ℕ∞)
 
7-Dec-2024nninfwlpor 7378 The Weak Limited Principle of Omniscience (WLPO) implies that equality for ℕ∞ is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.)
(ω ∈ WOmni → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦)
 
7-Dec-2024nninfwlporlem 7377 Lemma for nninfwlpor 7378. The result. (Contributed by Jim Kingdon, 7-Dec-2024.)
(𝜑 → 𝑋:ω⟶2o)    &   (𝜑 → 𝑌:ω⟶2o)    &   𝐷 = (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))    &   (𝜑 → ω ∈ WOmni)    ⇒   (𝜑 → DECID 𝑋 = 𝑌)
 
7-Dec-2024domssr 6956 If 𝐶 is a superset of 𝐵 and 𝐵 dominates 𝐴, then 𝐶 also dominates 𝐴. (Contributed by BTernaryTau, 7-Dec-2024.)
((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶 ∧ 𝐴 ≼ 𝐵) → 𝐴 ≼ 𝐶)
 
7-Dec-2024f1dom4g 6931 The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 6936 does not require the Axiom of Collection nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
(((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
 
7-Dec-2024f1oen4g 6930 The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 6935 does not require the Axiom of Collection nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
(((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
 
6-Dec-2024nninfwlporlemd 7376 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.)
(𝜑 → 𝑋:ω⟶2o)    &   (𝜑 → 𝑌:ω⟶2o)    &   𝐷 = (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))    ⇒   (𝜑 → (𝑋 = 𝑌 ↔ 𝐷 = (𝑖 ∈ ω ↦ 1o)))
 
3-Dec-2024nninfwlpo 7385 Decidability of equality for ℕ∞ is equivalent to the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 3-Dec-2024.)
(∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 ↔ ω ∈ WOmni)
 
3-Dec-2024nninfdcinf 7375 The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of ℕ∞ equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.)
(𝜑 → ω ∈ WOmni)    &   (𝜑 → 𝑁 ∈ ℕ∞)    ⇒   (𝜑 → DECID 𝑁 = (𝑖 ∈ ω ↦ 1o))
 
29-Nov-2024brdom2g 6923 Dominance relation. This variation of brdomg 6924 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 6924. (Revised by BTernaryTau, 29-Nov-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
 
28-Nov-2024basmexd 13166 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 28-Nov-2024.)
(𝜑 → 𝐵 = (Base‘𝐺))    &   (𝜑 → 𝐴 ∈ 𝐵)    ⇒   (𝜑 → 𝐺 ∈ V)
 
22-Nov-2024eliotaeu 5317 An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.)
(𝐴 ∈ (℩𝑥𝜑) → ∃!𝑥𝜑)
 
22-Nov-2024eliota 5316 An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.)
(𝐴 ∈ (℩𝑥𝜑) ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
 
18-Nov-2024basmex 13165 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 18-Nov-2024.)
𝐵 = (Base‘𝐺)    ⇒   (𝐴 ∈ 𝐵 → 𝐺 ∈ V)
 
14-Nov-2024dcand 940 A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.) (Revised by BJ, 14-Nov-2024.)
(𝜑 → DECID 𝜓)    &   (𝜑 → DECID 𝜒)    ⇒   (𝜑 → DECID (𝜓 ∧ 𝜒))
 
12-Nov-2024sravscag 14481 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.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
 
12-Nov-2024srascag 14480 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.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
 
12-Nov-2024slotsdifipndx 13281 The slot for the scalar is not the index of other slots. (Contributed by AV, 12-Nov-2024.)
(( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·𝑖‘ndx))
 
11-Nov-2024bj-con1st 16408 Contraposition when the antecedent is a negated stable proposition. See con1dc 863. (Contributed by BJ, 11-Nov-2024.)
(STAB 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))
 
11-Nov-2024slotsdifdsndx 13331 The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.)
((*𝑟‘ndx) ≠ (dist‘ndx) ∧ (le‘ndx) ≠ (dist‘ndx))
 
11-Nov-2024plendxnocndx 13320 The slot for the orthocomplementation is not the slot for the order in an extensible structure. (Contributed by AV, 11-Nov-2024.)
(le‘ndx) ≠ (oc‘ndx)
 
11-Nov-2024basendxnocndx 13319 The slot for the orthocomplementation is not the slot for the base set in an extensible structure. (Contributed by AV, 11-Nov-2024.)
(Base‘ndx) ≠ (oc‘ndx)
 
11-Nov-2024slotsdifplendx 13316 The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.)
((*𝑟‘ndx) ≠ (le‘ndx) ∧ (TopSet‘ndx) ≠ (le‘ndx))
 
11-Nov-2024tsetndxnstarvndx 13300 The slot for the topology is not the slot for the involution in an extensible structure. (Contributed by AV, 11-Nov-2024.)
(TopSet‘ndx) ≠ (*𝑟‘ndx)
 
11-Nov-2024ofeqd 6242 Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)
(𝜑 → 𝑅 = 𝑆)    ⇒   (𝜑 → ∘𝑓 𝑅 = ∘𝑓 𝑆)
 
11-Nov-2024const 859 Contraposition when the antecedent is a negated stable proposition. See comment of condc 860. (Contributed by BJ, 18-Nov-2023.) (Proof shortened by BJ, 11-Nov-2024.)
(STAB 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
 
10-Nov-2024slotsdifunifndx 13338 The index of the slot for the uniform set is not the index of other slots. (Contributed by AV, 10-Nov-2024.)
(((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx)))
 
7-Nov-2024ressbasd 13173 Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
(𝜑 → 𝑅 = (𝑊 ↾s 𝐴))    &   (𝜑 → 𝐵 = (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (𝐴 ∩ 𝐵) = (Base‘𝑅))
 
6-Nov-2024oppraddg 14113 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
𝑂 = (oppr‘𝑅)    &    + = (+g‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → + = (+g‘𝑂))
 
6-Nov-2024opprbasg 14112 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
𝑂 = (oppr‘𝑅)    &   𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → 𝐵 = (Base‘𝑂))
 
6-Nov-2024opprsllem 14111 Lemma for opprbasg 14112 and oppraddg 14113. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)
𝑂 = (oppr‘𝑅)    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝐸‘ndx) ≠ (.r‘ndx)    ⇒   (𝑅 ∈ 𝑉 → (𝐸‘𝑅) = (𝐸‘𝑂))
 
4-Nov-2024lgsfvalg 15763 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 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))    ⇒   ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝐹‘𝑀) = if(𝑀 ∈ ℙ, (if(𝑀 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑀 − 1) / 2)) + 1) mod 𝑀) − 1))↑(𝑀 pCnt 𝑁)), 1))
 
3-Nov-2024znmul 14680 The multiplicative structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    ⇒   (𝑁 ∈ ℕ0 → (.r‘𝑈) = (.r‘𝑌))
 
3-Nov-2024znadd 14679 The additive structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    ⇒   (𝑁 ∈ ℕ0 → (+g‘𝑈) = (+g‘𝑌))
 
3-Nov-2024znbas2 14678 The base set of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    ⇒   (𝑁 ∈ ℕ0 → (Base‘𝑈) = (Base‘𝑌))
 
3-Nov-2024znbaslemnn 14677 Lemma for znbas 14682. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 9-Sep-2021.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ∈ ℕ    &   (𝐸‘ndx) ≠ (le‘ndx)    ⇒   (𝑁 ∈ ℕ0 → (𝐸‘𝑈) = (𝐸‘𝑌))
 
3-Nov-2024zlmmulrg 14669 Ring operation of a ℤ-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &    · = (.r‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → · = (.r‘𝑊))
 
3-Nov-2024zlmplusgg 14668 Group operation of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &    + = (+g‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → + = (+g‘𝑊))
 
3-Nov-2024zlmbasg 14667 Base set of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &   𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → 𝐵 = (Base‘𝑊))
 
3-Nov-2024zlmlemg 14666 Lemma for zlmbasg 14667 and zlmplusgg 14668. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ∈ ℕ    &   (𝐸‘ndx) ≠ (Scalar‘ndx)    &   (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)    ⇒   (𝐺 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑊))
 
2-Nov-2024zlmsca 14670 Scalar ring of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) (Proof shortened by AV, 2-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → ℤring = (Scalar‘𝑊))
 
1-Nov-2024plendxnvscandx 13315 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.)
(le‘ndx) ≠ ( ·𝑠 ‘ndx)
 
1-Nov-2024plendxnscandx 13314 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.)
(le‘ndx) ≠ (Scalar‘ndx)
 
1-Nov-2024plendxnmulrndx 13313 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.)
(le‘ndx) ≠ (.r‘ndx)
 
1-Nov-2024qsqeqor 10918 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 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
 
31-Oct-2024dsndxnmulrndx 13328 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.)
(dist‘ndx) ≠ (.r‘ndx)
 
31-Oct-2024tsetndxnmulrndx 13299 The slot for the topology is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(TopSet‘ndx) ≠ (.r‘ndx)
 
31-Oct-2024tsetndxnbasendx 13297 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.)
(TopSet‘ndx) ≠ (Base‘ndx)
 
31-Oct-2024basendxlttsetndx 13296 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.)
(Base‘ndx) < (TopSet‘ndx)
 
31-Oct-2024tsetndxnn 13295 The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 31-Oct-2024.)
(TopSet‘ndx) ∈ ℕ
 
30-Oct-2024basendxltedgfndx 15890 The index value of the Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 30-Oct-2024.)
(Base‘ndx) < (.ef‘ndx)
 
30-Oct-2024plendxnbasendx 13311 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.)
(le‘ndx) ≠ (Base‘ndx)
 
30-Oct-2024basendxltplendx 13310 The index value of the Base slot is less than the index value of the le slot. (Contributed by AV, 30-Oct-2024.)
(Base‘ndx) < (le‘ndx)
 
30-Oct-2024plendxnn 13309 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.)
(le‘ndx) ∈ ℕ
 
29-Oct-2024sradsg 14486 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.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → (dist‘𝑊) = (dist‘𝐴))
 
29-Oct-2024sratsetg 14483 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.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → (TopSet‘𝑊) = (TopSet‘𝐴))
 
29-Oct-2024sramulrg 14479 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.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → (.r‘𝑊) = (.r‘𝐴))
 
29-Oct-2024sraaddgg 14478 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.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → (+g‘𝑊) = (+g‘𝐴))
 
29-Oct-2024srabaseg 14477 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.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    ⇒   (𝜑 → (Base‘𝑊) = (Base‘𝐴))
 
29-Oct-2024sralemg 14476 Lemma for srabaseg 14477 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   (𝜑 → 𝑊 ∈ 𝑋)    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (Scalar‘ndx) ≠ (𝐸‘ndx)    &   ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)    &   (·𝑖‘ndx) ≠ (𝐸‘ndx)    ⇒   (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴))
 
29-Oct-2024dsndxntsetndx 13330 The slot for the distance function is not the slot for the topology in an extensible structure. (Contributed by AV, 29-Oct-2024.)
(dist‘ndx) ≠ (TopSet‘ndx)
 
29-Oct-2024slotsdnscsi 13329 The slots Scalar, ·𝑠 and ·𝑖 are different from the slot dist. (Contributed by AV, 29-Oct-2024.)
((dist‘ndx) ≠ (Scalar‘ndx) ∧ (dist‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (dist‘ndx) ≠ (·𝑖‘ndx))
 
29-Oct-2024slotstnscsi 13301 The slots Scalar, ·𝑠 and ·𝑖 are different from the slot TopSet. (Contributed by AV, 29-Oct-2024.)
((TopSet‘ndx) ≠ (Scalar‘ndx) ∧ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (TopSet‘ndx) ≠ (·𝑖‘ndx))
 
29-Oct-2024ipndxnmulrndx 13280 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.)
(·𝑖‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024ipndxnplusgndx 13279 The slot for the inner product is not the slot for the group operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
(·𝑖‘ndx) ≠ (+g‘ndx)
 
29-Oct-2024vscandxnmulrndx 13267 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 ‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024scandxnmulrndx 13262 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.)
(Scalar‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024fiubnn 11100 A finite set of natural numbers has an upper bound which is a a natural number. (Contributed by Jim Kingdon, 29-Oct-2024.)
((𝐴 ⊆ ℕ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
 
29-Oct-2024fiubz 11099 A finite set of integers has an upper bound which is an integer. (Contributed by Jim Kingdon, 29-Oct-2024.)
((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
 
29-Oct-2024fiubm 11098 Lemma for fiubz 11099 and fiubnn 11100. A general form of those theorems. (Contributed by Jim Kingdon, 29-Oct-2024.)
(𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → 𝐵 ⊆ ℚ)    &   (𝜑 → 𝐶 ∈ 𝐵)    &   (𝜑 → 𝐴 ∈ Fin)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
 
28-Oct-2024edgfndxid 15889 The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 28-Oct-2024.)
(𝐺 ∈ 𝑉 → (.ef‘𝐺) = (𝐺‘(.ef‘ndx)))
 
28-Oct-2024unifndxntsetndx 13337 The slot for the uniform set is not the slot for the topology in an extensible structure. (Contributed by AV, 28-Oct-2024.)
(UnifSet‘ndx) ≠ (TopSet‘ndx)
 
28-Oct-2024basendxltunifndx 13335 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.)
(Base‘ndx) < (UnifSet‘ndx)
 
28-Oct-2024unifndxnn 13334 The index of the slot for the uniform set in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
(UnifSet‘ndx) ∈ ℕ
 
28-Oct-2024dsndxnbasendx 13326 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.)
(dist‘ndx) ≠ (Base‘ndx)
 
28-Oct-2024basendxltdsndx 13325 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.)
(Base‘ndx) < (dist‘ndx)
 
28-Oct-2024dsndxnn 13324 The index of the slot for the distance in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
(dist‘ndx) ∈ ℕ
 
27-Oct-2024bj-nnst 16400 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 16645 for the version not using the definition of stability. (Contributed by BJ, 9-Oct-2019.) Prove it in ( → , ¬ ) -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 ( → , ↔ , ¬ )-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)
¬ ¬ STAB 𝜑
 
27-Oct-2024bj-imnimnn 16395 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 16394 as its last step. (Contributed by BJ, 27-Oct-2024.)
(𝜑 → 𝜓)    &   (¬ 𝜑 → 𝜓)    ⇒    ¬ ¬ 𝜓
 
25-Oct-2024nnwosdc 12633 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.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ((∃𝑥 ∈ ℕ 𝜑 ∧ ∀𝑥 ∈ ℕ DECID 𝜑) → ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)))
 
23-Oct-2024nnwodc 12630 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 ⊆ ℕ ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
 
22-Oct-2024uzwodc 12631 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 ⊆ (ℤ≥‘𝑀) ∧ ∃𝑥 𝑥 ∈ 𝑆 ∧ ∀𝑥 ∈ (ℤ≥‘𝑀)DECID 𝑥 ∈ 𝑆) → ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘)
 
21-Oct-2024nnnotnotr 16645 Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 857, it can be proved with reference only to implication and negation (that is, without use of disjunction). (Contributed by Jim Kingdon, 21-Oct-2024.)
¬ ¬ (¬ ¬ 𝜑 → 𝜑)
 
21-Oct-2024unifndxnbasendx 13336 The slot for the uniform set is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(UnifSet‘ndx) ≠ (Base‘ndx)
 
21-Oct-2024ipndxnbasendx 13278 The slot for the inner product is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(·𝑖‘ndx) ≠ (Base‘ndx)
 
21-Oct-2024scandxnbasendx 13260 The slot for the scalar is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(Scalar‘ndx) ≠ (Base‘ndx)
 
20-Oct-2024isprm5lem 12736 Lemma for isprm5 12737. 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.)
(𝜑 → 𝑃 ∈ (ℤ≥‘2))    &   (𝜑 → ∀𝑧 ∈ ℙ ((𝑧↑2) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃))    &   (𝜑 → 𝑋 ∈ (2...(𝑃 − 1)))    ⇒   (𝜑 → ¬ 𝑋 ∥ 𝑃)
 
19-Oct-2024resseqnbasd 13179 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 = (𝑊 ↾s 𝐴)    &   𝐶 = (𝐸‘𝑊)    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝐸‘ndx) ≠ (Base‘ndx)    &   (𝜑 → 𝑊 ∈ 𝑋)    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → 𝐶 = (𝐸‘𝑅))
 
18-Oct-2024rmodislmod 14389 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 14327 of a left module, see also islmod 14329. (Contributed by AV, 3-Dec-2021.) (Proof shortened by AV, 18-Oct-2024.)
𝑉 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    · = ( ·𝑠 ‘𝑅)    &   𝐹 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐹)    &    ⨣ = (+g‘𝐹)    &    × = (.r‘𝐹)    &    1 = (1r‘𝐹)    &   (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑤 · 𝑟) ∈ 𝑉 ∧ ((𝑤 + 𝑥) · 𝑟) = ((𝑤 · 𝑟) + (𝑥 · 𝑟)) ∧ (𝑤 · (𝑞 ⨣ 𝑟)) = ((𝑤 · 𝑞) + (𝑤 · 𝑟))) ∧ ((𝑤 · (𝑞 × 𝑟)) = ((𝑤 · 𝑞) · 𝑟) ∧ (𝑤 · 1 ) = 𝑤)))    &    ∗ = (𝑠 ∈ 𝐾, 𝑣 ∈ 𝑉 ↦ (𝑣 · 𝑠))    &   𝐿 = (𝑅 sSet ⟨( ·𝑠 ‘ndx), ∗ ⟩)    ⇒   (𝐹 ∈ CRing → 𝐿 ∈ LMod)
 
18-Oct-2024mgpress 13968 Subgroup commutes with the multiplicative group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.)
𝑆 = (𝑅 ↾s 𝐴)    &   𝑀 = (mulGrp‘𝑅)    ⇒   ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (mulGrp‘𝑆))
 
18-Oct-2024dsndxnplusgndx 13327 The slot for the distance function is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(dist‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024plendxnplusgndx 13312 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.)
(le‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024tsetndxnplusgndx 13298 The slot for the topology is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(TopSet‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024vscandxnscandx 13268 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 ‘ndx) ≠ (Scalar‘ndx)
 
18-Oct-2024vscandxnplusgndx 13266 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 ‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024vscandxnbasendx 13265 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 ‘ndx) ≠ (Base‘ndx)
 
18-Oct-2024scandxnplusgndx 13261 The slot for the scalar field is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(Scalar‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024starvndxnmulrndx 13250 The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(*𝑟‘ndx) ≠ (.r‘ndx)
 
18-Oct-2024starvndxnplusgndx 13249 The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(*𝑟‘ndx) ≠ (+g‘ndx)
 
18-Oct-2024starvndxnbasendx 13248 The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(*𝑟‘ndx) ≠ (Base‘ndx)
 
17-Oct-2024basendxltplusgndx 13219 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.)
(Base‘ndx) < (+g‘ndx)
 
17-Oct-2024plusgndxnn 13217 The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 17-Oct-2024.)
(+g‘ndx) ∈ ℕ
 
17-Oct-2024elnndc 9851 Membership of an integer in ℕ is decidable. (Contributed by Jim Kingdon, 17-Oct-2024.)
(𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ)
 
14-Oct-20242zinfmin 11826 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 ∈ ℤ ∧ 𝐵 ∈ ℤ) → inf({𝐴, 𝐵}, ℝ, < ) = if(𝐴 ≤ 𝐵, 𝐴, 𝐵))
 
14-Oct-2024mingeb 11825 Equivalence of ≤ and being equal to the minimum of two reals. (Contributed by Jim Kingdon, 14-Oct-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ inf({𝐴, 𝐵}, ℝ, < ) = 𝐴))
 
13-Oct-2024edgfndxnn 15888 The index value of the edge function extractor 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, 21-Sep-2020.) (Proof shortened by AV, 13-Oct-2024.)
(.ef‘ndx) ∈ ℕ
 
13-Oct-2024edgfndx 15887 Index value of the df-edgf 15885 slot. (Contributed by AV, 13-Oct-2024.) (New usage is discouraged.)
(.ef‘ndx) = 18
 
13-Oct-2024prdsvallem 13378 Lemma for prdsval 13379. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 13379, dependency on df-hom 13207 removed. (Revised by AV, 13-Oct-2024.)
(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V
 
13-Oct-2024pcxnn0cl 12906 Extended nonnegative integer closure of the general prime count function. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 pCnt 𝑁) ∈ ℕ0*)
 
13-Oct-2024xnn0letri 10043 Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
 
13-Oct-2024xnn0dcle 10042 Decidability of ≤ for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → DECID 𝐴 ≤ 𝐵)
 
9-Oct-2024nn0leexp2 10978 Ordering law for exponentiation. (Contributed by Jim Kingdon, 9-Oct-2024.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ 1 < 𝐴) → (𝑀 ≤ 𝑁 ↔ (𝐴↑𝑀) ≤ (𝐴↑𝑁)))
 
8-Oct-2024pclemdc 12884 Lemma for the prime power pre-function's properties. (Contributed by Jim Kingdon, 8-Oct-2024.)
𝐴 = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁}    ⇒   ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)
 
8-Oct-2024elnn0dc 9850 Membership of an integer in ℕ0 is decidable. (Contributed by Jim Kingdon, 8-Oct-2024.)
(𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ0)
 
7-Oct-2024pclemub 12883 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 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁}    ⇒   ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
 
7-Oct-2024pclem0 12882 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 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁}    ⇒   ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 0 ∈ 𝐴)
 
7-Oct-2024nn0ltexp2 10977 Special case of ltexp2 15694 which we use here because we haven't yet defined df-rpcxp 15612 which is used in the current proof of ltexp2 15694. (Contributed by Jim Kingdon, 7-Oct-2024.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁)))
 
6-Oct-2024suprzcl2dc 10505 The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 8158.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 6-Oct-2024.)
(𝜑 → 𝐴 ⊆ ℤ)    &   (𝜑 → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    &   (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    ⇒   (𝜑 → sup(𝐴, ℝ, < ) ∈ 𝐴)
 
5-Oct-2024zsupssdc 10504 An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 8158.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
(𝜑 → 𝐴 ⊆ ℤ)    &   (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
 
5-Oct-2024suprzubdc 10502 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.)
(𝜑 → 𝐴 ⊆ ℤ)    &   (𝜑 → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    &   (𝜑 → 𝐵 ∈ 𝐴)    ⇒   (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < ))
 
1-Oct-2024infex2g 7238 Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
(𝐴 ∈ 𝐶 → inf(𝐵, 𝐴, 𝑅) ∈ V)
 
30-Sep-2024unbendc 13098 An unbounded decidable set of positive integers is infinite. (Contributed by NM, 5-May-2005.) (Revised by Jim Kingdon, 30-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → 𝐴 ≈ ℕ)
 
30-Sep-2024prmdc 12725 Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.)
(𝑁 ∈ ℕ → DECID 𝑁 ∈ ℙ)
 
30-Sep-2024dcfi 7185 Decidability of a family of propositions indexed by a finite set. (Contributed by Jim Kingdon, 30-Sep-2024.)
((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) → DECID ∀𝑥 ∈ 𝐴 𝜑)
 
30-Sep-2024cbvriotavw 5987 Change bound variable in a restricted description binder. Version of cbvriotav 5989 with a disjoint variable condition. (Contributed by NM, 18-Mar-2013.) (Revised by GG, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
 
30-Sep-2024cbviotavw 5294 Change bound variables in a description binder. Version of cbviotav 5295 with a disjoint variable condition. (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by GG, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (℩𝑥𝜑) = (℩𝑦𝜓)
 
29-Sep-2024ssnnct 13091 A decidable subset of ℕ is countable. (Contributed by Jim Kingdon, 29-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o))
 
29-Sep-2024ssnnctlemct 13090 Lemma for ssnnct 13091. The result. (Contributed by Jim Kingdon, 29-Sep-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 1)    ⇒   ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o))
 
28-Sep-2024nninfdcex 10503 A decidable set of natural numbers has an infimum. (Contributed by Jim Kingdon, 28-Sep-2024.)
(𝜑 → 𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑦 𝑦 ∈ 𝐴)    ⇒   (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
 
27-Sep-2024infregelbex 9837 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.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))    &   (𝜑 → 𝐴 ⊆ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    ⇒   (𝜑 → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧))
 
26-Sep-2024nninfdclemp1 13094 Lemma for nninfdc 13097. Each element of the sequence 𝐹 is greater than the previous element. (Contributed by Jim Kingdon, 26-Sep-2024.)
(𝜑 → 𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽 ∈ 𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ≥‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))    &   (𝜑 → 𝑈 ∈ ℕ)    ⇒   (𝜑 → (𝐹‘𝑈) < (𝐹‘(𝑈 + 1)))
 
26-Sep-2024nnminle 12629 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 12628. (Contributed by Jim Kingdon, 26-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝐵)
 
25-Sep-2024nninfdclemcl 13092 Lemma for nninfdc 13097. (Contributed by Jim Kingdon, 25-Sep-2024.)
(𝜑 → 𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛)    &   (𝜑 → 𝑃 ∈ 𝐴)    &   (𝜑 → 𝑄 ∈ 𝐴)    ⇒   (𝜑 → (𝑃(𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ≥‘(𝑦 + 1))), ℝ, < ))𝑄) ∈ 𝐴)
 
24-Sep-2024nninfdclemlt 13095 Lemma for nninfdc 13097. The function from nninfdclemf 13093 is strictly monotonic. (Contributed by Jim Kingdon, 24-Sep-2024.)
(𝜑 → 𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽 ∈ 𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ≥‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))    &   (𝜑 → 𝑈 ∈ ℕ)    &   (𝜑 → 𝑉 ∈ ℕ)    &   (𝜑 → 𝑈 < 𝑉)    ⇒   (𝜑 → (𝐹‘𝑈) < (𝐹‘𝑉))
 
23-Sep-2024nninfdc 13097 An unbounded decidable set of positive integers is infinite. (Contributed by Jim Kingdon, 23-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛) → ω ≼ 𝐴)
 
23-Sep-2024nninfdclemf1 13096 Lemma for nninfdc 13097. The function from nninfdclemf 13093 is one-to-one. (Contributed by Jim Kingdon, 23-Sep-2024.)
(𝜑 → 𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽 ∈ 𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ≥‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))    ⇒   (𝜑 → 𝐹:ℕ–1-1→𝐴)
 
23-Sep-2024nninfdclemf 13093 Lemma for nninfdc 13097. A function from the natural numbers into 𝐴. (Contributed by Jim Kingdon, 23-Sep-2024.)
(𝜑 → 𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛 ∈ 𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽 ∈ 𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ≥‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))    ⇒   (𝜑 → 𝐹:ℕ⟶𝐴)
 
23-Sep-2024nnmindc 12628 An inhabited decidable subset of the natural numbers has a minimum. (Contributed by Jim Kingdon, 23-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐴) → inf(𝐴, ℝ, < ) ∈ 𝐴)
 
23-Sep-2024breng 6921 Equinumerosity relation. This variation of bren 6922 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 6922. (Revised by BTernaryTau, 23-Sep-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
 
19-Sep-2024ssomct 13089 A decidable subset of ω is countable. (Contributed by Jim Kingdon, 19-Sep-2024.)
((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω DECID 𝑥 ∈ 𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o))
 
19-Sep-20242oex 6604 2o is a set. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by Zhi Wang, 19-Sep-2024.)
2o ∈ V
 
19-Sep-2024ecase2d 1387 Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Sep-2024.)
(𝜑 → 𝜓)    &   (𝜑 → ¬ (𝜓 ∧ 𝜒))    &   (𝜑 → ¬ (𝜓 ∧ 𝜃))    &   (𝜑 → (𝜏 ∨ (𝜒 ∨ 𝜃)))    ⇒   (𝜑 → 𝜏)
 
18-Sep-2024fcof 5836 Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 5502. (Contributed by AV, 18-Sep-2024.)
((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
 
17-Sep-2024fncofn 5835 Composition of a function with domain and a function as a function with domain. Generalization of fnco 5442. (Contributed by AV, 17-Sep-2024.)
((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))
 
14-Sep-2024nnpredlt 4724 The predecessor (see nnpredcl 4723) of a nonzero natural number is less than (see df-iord 4465) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴)
 
13-Sep-2024nninfisollemeq 7336 Lemma for nninfisol 7337. The case where 𝑁 is a successor and 𝑁 and 𝑋 are equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
(𝜑 → 𝑋 ∈ ℕ∞)    &   (𝜑 → (𝑋‘𝑁) = ∅)    &   (𝜑 → 𝑁 ∈ ω)    &   (𝜑 → 𝑁 ≠ ∅)    &   (𝜑 → (𝑋‘∪ 𝑁) = 1o)    ⇒   (𝜑 → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
 
13-Sep-2024nninfisollemne 7335 Lemma for nninfisol 7337. A case where 𝑁 is a successor and 𝑁 and 𝑋 are not equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
(𝜑 → 𝑋 ∈ ℕ∞)    &   (𝜑 → (𝑋‘𝑁) = ∅)    &   (𝜑 → 𝑁 ∈ ω)    &   (𝜑 → 𝑁 ≠ ∅)    &   (𝜑 → (𝑋‘∪ 𝑁) = ∅)    ⇒   (𝜑 → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
 
13-Sep-2024nninfisollem0 7334 Lemma for nninfisol 7337. The case where 𝑁 is zero. (Contributed by Jim Kingdon, 13-Sep-2024.)
(𝜑 → 𝑋 ∈ ℕ∞)    &   (𝜑 → (𝑋‘𝑁) = ∅)    &   (𝜑 → 𝑁 ∈ ω)    &   (𝜑 → 𝑁 = ∅)    ⇒   (𝜑 → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
 
12-Sep-2024nninfisol 7337 Finite elements of ℕ∞ are isolated. That is, given a natural number and any element of ℕ∞, it is decidable whether the natural number (when converted to an element of ℕ∞) is equal to the given element of ℕ∞. 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).

By contrast, the point at infinity being isolated is equivalent to the Weak Limited Principle of Omniscience (WLPO) (nninfinfwlpo 7384). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.)

((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
 
8-Sep-2024relopabv 4856 A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopab 4858. (Contributed by SN, 8-Sep-2024.)
Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
7-Sep-2024eulerthlemfi 12823 Lemma for eulerth 12828. The set 𝑆 is finite. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.)
(𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    ⇒   (𝜑 → 𝑆 ∈ Fin)
 
7-Sep-2024modqexp 10934 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.)
(𝜑 → 𝐴 ∈ ℤ)    &   (𝜑 → 𝐵 ∈ ℤ)    &   (𝜑 → 𝐶 ∈ ℕ0)    &   (𝜑 → 𝐷 ∈ ℚ)    &   (𝜑 → 0 < 𝐷)    &   (𝜑 → (𝐴 mod 𝐷) = (𝐵 mod 𝐷))    ⇒   (𝜑 → ((𝐴↑𝐶) mod 𝐷) = ((𝐵↑𝐶) mod 𝐷))
 
5-Sep-2024eulerthlemh 12826 Lemma for eulerth 12828. A permutation of (1...(ϕ‘𝑁)). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 5-Sep-2024.)
(𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)    &   𝐻 = (◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)))    ⇒   (𝜑 → 𝐻:(1...(ϕ‘𝑁))–1-1-onto→(1...(ϕ‘𝑁)))
 
2-Sep-2024eulerthlemth 12827 Lemma for eulerth 12828. The result. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
(𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)    ⇒   (𝜑 → ((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁))
 
2-Sep-2024eulerthlema 12825 Lemma for eulerth 12828. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
(𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)    ⇒   (𝜑 → (((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) mod 𝑁) = (∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) mod 𝑁))
 
2-Sep-2024eulerthlemrprm 12824 Lemma for eulerth 12828. 𝑁 and ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) are relatively prime. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
(𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))    &   𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}    &   (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)    ⇒   (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = 1)
 
1-Sep-2024qusmul2 14567 Value of the ring operation in a quotient ring. (Contributed by Thierry Arnoux, 1-Sep-2024.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    × = (.r‘𝑄)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ([𝑋](𝑅 ~QG 𝐼) × [𝑌](𝑅 ~QG 𝐼)) = [(𝑋 · 𝑌)](𝑅 ~QG 𝐼))
 
30-Aug-2024fprodap0f 12220 A finite product of terms apart from zero is apart from zero. A version of fprodap0 12205 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by Jim Kingdon, 30-Aug-2024.)
Ⅎ𝑘𝜑    &   (𝜑 → 𝐴 ∈ Fin)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 # 0)    ⇒   (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 # 0)
 
28-Aug-2024fprodrec 12213 The finite product of reciprocals is the reciprocal of the product. (Contributed by Jim Kingdon, 28-Aug-2024.)
(𝜑 → 𝐴 ∈ Fin)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 # 0)    ⇒   (𝜑 → ∏𝑘 ∈ 𝐴 (1 / 𝐵) = (1 / ∏𝑘 ∈ 𝐴 𝐵))
 
26-Aug-2024exmidontri2or 7466 Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
(EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
 
26-Aug-2024exmidontri 7462 Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
(EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
 
26-Aug-2024ontri2orexmidim 4672 Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4671. (Contributed by Jim Kingdon, 26-Aug-2024.)
(∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → DECID 𝜑)
 
26-Aug-2024ontriexmidim 4622 Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4621. (Contributed by Jim Kingdon, 26-Aug-2024.)
(∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → DECID 𝜑)
 
25-Aug-2024onntri2or 7469 Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
(¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
 
25-Aug-2024onntri3or 7468 Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
(¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
 
25-Aug-2024csbcow 3137 Composition law for chained substitutions into a class. Version of csbco 3136 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-Nov-2005.) (Revised by GG, 25-Aug-2024.)
⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵
 
25-Aug-2024cbvreuvw 2772 Version of cbvreuv 2768 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
 
25-Aug-2024cbvrexvw 2771 Version of cbvrexv 2767 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
 
25-Aug-2024cbvralvw 2770 Version of cbvralv 2766 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
 
25-Aug-2024cbvabw 2353 Version of cbvab 2354 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
 
25-Aug-2024nfsbv 1999 If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑧 is distinct from 𝑥 and 𝑦. Version of nfsb 1998 requiring more disjoint variables. (Contributed by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
Ⅎ𝑧𝜑    ⇒   Ⅎ𝑧[𝑦 / 𝑥]𝜑
 
25-Aug-2024cbvexvw 1968 Change bound variable. See cbvexv 1966 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1496. (Revised by GG, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥𝜑 ↔ ∃𝑦𝜓)
 
25-Aug-2024cbvalvw 1967 Change bound variable. See cbvalv 1965 for a version with fewer disjoint variable conditions. (Contributed by NM, 9-Apr-2017.) Avoid ax-7 1496. (Revised by GG, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 
25-Aug-2024nfal 1624 If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-4 1558. (Revised by GG, 25-Aug-2024.)
Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∀𝑦𝜑
 
24-Aug-2024gcdcomd 12568 The gcd operator is commutative, deduction version. (Contributed by SN, 24-Aug-2024.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀))
 
21-Aug-2024dvds2addd 12413 Deduction form of dvds2add 12409. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐾 ∈ ℤ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ∥ 𝑀)    &   (𝜑 → 𝐾 ∥ 𝑁)    ⇒   (𝜑 → 𝐾 ∥ (𝑀 + 𝑁))
 
18-Aug-2024prdsmulr 13384 Multiplication in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    · = (.r‘𝑃)    ⇒   (𝜑 → · = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
 
18-Aug-2024prdsplusg 13383 Addition in a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    + = (+g‘𝑃)    ⇒   (𝜑 → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
 
18-Aug-2024prdsbas 13382 Base set of a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    ⇒   (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
 
18-Aug-2024prdssca 13381 Scalar ring of a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    ⇒   (𝜑 → 𝑆 = (Scalar‘𝑃))
 
18-Aug-2024prdsval 13379 Value of the structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   𝐾 = (Base‘𝑆)    &   (𝜑 → dom 𝑅 = 𝐼)    &   (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))    &   (𝜑 → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))    &   (𝜑 → × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))))    &   (𝜑 → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))    &   (𝜑 → , = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))))    &   (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))    &   (𝜑 → ≤ = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})    &   (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )))    &   (𝜑 → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))    &   (𝜑 → ∙ = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))    &   (𝜑 → 𝑆 ∈ 𝑊)    &   (𝜑 → 𝑅 ∈ 𝑍)    ⇒   (𝜑 → 𝑃 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})))
 
18-Aug-2024df-prds 13373 Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ ⦋X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ ((2nd ‘𝑎)ℎ𝑐), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))
 
17-Aug-2024fprodcl2lem 12189 Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.) (Revised by Jim Kingdon, 17-Aug-2024.)
(𝜑 → 𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   (𝜑 → 𝐴 ∈ Fin)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆)    &   (𝜑 → 𝐴 ≠ ∅)    ⇒   (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)
 
16-Aug-2024fprodunsn 12188 Multiply in an additional term in a finite product. See also fprodsplitsn 12217 which is the same but with a Ⅎ𝑘𝜑 hypothesis in place of the distinct variable condition between 𝜑 and 𝑘. (Contributed by Jim Kingdon, 16-Aug-2024.)
Ⅎ𝑘𝐷    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → ¬ 𝐵 ∈ 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐷 ∈ ℂ)    &   (𝑘 = 𝐵 → 𝐶 = 𝐷)    ⇒   (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷))
 
16-Aug-2024if0ab 3607 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. (Contributed by BJ, 16-Aug-2024.)
if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}
 
15-Aug-2024bj-charfundcALT 16464 Alternate proof of bj-charfundc 16463. It was expected to be much shorter since it uses bj-charfun 16462 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.)
(𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    &   (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)    ⇒   (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
 
15-Aug-2024bj-charfun 16462 Properties of the characteristic function on the class 𝑋 of the class 𝐴. (Contributed by BJ, 15-Aug-2024.)
(𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    ⇒   (𝜑 → ((𝐹:𝑋⟶𝒫 1o ∧ (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))):((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))⟶2o) ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
 
15-Aug-2024cnstab 8830 Equality of complex numbers is stable. Stability here means ¬ ¬ 𝐴 = 𝐵 → 𝐴 = 𝐵 as defined at df-stab 838. 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 ∈ ℂ ∧ 𝐵 ∈ ℂ) → STAB 𝐴 = 𝐵)
 
15-Aug-2024subap0d 8829 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.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐴 # 𝐵)    ⇒   (𝜑 → (𝐴 − 𝐵) # 0)
 
15-Aug-2024ifexd 4583 Existence of a conditional class (deduction form). (Contributed by BJ, 15-Aug-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V)
 
15-Aug-2024ifelpwun 4582 Existence of a conditional class, quantitative version (inference form). (Contributed by BJ, 15-Aug-2024.)
𝐴 ∈ V    &   𝐵 ∈ V    ⇒   if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵)
 
15-Aug-2024ifelpwund 4581 Existence of a conditional class, quantitative version (deduction form). (Contributed by BJ, 15-Aug-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵))
 
15-Aug-2024ifelpwung 4580 Existence of a conditional class, quantitative version (closed form). (Contributed by BJ, 15-Aug-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵))
 
15-Aug-2024ifidss 3622 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(𝜑, 𝐴, 𝐴) ⊆ 𝐴
 
15-Aug-2024ifssun 3621 A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.)
if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)
 
12-Aug-2024exmidontriimlem2 7442 Lemma for exmidontriim 7445. (Contributed by Jim Kingdon, 12-Aug-2024.)
(𝜑 → 𝐵 ∈ On)    &   (𝜑 → EXMID)    &   (𝜑 → ∀𝑦 ∈ 𝐵 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴))    ⇒   (𝜑 → (𝐴 ∈ 𝐵 ∨ ∀𝑦 ∈ 𝐵 𝑦 ∈ 𝐴))
 
12-Aug-2024exmidontriimlem1 7441 Lemma for exmidontriim 7445. A variation of r19.30dc 2679. (Contributed by Jim Kingdon, 12-Aug-2024.)
((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓 ∨ 𝜒) ∧ EXMID) → (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜒))
 
11-Aug-2024nndc 858 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 ¬ ¬ EXMID is provable in intuitionistic logic. Indeed, if we could quantify over formula metavariables, then generalizing nnexmid 857 over 𝜑 would give "∀𝜑¬ ¬ DECID 𝜑", but EXMID is "∀𝜑DECID 𝜑", so proving ¬ ¬ EXMID would amount to proving "¬ ¬ ∀𝜑DECID 𝜑", which is not implied by the above theorem. Indeed, the converse of nnal 1697 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 4287): then, we can prove ∀𝑥 ∈ 𝒫 1o¬ ¬ DECID 𝑥 = 1o but we cannot prove ¬ ¬ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o because the converse of nnral 2521 does not hold.

Actually, ¬ ¬ EXMID is not provable in intuitionistic logic since intuitionistic logic has models satisfying ¬ EXMID and noncontradiction holds (pm3.24 700). (Contributed by BJ, 9-Oct-2019.) Add explanation on non-provability of ¬ ¬ EXMID. (Revised by BJ, 11-Aug-2024.)

¬ ¬ DECID 𝜑
 
10-Aug-2024exmidontriim 7445 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 → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
 
10-Aug-2024exmidontriimlem4 7444 Lemma for exmidontriim 7445. The induction step for the induction on 𝐴. (Contributed by Jim Kingdon, 10-Aug-2024.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → EXMID)    &   (𝜑 → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))    ⇒   (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
 
10-Aug-2024exmidontriimlem3 7443 Lemma for exmidontriim 7445. What we get to do based on induction on both 𝐴 and 𝐵. (Contributed by Jim Kingdon, 10-Aug-2024.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → EXMID)    &   (𝜑 → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))    &   (𝜑 → ∀𝑦 ∈ 𝐵 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴))    ⇒   (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
 
10-Aug-2024nnnninf2 7331 Canonical embedding of suc ω into ℕ∞. (Contributed by BJ, 10-Aug-2024.)
(𝑁 ∈ suc ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) ∈ ℕ∞)
 
10-Aug-2024infnninf 7328 The point at infinity in ℕ∞ is the constant sequence equal to 1o. Note that with our encoding of functions, that constant function can also be expressed as (ω × {1o}), as fconstmpt 4775 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
(𝑖 ∈ ω ↦ 1o) ∈ ℕ∞
 
9-Aug-2024ss1o0el1o 7110 Reformulation of ss1o0el1 4289 using 1o instead of {∅}. (Contributed by BJ, 9-Aug-2024.)
(𝐴 ⊆ 1o → (∅ ∈ 𝐴 ↔ 𝐴 = 1o))
 
9-Aug-2024pw1dc0el 7108 Another equivalent of excluded middle, which is a mere reformulation of the definition. (Contributed by BJ, 9-Aug-2024.)
(EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥)
 
9-Aug-2024ss1o0el1 4289 A subclass of {∅} contains the empty set if and only if it equals {∅}. (Contributed by BJ and Jim Kingdon, 9-Aug-2024.)
(𝐴 ⊆ {∅} → (∅ ∈ 𝐴 ↔ 𝐴 = {∅}))
 
8-Aug-2024pw1dc1 7111 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 ↔ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o)
 
8-Aug-2024fdmexb 5881 The domain of a function is a set iff the function is a set. (Contributed by AV, 8-Aug-2024.)
(𝐹:𝐴⟶𝐵 → (𝐴 ∈ V ↔ 𝐹 ∈ V))
 
8-Aug-2024fndmexb 5880 The domain of a function is a set iff the function is a set. (Contributed by AV, 8-Aug-2024.)
(𝐹 Fn 𝐴 → (𝐴 ∈ V ↔ 𝐹 ∈ V))
 
7-Aug-2024pw1fin 7107 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 4705 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 4706. (Revised by BJ, 7-Aug-2024.)
(𝐴 ∈ ω → 𝐴 ⊆ ω)
 
6-Aug-2024bj-charfunbi 16466 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.)

(𝜑 → 𝑋 ∈ 𝑉)    &   (𝜑 → ∀𝑥 ∈ 𝑋 STAB 𝑥 ∈ 𝐴)    ⇒   (𝜑 → (∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴 ↔ ∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)))
 
6-Aug-2024bj-charfunr 16465 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 (𝜑 → (𝐹:𝑋⟶ω ∧ ...)) 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 ∖ 𝐴) is sent. (Contributed by BJ, 6-Aug-2024.)

(𝜑 → ∃𝑓 ∈ (ω ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅))    ⇒   (𝜑 → ∀𝑥 ∈ 𝑋 DECID ¬ 𝑥 ∈ 𝐴)
 
6-Aug-2024bj-charfundc 16463 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.)
(𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    &   (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)    ⇒   (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
 
6-Aug-2024psrbagconf1o 14716 Bag complementation is a bijection on the set of bags dominated by a given bag 𝐹. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 6-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝐹}    ⇒   (𝐹 ∈ 𝐷 → (𝑥 ∈ 𝑆 ↦ (𝐹 ∘𝑓 − 𝑥)):𝑆–1-1-onto→𝑆)
 
6-Aug-2024psrbagconcl 14715 The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 6-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝐹}    ⇒   ((𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → (𝐹 ∘𝑓 − 𝑋) ∈ 𝑆)
 
6-Aug-2024prodssdc 12173 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.)
(𝜑 → 𝐴 ⊆ 𝐵)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   (𝜑 → ∃𝑛 ∈ (ℤ≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐵, 𝐶, 1))) ⇝ 𝑦))    &   (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴)    &   (𝜑 → 𝑀 ∈ ℤ)    &   ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 1)    &   (𝜑 → 𝐵 ⊆ (ℤ≥‘𝑀))    &   (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐵)    ⇒   (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)
 
5-Aug-2024fnmptd 16461 The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.)
(𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   (𝜑 → 𝐹 Fn 𝐴)
 
5-Aug-2024funmptd 16460 The maps-to notation defines a function (deduction form).

Note: one should similarly prove a deduction form of funopab4 5365, then prove funmptd 16460 from it, and then prove funmpt 5366 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.)

(𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    ⇒   (𝜑 → Fun 𝐹)
 
5-Aug-2024bj-dcfal 16412 The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID ⊥
 
5-Aug-2024bj-dctru 16410 The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID ⊤
 
5-Aug-2024bj-stfal 16399 The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)
STAB ⊥
 
5-Aug-2024bj-sttru 16397 The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)
STAB ⊤
 
5-Aug-2024psrbagcon 14714 The analogue of the statement "0 ≤ 𝐺 ≤ 𝐹 implies 0 ≤ 𝐹 − 𝐺 ≤ 𝐹 " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → ((𝐹 ∘𝑓 − 𝐺) ∈ 𝐷 ∧ (𝐹 ∘𝑓 − 𝐺) ∘𝑟 ≤ 𝐹))
 
5-Aug-2024psrbaglecl 14713 The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐺 ∈ 𝐷)
 
5-Aug-2024psrbaglesupp 14711 The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))
 
5-Aug-2024prod1dc 12170 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 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∨ 𝐴 ∈ Fin) → ∏𝑘 ∈ 𝐴 1 = 1)
 
5-Aug-20242ssom 6697 The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
2o ⊆ ω
 
2-Aug-2024onntri52 7467 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
 
2-Aug-2024onntri24 7465 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
 
2-Aug-2024onntri45 7464 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ ¬ EXMID)
 
2-Aug-2024onntri51 7463 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
 
2-Aug-2024onntri13 7461 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
 
2-Aug-2024onntri35 7460 Double negated ordinal trichotomy.

There are five equivalent statements: (1) ¬ ¬ ∀𝑥 ∈ On∀𝑦 ∈ On(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥), (2) ¬ ¬ ∀𝑥 ∈ On∀𝑦 ∈ On(𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥), (3) ∀𝑥 ∈ On∀𝑦 ∈ On¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥), (4) ∀𝑥 ∈ On∀𝑦 ∈ On¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥), and (5) ¬ ¬ EXMID. That these are all equivalent is expressed by (1) implies (3) (onntri13 7461), (3) implies (5) (onntri35 7460), (5) implies (1) (onntri51 7463), (2) implies (4) (onntri24 7465), (4) implies (5) (onntri45 7464), and (5) implies (2) (onntri52 7467).

Another way of stating this is that EXMID is equivalent to trichotomy, either the 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 or the 𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥 form, as shown in exmidontri 7462 and exmidontri2or 7466, respectively. Thus ¬ ¬ EXMID is equivalent to (1) or (2). In addition, ¬ ¬ EXMID is equivalent to (3) by onntri3or 7468 and (4) by onntri2or 7469.

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

(∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ EXMID)
 
1-Aug-2024nnral 2521 The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1697. (Contributed by Jim Kingdon, 1-Aug-2024.)
(¬ ¬ ∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑)
 
31-Jul-20243nsssucpw1 7459 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.)
(¬ EXMID → ¬ 3o ⊆ suc 𝒫 1o)
 
31-Jul-2024sucpw1nss3 7458 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.)
(¬ EXMID → ¬ suc 𝒫 1o ⊆ 3o)
 
30-Jul-2024psrbagf 14708 A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 30-Jul-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
 
30-Jul-20243nelsucpw1 7457 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.)
¬ 3o ∈ suc 𝒫 1o
 
30-Jul-2024sucpw1nel3 7456 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.)
¬ suc 𝒫 1o ∈ 3o
 
30-Jul-2024sucpw1ne3 7455 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.)
(¬ EXMID → suc 𝒫 1o ≠ 3o)
 
30-Jul-2024pw1nel3 7454 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.)
(¬ EXMID → ¬ 𝒫 1o ∈ 3o)
 
30-Jul-2024pw1ne3 7453 The power set of 1o is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
𝒫 1o ≠ 3o
 
30-Jul-2024pw1ne1 7452 The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
𝒫 1o ≠ 1o
 
30-Jul-2024pw1ne0 7451 The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
𝒫 1o ≠ ∅
 
29-Jul-2024grpcld 13620 Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
 
29-Jul-2024pw1on 7449 The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
𝒫 1o ∈ On
 
28-Jul-2024exmidpweq 7106 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 16734 Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 16733 by means of dceqnconst 16732. (Contributed by Jim Kingdon, 27-Jul-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0))
 
27-Jul-2024reap0 16730 Real number trichotomy is equivalent to decidability of apartness from zero. (Contributed by Jim Kingdon, 27-Jul-2024.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑧 ∈ ℝ DECID 𝑧 # 0)
 
26-Jul-2024nconstwlpolemgt0 16736 Lemma for nconstwlpo 16738. If one of the terms of series is positive, so is the sum. (Contributed by Jim Kingdon, 26-Jul-2024.)
(𝜑 → 𝐺:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺‘𝑖))    &   (𝜑 → ∃𝑥 ∈ ℕ (𝐺‘𝑥) = 1)    ⇒   (𝜑 → 0 < 𝐴)
 
26-Jul-2024nconstwlpolem0 16735 Lemma for nconstwlpo 16738. If all the terms of the series are zero, so is their sum. (Contributed by Jim Kingdon, 26-Jul-2024.)
(𝜑 → 𝐺:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺‘𝑖))    &   (𝜑 → ∀𝑥 ∈ ℕ (𝐺‘𝑥) = 0)    ⇒   (𝜑 → 𝐴 = 0)
 
24-Jul-2024tridceq 16728 Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 16714 and redcwlpo 16727). Thus, this is an analytic analogue to lpowlpo 7372. (Contributed by Jim Kingdon, 24-Jul-2024.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦)
 
24-Jul-2024iswomni0 16723 Weak omniscience stated in terms of equality with 0. Like iswomninn 16722 but with zero in place of one. (Contributed by Jim Kingdon, 24-Jul-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0))
 
24-Jul-2024lpowlpo 7372 LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7371. There is an analogue in terms of analytic omniscience principles at tridceq 16728. (Contributed by Jim Kingdon, 24-Jul-2024.)
(ω ∈ Omni → ω ∈ WOmni)
 
23-Jul-2024nconstwlpolem 16737 Lemma for nconstwlpo 16738. (Contributed by Jim Kingdon, 23-Jul-2024.)
(𝜑 → 𝐹:ℝ⟶ℤ)    &   (𝜑 → (𝐹‘0) = 0)    &   ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝐹‘𝑥) ≠ 0)    &   (𝜑 → 𝐺:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺‘𝑖))    ⇒   (𝜑 → (∀𝑦 ∈ ℕ (𝐺‘𝑦) = 0 ∨ ¬ ∀𝑦 ∈ ℕ (𝐺‘𝑦) = 0))
 
23-Jul-2024dceqnconst 16732 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 16727 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.)
(∀𝑥 ∈ ℝ DECID 𝑥 = 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0))
 
23-Jul-2024redc0 16729 Two ways to express decidability of real number equality. (Contributed by Jim Kingdon, 23-Jul-2024.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ↔ ∀𝑧 ∈ ℝ DECID 𝑧 = 0)
 
23-Jul-2024canth 5974 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 1548 if you want the form ¬ ∃𝑓𝑓:𝐴–onto→𝒫 𝐴.) (Contributed by NM, 7-Aug-1994.) (Revised by Noah R Kingdon, 23-Jul-2024.)
𝐴 ∈ V    ⇒    ¬ 𝐹:𝐴–onto→𝒫 𝐴
 
22-Jul-2024nconstwlpo 16738 Existence of a certain non-constant function from reals to integers implies ω ∈ 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.)
(𝜑 → 𝐹:ℝ⟶ℤ)    &   (𝜑 → (𝐹‘0) = 0)    &   ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝐹‘𝑥) ≠ 0)    ⇒   (𝜑 → ω ∈ WOmni)
 
15-Jul-2024fprodseq 12167 The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.) (Revised by Jim Kingdon, 15-Jul-2024.)
(𝑘 = (𝐹‘𝑛) → 𝐵 = 𝐶)    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = 𝐶)    ⇒   (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 1)))‘𝑀))
 
14-Jul-2024rexbid2 2536 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))
 
14-Jul-2024ralbid2 2535 Formula-building rule for restricted universal quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒)))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
 
12-Jul-20242irrexpqap 15731 There exist real numbers 𝑎 and 𝑏 which are irrational (in the sense of being apart from any rational number) such that (𝑎↑𝑏) 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 (√‘2) and (2 logb 9), see sqrt2irrap 12775, 2logb9irrap 15730 and sqrt2cxp2logb9e3 15728. Therefore, this proof is acceptable/usable in intuitionistic logic. (Contributed by Jim Kingdon, 12-Jul-2024.)
∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ (∀𝑝 ∈ ℚ 𝑎 # 𝑝 ∧ ∀𝑞 ∈ ℚ 𝑏 # 𝑞 ∧ (𝑎↑𝑐𝑏) ∈ ℚ)
 
12-Jul-20242logb9irrap 15730 Example for logbgcd1irrap 15723. 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 ∈ ℚ → (2 logb 9) # 𝑄)
 
12-Jul-2024erlecpbl 13438 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.)
(𝜑 → ∼ Er 𝑉)    &   (𝜑 → 𝑉 ∈ 𝑊)    &   𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   (𝜑 → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴𝑁𝐵 ↔ 𝐶𝑁𝐷)))    ⇒   ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐴𝑁𝐵 ↔ 𝐶𝑁𝐷)))
 
12-Jul-2024ercpbl 13437 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.)
(𝜑 → ∼ Er 𝑉)    &   (𝜑 → 𝑉 ∈ 𝑊)    &   𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 + 𝑏) ∈ 𝑉)    &   (𝜑 → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷)))    ⇒   ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷))))
 
12-Jul-2024ercpbllemg 13436 Lemma for ercpbl 13437. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by AV, 12-Jul-2024.)
(𝜑 → ∼ Er 𝑉)    &   (𝜑 → 𝑉 ∈ 𝑊)    &   𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑉)    ⇒   (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ 𝐴 ∼ 𝐵))
 
12-Jul-2024divsfvalg 13435 Value of the function in qusval 13429. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
(𝜑 → ∼ Er 𝑉)    &   (𝜑 → 𝑉 ∈ 𝑊)    &   𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )
 
12-Jul-2024divsfval 13434 Value of the function in qusval 13429. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
(𝜑 → ∼ Er 𝑉)    &   (𝜑 → 𝑉 ∈ 𝑊)    &   𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    ⇒   (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )
 
11-Jul-2024logbgcd1irraplemexp 15721 Lemma for logbgcd1irrap 15723. Apartness of 𝑋↑𝑁 and 𝐵↑𝑀. (Contributed by Jim Kingdon, 11-Jul-2024.)
(𝜑 → 𝑋 ∈ (ℤ≥‘2))    &   (𝜑 → 𝐵 ∈ (ℤ≥‘2))    &   (𝜑 → (𝑋 gcd 𝐵) = 1)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℕ)    ⇒   (𝜑 → (𝑋↑𝑁) # (𝐵↑𝑀))
 
11-Jul-2024reapef 15531 Apartness and the exponential function for reals. (Contributed by Jim Kingdon, 11-Jul-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (exp‘𝐴) # (exp‘𝐵)))
 
10-Jul-2024apcxp2 15692 Apartness and real exponentiation. (Contributed by Jim Kingdon, 10-Jul-2024.)
(((𝐴 ∈ ℝ+ ∧ 𝐴 # 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 # 𝐶 ↔ (𝐴↑𝑐𝐵) # (𝐴↑𝑐𝐶)))
 
9-Jul-2024logbgcd1irraplemap 15722 Lemma for logbgcd1irrap 15723. The result, with the rational number expressed as numerator and denominator. (Contributed by Jim Kingdon, 9-Jul-2024.)
(𝜑 → 𝑋 ∈ (ℤ≥‘2))    &   (𝜑 → 𝐵 ∈ (ℤ≥‘2))    &   (𝜑 → (𝑋 gcd 𝐵) = 1)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℕ)    ⇒   (𝜑 → (𝐵 logb 𝑋) # (𝑀 / 𝑁))
 
9-Jul-2024apexp1 10986 Exponentiation and apartness. (Contributed by Jim Kingdon, 9-Jul-2024.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) # (𝐵↑𝑁) → 𝐴 # 𝐵))
 
5-Jul-2024logrpap0 15630 The logarithm is apart from 0 if its argument is apart from 1. (Contributed by Jim Kingdon, 5-Jul-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐴 # 1) → (log‘𝐴) # 0)
 
3-Jul-2024rplogbval 15698 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.)
((𝐵 ∈ ℝ+ ∧ 𝐵 # 1 ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵)))
 
3-Jul-2024logrpap0d 15631 Deduction form of logrpap0 15630. (Contributed by Jim Kingdon, 3-Jul-2024.)
(𝜑 → 𝐴 ∈ ℝ+)    &   (𝜑 → 𝐴 # 1)    ⇒   (𝜑 → (log‘𝐴) # 0)
 
3-Jul-2024logrpap0b 15629 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+ → (𝐴 # 1 ↔ (log‘𝐴) # 0))
 
28-Jun-20242o01f 16653 Mapping zero and one between ω and ℕ0 style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   (𝐺 ↾ 2o):2o⟶{0, 1}
 
28-Jun-2024012of 16652 Mapping zero and one between ℕ0 and ω style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   (◡𝐺 ↾ {0, 1}):{0, 1}⟶2o
 
27-Jun-2024iooreen 16706 An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.)
(0(,)1) ≈ ℝ
 
27-Jun-2024iooref1o 16705 A one-to-one mapping from the real numbers onto the open unit interval. (Contributed by Jim Kingdon, 27-Jun-2024.)
𝐹 = (𝑥 ∈ ℝ ↦ (1 / (1 + (exp‘𝑥))))    ⇒   𝐹:ℝ–1-1-onto→(0(,)1)
 
25-Jun-2024neapmkvlem 16739 Lemma for neapmkv 16740. The result, with a few hypotheses broken out for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.)
(𝜑 → 𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖))    &   ((𝜑 ∧ 𝐴 ≠ 1) → 𝐴 # 1)    ⇒   (𝜑 → (¬ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1 → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0))
 
25-Jun-2024ismkvnn 16725 The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 25-Jun-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1 → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0)))
 
25-Jun-2024ismkvnnlem 16724 Lemma for ismkvnn 16725. The result, with a hypothesis to give a name to an expression for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1 → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0)))
 
25-Jun-2024enmkvlem 7365 Lemma for enmkv 7366. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ Markov → 𝐵 ∈ Markov))
 
24-Jun-2024neapmkv 16740 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.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≠ 𝑦 → 𝑥 # 𝑦) → ω ∈ Markov)
 
24-Jun-2024dcapnconst 16733 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 16714 for more discussion of decidability of real number apartness.

This is a weaker form of dceqnconst 16732 and in fact this theorem can be proved using dceqnconst 16732 as shown at dcapnconstALT 16734. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.)

(∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0))
 
24-Jun-2024enmkv 7366 Being Markov is invariant with respect to equinumerosity. For example, this means that we can express the Markov's Principle as either ω ∈ Markov or ℕ0 ∈ Markov. The former is a better match to conventional notation in the sense that df2o3 6602 says that 2o = {∅, 1o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 24-Jun-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ Markov ↔ 𝐵 ∈ Markov))
 
21-Jun-2024redcwlpolemeq1 16726 Lemma for redcwlpo 16727. A biconditionalized version of trilpolemeq1 16711. (Contributed by Jim Kingdon, 21-Jun-2024.)
(𝜑 → 𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖))    ⇒   (𝜑 → (𝐴 = 1 ↔ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1))
 
20-Jun-2024redcwlpo 16727 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 16726). 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 10510 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)

(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ω ∈ WOmni)
 
20-Jun-2024iswomninn 16722 Weak omniscience stated in terms of natural numbers. Similar to iswomnimap 7370 but it will sometimes be more convenient to use 0 and 1 rather than ∅ and 1o. (Contributed by Jim Kingdon, 20-Jun-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1))
 
20-Jun-2024iswomninnlem 16721 Lemma for iswomnimap 7370. The result, with a hypothesis for convenience. (Contributed by Jim Kingdon, 20-Jun-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1))
 
20-Jun-2024enwomni 7374 Weak omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Weak Limited Principle of Omniscience as either ω ∈ WOmni or ℕ0 ∈ WOmni. The former is a better match to conventional notation in the sense that df2o3 6602 says that 2o = {∅, 1o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 20-Jun-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni ↔ 𝐵 ∈ WOmni))
 
20-Jun-2024enwomnilem 7373 Lemma for enwomni 7374. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni → 𝐵 ∈ WOmni))
 
19-Jun-2024rpabscxpbnd 15693 Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Jim Kingdon, 19-Jun-2024.)
(𝜑 → 𝐴 ∈ ℝ+)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 0 < (ℜ‘𝐵))    &   (𝜑 → 𝑀 ∈ ℝ)    &   (𝜑 → (abs‘𝐴) ≤ 𝑀)    ⇒   (𝜑 → (abs‘(𝐴↑𝑐𝐵)) ≤ ((𝑀↑𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
 
16-Jun-2024rpcxpsqrt 15675 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+ → (𝐴↑𝑐(1 / 2)) = (√‘𝐴))
 
16-Jun-2024biadanid 618 Deduction associated with biadani 616. Add a conjunction to an equivalence. (Contributed by Thierry Arnoux, 16-Jun-2024.)
((𝜑 ∧ 𝜓) → 𝜒)    &   ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃))    ⇒   (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))
 
13-Jun-2024rpcxpadd 15658 Sum of exponents law for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 13-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴↑𝑐(𝐵 + 𝐶)) = ((𝐴↑𝑐𝐵) · (𝐴↑𝑐𝐶)))
 
12-Jun-2024cxpap0 15657 Complex exponentiation is apart from zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) # 0)
 
12-Jun-2024rpcncxpcl 15655 Closure of the complex power function. (Contributed by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) ∈ ℂ)
 
12-Jun-2024rpcxp0 15651 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+ → (𝐴↑𝑐0) = 1)
 
12-Jun-2024cxpexpnn 15649 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 ∈ ℕ ∧ 𝐵 ∈ ℤ) → (𝐴↑𝑐𝐵) = (𝐴↑𝐵))
 
12-Jun-2024cxpexprp 15648 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+ ∧ 𝐵 ∈ ℤ) → (𝐴↑𝑐𝐵) = (𝐴↑𝐵))
 
12-Jun-2024rpcxpef 15647 Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
 
12-Jun-2024df-rpcxp 15612 Define the power function on complex numbers. Because df-relog 15611 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 = (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥))))
 
10-Jun-2024trirec0xor 16716 Version of trirec0 16715 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.)

(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ⊻ 𝑥 = 0))
 
10-Jun-2024trirec0 16715 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 16714). (Contributed by Jim Kingdon, 10-Jun-2024.)

(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
 
9-Jun-2024omniwomnimkv 7371 A set is omniscient if and only if it is weakly omniscient and Markov. The case 𝐴 = ω 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 7370 The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
 
9-Jun-2024iswomni 7369 The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
 
9-Jun-2024df-womni 7368 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, ω ∈ 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 ∣ ∀𝑓(𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)}
 
1-Jun-2024ringcmnd 14072 A ring is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → 𝑅 ∈ CMnd)
 
1-Jun-2024ringabld 14071 A ring is an Abelian group. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → 𝑅 ∈ Abel)
 
1-Jun-2024cmnmndd 13918 A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝐺 ∈ CMnd)    ⇒   (𝜑 → 𝐺 ∈ Mnd)
 
1-Jun-2024ablcmnd 13902 An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝐺 ∈ Abel)    ⇒   (𝜑 → 𝐺 ∈ CMnd)
 
1-Jun-2024grpmndd 13619 A group is a monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝐺 ∈ Grp)    ⇒   (𝜑 → 𝐺 ∈ Mnd)
 
1-Jun-2024fndmi 5432 The domain of a function. (Contributed by Wolf Lammen, 1-Jun-2024.)
𝐹 Fn 𝐴    ⇒   dom 𝐹 = 𝐴
 
29-May-2024pw1nct 16664 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.)
(∀𝑟(𝑟 ⊆ (𝒫 1o × ω) → (∀𝑝 ∈ 𝒫 1o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1o𝑞𝑟𝑚)) → ¬ ∃𝑓 𝑓:ω–onto→(𝒫 1o ⊔ 1o))
 
28-May-2024sssneq 16663 Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.)
(𝐴 ⊆ {𝐵} → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑦 = 𝑧)
 
26-May-2024elpwi2 4249 Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof shortened by Wolf Lammen, 26-May-2024.)
𝐵 ∈ 𝑉    &   𝐴 ⊆ 𝐵    ⇒   𝐴 ∈ 𝒫 𝐵
 
25-May-2024mplnegfi 14748 The negative function on multivariate polynomials. (Contributed by SN, 25-May-2024.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑁 = (invg‘𝑅)    &   𝑀 = (invg‘𝑃)    &   (𝜑 → 𝐼 ∈ Fin)    &   (𝜑 → 𝑅 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑀‘𝑋) = (𝑁 ∘ 𝑋))
 
24-May-2024dvmptcjx 15477 Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 24-May-2024.)
((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    &   (𝜑 → 𝑋 ⊆ ℝ)    ⇒   (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (∗‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (∗‘𝐵)))
 
23-May-2024cbvralfw 2755 Rule used to change bound variables, using implicit substitution. Version of cbvralf 2757 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 1555 and ax-bndl 1557 in the proof. (Contributed by NM, 7-Mar-2004.) (Revised by GG, 23-May-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑦𝐴    &   Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
 
23-May-2024cbvrmow 2715 Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo 2765 with a disjoint variable condition. (Contributed by NM, 16-Jun-2017.) (Revised by GG, 23-May-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
 
23-May-2024cbvmow 2119 Rule used to change bound variables, using implicit substitution. Version of cbvmo 2118 with a disjoint variable condition. (Contributed by NM, 9-Mar-1995.) (Revised by GG, 23-May-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
 
22-May-2024efltlemlt 15527 Lemma for eflt 15528. The converse of efltim 12282 plus the epsilon-delta setup. (Contributed by Jim Kingdon, 22-May-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → (exp‘𝐴) < (exp‘𝐵))    &   (𝜑 → 𝐷 ∈ ℝ+)    &   (𝜑 → ((abs‘(𝐴 − 𝐵)) < 𝐷 → (abs‘((exp‘𝐴) − (exp‘𝐵))) < ((exp‘𝐵) − (exp‘𝐴))))    ⇒   (𝜑 → 𝐴 < 𝐵)
 
21-May-2024eflt 15528 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 ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (exp‘𝐴) < (exp‘𝐵)))
 
20-May-2024nsyl5 655 A negated syllogism inference. (Contributed by Wolf Lammen, 20-May-2024.)
(𝜑 → 𝜓)    &   (¬ 𝜑 → 𝜒)    ⇒   (¬ 𝜓 → 𝜒)
 
19-May-2024apdifflemr 16718 Lemma for apdiff 16719. (Contributed by Jim Kingdon, 19-May-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝑆 ∈ ℚ)    &   (𝜑 → (abs‘(𝐴 − -1)) # (abs‘(𝐴 − 1)))    &   ((𝜑 ∧ 𝑆 ≠ 0) → (abs‘(𝐴 − 0)) # (abs‘(𝐴 − (2 · 𝑆))))    ⇒   (𝜑 → 𝐴 # 𝑆)
 
18-May-2024apdifflemf 16717 Lemma for apdiff 16719. 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.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝑄 ∈ ℚ)    &   (𝜑 → 𝑅 ∈ ℚ)    &   (𝜑 → 𝑄 < 𝑅)    &   (𝜑 → ((𝑄 + 𝑅) / 2) # 𝐴)    ⇒   (𝜑 → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))
 
17-May-2024apdiff 16719 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 → (∀𝑞 ∈ ℚ 𝐴 # 𝑞 ↔ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) # (abs‘(𝐴 − 𝑟)))))
 
16-May-2024lmodgrpd 14335 A left module is a group. (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑊 ∈ LMod)    ⇒   (𝜑 → 𝑊 ∈ Grp)
 
16-May-2024crnggrpd 14047 A commutative ring is a group. (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑅 ∈ CRing)    ⇒   (𝜑 → 𝑅 ∈ Grp)
 
16-May-2024crngringd 14046 A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑅 ∈ CRing)    ⇒   (𝜑 → 𝑅 ∈ Ring)
 
16-May-2024ringgrpd 14042 A ring is a group. (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → 𝑅 ∈ Grp)
 
15-May-2024reeff1oleme 15525 Lemma for reeff1o 15526. (Contributed by Jim Kingdon, 15-May-2024.)
(𝑈 ∈ (0(,)e) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈)
 
14-May-2024df-relog 15611 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 = ◡(exp ↾ ℝ)
 
14-May-2024fvmpopr2d 6163 Value of an operation given by maps-to notation. (Contributed by Rohan Ridenour, 14-May-2024.)
(𝜑 → 𝐹 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶))    &   (𝜑 → 𝑃 = ⟨𝑎, 𝑏⟩)    &   ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)    ⇒   ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = 𝐶)
 
13-May-2024fndmexd 5528 If a function is a set, its domain is a set. (Contributed by Rohan Ridenour, 13-May-2024.)
(𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝐹 Fn 𝐷)    ⇒   (𝜑 → 𝐷 ∈ V)
 
12-May-2024dvdstrd 12414 The divides relation is transitive, a deduction version of dvdstr 12412. (Contributed by metakunt, 12-May-2024.)
(𝜑 → 𝐾 ∈ ℤ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ∥ 𝑀)    &   (𝜑 → 𝑀 ∥ 𝑁)    ⇒   (𝜑 → 𝐾 ∥ 𝑁)
 
7-May-2024ioocosf1o 15607 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 ↾ (0(,)π)):(0(,)π)–1-1-onto→(-1(,)1)
 
7-May-2024cos0pilt1 15605 Cosine is between minus one and one on the open interval between zero and π. (Contributed by Jim Kingdon, 7-May-2024.)
(𝐴 ∈ (0(,)π) → (cos‘𝐴) ∈ (-1(,)1))
 
6-May-2024cos11 15606 Cosine is one-to-one over the closed interval from 0 to π. (Contributed by Paul Chapman, 16-Mar-2008.) (Revised by Jim Kingdon, 6-May-2024.)
((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → (𝐴 = 𝐵 ↔ (cos‘𝐴) = (cos‘𝐵)))
 
5-May-2024omiunct 13088 The union of a countably infinite collection of countable sets is countable. Theorem 8.1.28 of [AczelRathjen], p. 78. Compare with ctiunct 13084 which has a stronger hypothesis but does not require countable choice. (Contributed by Jim Kingdon, 5-May-2024.)
(𝜑 → CCHOICE)    &   ((𝜑 ∧ 𝑥 ∈ ω) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o))    ⇒   (𝜑 → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ ω 𝐵 ⊔ 1o))
 
5-May-2024ctiunctal 13085 Variation of ctiunct 13084 which allows 𝑥 to be present in 𝜑. (Contributed by Jim Kingdon, 5-May-2024.)
(𝜑 → 𝐹:ω–onto→(𝐴 ⊔ 1o))    &   (𝜑 → ∀𝑥 ∈ 𝐴 𝐺:ω–onto→(𝐵 ⊔ 1o))    ⇒   (𝜑 → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o))
 
5-May-2024ifpnst 996 Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019.) (Proof shortened by Wolf Lammen, 5-May-2024.)
(STAB 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓)))
 
3-May-2024cc4n 7495 Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7494, the hypotheses only require an A(n) for each value of 𝑛, not a single set 𝐴 which suffices for every 𝑛 ∈ ω. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
(𝜑 → CCHOICE)    &   (𝜑 → ∀𝑛 ∈ 𝑁 {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ 𝑉)    &   (𝜑 → 𝑁 ≈ ω)    &   (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))    &   (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)    ⇒   (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 𝜒))
 
3-May-2024cc4f 7493 Countable choice by showing the existence of a function 𝑓 which can choose a value at each index 𝑛 such that 𝜒 holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
(𝜑 → CCHOICE)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   Ⅎ𝑛𝐴    &   (𝜑 → 𝑁 ≈ ω)    &   (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))    &   (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)    ⇒   (𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒))
 
1-May-2024cc4 7494 Countable choice by showing the existence of a function 𝑓 which can choose a value at each index 𝑛 such that 𝜒 holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 1-May-2024.)
(𝜑 → CCHOICE)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝑁 ≈ ω)    &   (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))    &   (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)    ⇒   (𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒))
 
30-Apr-2024ifpdfbidc 993 Define the biconditional as conditional logic operator. (Contributed by RP, 20-Apr-2020.) (Proof shortened by Wolf Lammen, 30-Apr-2024.)
(DECID 𝜑 → ((𝜑 ↔ 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓)))
 
29-Apr-2024cc3 7492 Countable choice using a sequence F(n) . (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Jim Kingdon, 29-Apr-2024.)
(𝜑 → CCHOICE)    &   (𝜑 → ∀𝑛 ∈ 𝑁 𝐹 ∈ V)    &   (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑤 𝑤 ∈ 𝐹)    &   (𝜑 → 𝑁 ≈ ω)    ⇒   (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹))
 
28-Apr-2024ifpbi23d 1001 Equivalence deduction for conditional operator for propositions. Convenience theorem for a frequent case. (Contributed by Wolf Lammen, 28-Apr-2024.)
(𝜑 → (𝜒 ↔ 𝜂))    &   (𝜑 → (𝜃 ↔ 𝜁))    ⇒   (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜓, 𝜂, 𝜁)))
 
27-Apr-2024cc2 7491 Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
(𝜑 → CCHOICE)    &   (𝜑 → 𝐹 Fn ω)    &   (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑥))    ⇒   (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔‘𝑛) ∈ (𝐹‘𝑛)))
 
27-Apr-2024cc2lem 7490 Lemma for cc2 7491. (Contributed by Jim Kingdon, 27-Apr-2024.)
(𝜑 → CCHOICE)    &   (𝜑 → 𝐹 Fn ω)    &   (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑥))    &   𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐹‘𝑛)))    &   𝐺 = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘(𝐴‘𝑛))))    ⇒   (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔‘𝑛) ∈ (𝐹‘𝑛)))
 
27-Apr-2024cc1 7489 Countable choice in terms of a choice function on a countably infinite set of inhabited sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
(CCHOICE → ∀𝑥((𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑓‘𝑧) ∈ 𝑧))
 
24-Apr-2024lsppropd 14470 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.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝐵 = (Base‘𝐿))    &   (𝜑 → 𝐵 ⊆ 𝑊)    &   ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)    &   ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    &   (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾)))    &   (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))    &   (𝜑 → 𝐾 ∈ 𝑋)    &   (𝜑 → 𝐿 ∈ 𝑌)    ⇒   (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
 
19-Apr-2024omctfn 13087 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.)
(𝜑 → CCHOICE)    &   ((𝜑 ∧ 𝑥 ∈ ω) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o))    ⇒   (𝜑 → ∃𝑓(𝑓 Fn ω ∧ ∀𝑥 ∈ ω (𝑓‘𝑥):ω–onto→(𝐵 ⊔ 1o)))
 
17-Apr-2024ifpbi123d 1000 Equivalence deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 17-Apr-2024.)
(𝜑 → (𝜓 ↔ 𝜏))    &   (𝜑 → (𝜒 ↔ 𝜂))    &   (𝜑 → (𝜃 ↔ 𝜁))    ⇒   (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))
 
13-Apr-2024prodmodclem2 12161 Lemma for prodmodc 12162. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 13-Apr-2024.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1))    ⇒   ((𝜑 ∧ ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
 
11-Apr-2024prodmodclem2a 12160 Lemma for prodmodc 12162. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1))    &   𝐻 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝐾‘𝑗) / 𝑘⦌𝐵, 1))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))    &   (𝜑 → 𝑓:(1...𝑁)–1-1-onto→𝐴)    &   (𝜑 → 𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴))    ⇒   (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑁))
 
11-Apr-2024prodmodclem3 12159 Lemma for prodmodc 12162. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1))    &   𝐻 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝐾‘𝑗) / 𝑘⦌𝐵, 1))    &   (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))    &   (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)    &   (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)    ⇒   (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
 
10-Apr-2024jcnd 658 Deduction joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Apr-2024.)
(𝜑 → 𝜓)    &   (𝜑 → ¬ 𝜒)    ⇒   (𝜑 → ¬ (𝜓 → 𝜒))
 
4-Apr-2024prodrbdclem 12155 Lemma for prodrbdc 12158. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 4-Apr-2024.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   ((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) → (seq𝑀( · , 𝐹) ↾ (ℤ≥‘𝑁)) = seq𝑁( · , 𝐹))
 
24-Mar-2024prodfdivap 12131 The quotient of two products. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) # 0)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘) / (𝐺‘𝑘)))    ⇒   (𝜑 → (seq𝑀( · , 𝐻)‘𝑁) = ((seq𝑀( · , 𝐹)‘𝑁) / (seq𝑀( · , 𝐺)‘𝑁)))
 
24-Mar-2024prodfrecap 12130 The reciprocal of a finite product. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) # 0)    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) = (1 / (𝐹‘𝑘)))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ)    ⇒   (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))
 
23-Mar-2024prodfap0 12129 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.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) # 0)    ⇒   (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) # 0)
 
22-Mar-2024prod3fmul 12125 The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017.) (Revised by Jim Kingdon, 22-Mar-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘)))    ⇒   (𝜑 → (seq𝑀( · , 𝐻)‘𝑁) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq𝑀( · , 𝐺)‘𝑁)))
 
21-Mar-2024df-proddc 12135 Define the product of a series with an index set of integers 𝐴. This definition takes most of the aspects of df-sumdc 11937 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.)
∏𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))))
 
19-Mar-2024cos02pilt1 15604 Cosine is less than one between zero and 2 · π. (Contributed by Jim Kingdon, 19-Mar-2024.)
(𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) < 1)
 
19-Mar-2024cosq34lt1 15603 Cosine is less than one in the third and fourth quadrants. (Contributed by Jim Kingdon, 19-Mar-2024.)
(𝐴 ∈ (π[,)(2 · π)) → (cos‘𝐴) < 1)
 
14-Mar-2024coseq0q4123 15587 Location of the zeroes of cosine in (-(π / 2)(,)(3 · (π / 2))). (Contributed by Jim Kingdon, 14-Mar-2024.)
(𝐴 ∈ (-(π / 2)(,)(3 · (π / 2))) → ((cos‘𝐴) = 0 ↔ 𝐴 = (π / 2)))
 
14-Mar-2024cosq23lt0 15586 The cosine of a number in the second and third quadrants is negative. (Contributed by Jim Kingdon, 14-Mar-2024.)
(𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (cos‘𝐴) < 0)
 
9-Mar-2024pilem3 15536 Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.)
(π ∈ (2(,)4) ∧ (sin‘π) = 0)
 
9-Mar-2024exmidonfin 7410 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 7064 and nnon 4710. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
(ω = (On ∩ Fin) → EXMID)
 
9-Mar-2024exmidonfinlem 7409 Lemma for exmidonfin 7410. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
𝐴 = {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}    ⇒   (ω = (On ∩ Fin) → DECID 𝜑)
 
8-Mar-2024sin0pilem2 15535 Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.)
∃𝑞 ∈ (2(,)4)((sin‘𝑞) = 0 ∧ ∀𝑥 ∈ (0(,)𝑞)0 < (sin‘𝑥))
 
8-Mar-2024sin0pilem1 15534 Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.)
∃𝑝 ∈ (1(,)2)((cos‘𝑝) = 0 ∧ ∀𝑥 ∈ (𝑝(,)(2 · 𝑝))0 < (sin‘𝑥))
 
7-Mar-2024cosz12 15533 Cosine has a zero between 1 and 2. (Contributed by Mario Carneiro and Jim Kingdon, 7-Mar-2024.)
∃𝑝 ∈ (1(,)2)(cos‘𝑝) = 0
 
6-Mar-2024cos12dec 12352 Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.)
((𝐴 ∈ (1[,]2) ∧ 𝐵 ∈ (1[,]2) ∧ 𝐴 < 𝐵) → (cos‘𝐵) < (cos‘𝐴))
 
2-Mar-2024clwwlknonmpo 16308 (ClWWalksNOn‘𝐺) is an operator mapping a vertex 𝑣 and a nonnegative integer 𝑛 to the set of closed walks on 𝑣 of length 𝑛 as words over the set of vertices in a graph 𝐺. (Contributed by AV, 25-Feb-2022.) (Proof shortened by AV, 2-Mar-2024.)
(ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
 
2-Mar-2024scaffvalg 14344 The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    ∙ = ( ·sf ‘𝑊)    &    · = ( ·𝑠 ‘𝑊)    ⇒   (𝑊 ∈ 𝑉 → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
 
2-Mar-2024dvrfvald 14171 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.)
(𝜑 → 𝐵 = (Base‘𝑅))    &   (𝜑 → · = (.r‘𝑅))    &   (𝜑 → 𝑈 = (Unit‘𝑅))    &   (𝜑 → 𝐼 = (invr‘𝑅))    &   (𝜑 → / = (/r‘𝑅))    &   (𝜑 → 𝑅 ∈ SRing)    ⇒   (𝜑 → / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))))
 
2-Mar-2024plusffvalg 13468 The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    ⨣ = (+𝑓‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
 
25-Feb-2024insubm 13591 The intersection of two submonoids is a submonoid. (Contributed by AV, 25-Feb-2024.)
((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀))
 
25-Feb-2024mul2lt0pn 10004 The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐴 < 0)    &   (𝜑 → 0 < 𝐵)    ⇒   (𝜑 → (𝐵 · 𝐴) < 0)
 
25-Feb-2024mul2lt0np 10003 The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐴 < 0)    &   (𝜑 → 0 < 𝐵)    ⇒   (𝜑 → (𝐴 · 𝐵) < 0)
 
25-Feb-2024lt0ap0 8833 A number which is less than zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 # 0)
 
25-Feb-2024negap0d 8816 The negative of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → -𝐴 # 0)
 
24-Feb-2024lt0ap0d 8834 A real number less than zero is apart from zero. Deduction form. (Contributed by Jim Kingdon, 24-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 < 0)    ⇒   (𝜑 → 𝐴 # 0)
 
20-Feb-2024ivthdec 15397 The intermediate value theorem, decreasing case, for a strictly monotonic function. (Contributed by Jim Kingdon, 20-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑦) < (𝐹‘𝑥))    ⇒   (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)
 
20-Feb-2024ivthinclemex 15395 Lemma for ivthinc 15396. Existence of a number between the lower cut and the upper cut. (Contributed by Jim Kingdon, 20-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    ⇒   (𝜑 → ∃!𝑧 ∈ (𝐴(,)𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑧 ∧ ∀𝑟 ∈ 𝑅 𝑧 < 𝑟))
 
19-Feb-2024ivthinclemuopn 15391 Lemma for ivthinc 15396. The upper cut is open. (Contributed by Jim Kingdon, 19-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    &   (𝜑 → 𝑆 ∈ 𝑅)    ⇒   (𝜑 → ∃𝑞 ∈ 𝑅 𝑞 < 𝑆)
 
19-Feb-2024dedekindicc 15386 A Dedekind cut identifies a unique real number. Similar to df-inp 7691 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.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐴 < 𝐵)    ⇒   (𝜑 → ∃!𝑥 ∈ (𝐴(,)𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟))
 
19-Feb-2024grpsubfvalg 13651 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.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   𝐼 = (invg‘𝐺)    &    − = (-g‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))))
 
18-Feb-2024ivthinclemloc 15394 Lemma for ivthinc 15396. Locatedness. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    ⇒   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑅)))
 
18-Feb-2024ivthinclemdisj 15393 Lemma for ivthinc 15396. The lower and upper cuts are disjoint. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    ⇒   (𝜑 → (𝐿 ∩ 𝑅) = ∅)
 
18-Feb-2024ivthinclemur 15392 Lemma for ivthinc 15396. The upper cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    ⇒   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑅 ↔ ∃𝑞 ∈ 𝑅 𝑞 < 𝑟))
 
18-Feb-2024ivthinclemlr 15390 Lemma for ivthinc 15396. The lower cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    ⇒   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
 
18-Feb-2024ivthinclemum 15388 Lemma for ivthinc 15396. The upper cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    ⇒   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑅)
 
18-Feb-2024ivthinclemlm 15387 Lemma for ivthinc 15396. The lower cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    ⇒   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)
 
17-Feb-20240subm 13590 The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)
0 = (0g‘𝐺)    ⇒   (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺))
 
17-Feb-2024mndissubm 13581 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.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)    &    0 = (0g‘𝐺)    ⇒   ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubMnd‘𝐺)))
 
17-Feb-2024mgmsscl 13467 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.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)    ⇒   (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐺)𝑌) ∈ 𝑆)
 
15-Feb-2024dedekindicclemeu 15384 Lemma for dedekindicc 15386. Part of proving uniqueness. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))    &   (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐶 ∧ ∀𝑟 ∈ 𝑈 𝐶 < 𝑟))    &   (𝜑 → 𝐷 ∈ (𝐴[,]𝐵))    &   (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐷 ∧ ∀𝑟 ∈ 𝑈 𝐷 < 𝑟))    &   (𝜑 → 𝐶 < 𝐷)    ⇒   (𝜑 → ⊥)
 
15-Feb-2024dedekindicclemlu 15383 Lemma for dedekindicc 15386. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐴 < 𝐵)    ⇒   (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟))
 
15-Feb-2024dedekindicclemlub 15382 Lemma for dedekindicc 15386. The set L has a least upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐴 < 𝐵)    ⇒   (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)(∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧)))
 
15-Feb-2024dedekindicclemloc 15381 Lemma for dedekindicc 15386. The set L is located. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    ⇒   (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (∃𝑧 ∈ 𝐿 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐿 𝑧 < 𝑦)))
 
15-Feb-2024dedekindicclemub 15380 Lemma for dedekindicc 15386. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    ⇒   (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ 𝐿 𝑦 < 𝑥)
 
15-Feb-2024dedekindicclemuub 15379 Lemma for dedekindicc 15386. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐶 ∈ 𝑈)    ⇒   (𝜑 → ∀𝑧 ∈ 𝐿 𝑧 < 𝐶)
 
14-Feb-2024suplociccex 15378 An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8257 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
(𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐶 ∈ ℝ)    &   (𝜑 → 𝐵 < 𝐶)    &   (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶))    &   (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ (𝐵[,]𝐶)(𝑥 < 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 < 𝑦)))    ⇒   (𝜑 → ∃𝑥 ∈ (𝐵[,]𝐶)(∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐵[,]𝐶)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
 
14-Feb-2024suplociccreex 15377 An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8257 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
(𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐶 ∈ ℝ)    &   (𝜑 → 𝐵 < 𝐶)    &   (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶))    &   (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ (𝐵[,]𝐶)(𝑥 < 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 < 𝑦)))    ⇒   (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
 
10-Feb-2024cbvexdvaw 1979 Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Version of cbvexdva 1977 with a disjoint variable condition. (Contributed by David Moews, 1-May-2017.) (Revised by GG, 10-Jan-2024.) (Revised by Wolf Lammen, 10-Feb-2024.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
 
10-Feb-2024cbvaldvaw 1978 Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva 1976 with a disjoint variable condition. (Contributed by David Moews, 1-May-2017.) (Revised by GG, 10-Jan-2024.) (Revised by Wolf Lammen, 10-Feb-2024.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 
6-Feb-2024ivthinclemlopn 15389 Lemma for ivthinc 15396. The lower cut is open. (Contributed by Jim Kingdon, 6-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}    &   (𝜑 → 𝑄 ∈ 𝐿)    ⇒   (𝜑 → ∃𝑟 ∈ 𝐿 𝑄 < 𝑟)
 
5-Feb-2024ivthinc 15396 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.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))    ⇒   (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)
 
2-Feb-2024dedekindeulemuub 15370 Lemma for dedekindeu 15376. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-2024.)
(𝜑 → 𝐿 ⊆ ℝ)    &   (𝜑 → 𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐴 ∈ 𝑈)    ⇒   (𝜑 → ∀𝑧 ∈ 𝐿 𝑧 < 𝐴)
 
31-Jan-2024dedekindeulemeu 15375 Lemma for dedekindeu 15376. Part of proving uniqueness. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑 → 𝐿 ⊆ ℝ)    &   (𝜑 → 𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐴 ∧ ∀𝑟 ∈ 𝑈 𝐴 < 𝑟))    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐵 ∧ ∀𝑟 ∈ 𝑈 𝐵 < 𝑟))    &   (𝜑 → 𝐴 < 𝐵)    ⇒   (𝜑 → ⊥)
 
31-Jan-2024dedekindeulemlu 15374 Lemma for dedekindeu 15376. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑 → 𝐿 ⊆ ℝ)    &   (𝜑 → 𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    ⇒   (𝜑 → ∃𝑥 ∈ ℝ (∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟))
 
31-Jan-2024dedekindeulemlub 15373 Lemma for dedekindeu 15376. The set L has a least upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑 → 𝐿 ⊆ ℝ)    &   (𝜑 → 𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    ⇒   (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧)))
 
31-Jan-2024dedekindeulemloc 15372 Lemma for dedekindeu 15376. The set L is located. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑 → 𝐿 ⊆ ℝ)    &   (𝜑 → 𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    ⇒   (𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧 ∈ 𝐿 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐿 𝑧 < 𝑦)))
 
31-Jan-2024dedekindeulemub 15371 Lemma for dedekindeu 15376. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑 → 𝐿 ⊆ ℝ)    &   (𝜑 → 𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    ⇒   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐿 𝑦 < 𝑥)
 
30-Jan-2024axsuploc 8257 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 8158 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.)
(((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 < 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
 
30-Jan-2024iotam 5320 Representation of "the unique element such that 𝜑 " with a class expression 𝐴 which is inhabited (that means that "the unique element such that 𝜑 " exists). (Contributed by AV, 30-Jan-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ((𝐴 ∈ 𝑉 ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)
 
29-Jan-2024sgrpidmndm 13526 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.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    ⇒   ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (∃𝑤 𝑤 ∈ 𝑒 ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd)
 
29-Jan-2024ccatw2s1p1g 11231 Extract the symbol of the first singleton word of a word concatenated with this singleton word and another singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.) (Revised by AV, 1-May-2020.) (Revised by AV, 29-Jan-2024.)
(((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝑁) = 𝑋)
 
28-Jan-2024ccat2s1fstg 11234 The first symbol of the concatenation of a word with two single symbols. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by AV, 28-Jan-2024.)
(((𝑊 ∈ Word 𝑉 ∧ 0 < (♯‘𝑊)) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘0) = (𝑊‘0))
 
28-Jan-2024ccat2s1fvwd 11233 Extract a symbol of a word from the concatenation of the word with two single symbols. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 13-Jan-2020.) (Proof shortened by AV, 1-May-2020.) (Revised by AV, 28-Jan-2024.)
(𝜑 → 𝑊 ∈ Word 𝑉)    &   (𝜑 → 𝐼 ∈ ℕ0)    &   (𝜑 → 𝐼 < (♯‘𝑊))    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊‘𝐼))
 
26-Jan-2024elovmporab1w 6228 Implications for the value of an operation, defined by the maps-to notation with a class abstraction as a result, having an element. Here, the base set of the class abstraction depends on the first operand. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by GG, 26-Jan-2024.)
𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})    &   ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)    ⇒   (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀))
 
26-Jan-2024opabidw 4353 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 4352 with a disjoint variable condition. (Contributed by NM, 14-Apr-1995.) (Revised by GG, 26-Jan-2024.)
(⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
 
26-Jan-2024invdisjrab 4083 The restricted class abstractions {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} for distinct 𝑦 ∈ 𝐴 are disjoint. (Contributed by AV, 6-May-2020.) (Proof shortened by GG, 26-Jan-2024.)
Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}
 
24-Jan-2024axpre-suploclemres 8126 Lemma for axpre-suploc 8127. The result. The proof just needs to define 𝐵 as basically the same set as 𝐴 (but expressed as a subset of R rather than a subset of ℝ), and apply suplocsr 8034. (Contributed by Jim Kingdon, 24-Jan-2024.)
(𝜑 → 𝐴 ⊆ ℝ)    &   (𝜑 → 𝐶 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)    &   (𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))    &   𝐵 = {𝑤 ∈ R ∣ ⟨𝑤, 0R⟩ ∈ 𝐴}    ⇒   (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
 
23-Jan-2024ax-pre-suploc 8158 An inhabited, bounded-above, located set of reals has a supremum.

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

Although this and ax-caucvg 8157 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 8157.

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

(((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
 
23-Jan-2024axpre-suploc 8127 An inhabited, bounded-above, located set of reals has a supremum.

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

This construction-dependent theorem should not be referenced directly; instead, use ax-pre-suploc 8158. (Contributed by Jim Kingdon, 23-Jan-2024.) (New usage is discouraged.)

(((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 <ℝ 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <ℝ 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <ℝ 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
 
22-Jan-2024suplocsr 8034 An inhabited, bounded, located set of signed reals has a supremum. (Contributed by Jim Kingdon, 22-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦)))    ⇒   (𝜑 → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
 
21-Jan-2024bj-el2oss1o 16431 Shorter proof of el2oss1o 6616 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ 2o → 𝐴 ⊆ 1o)
 
21-Jan-2024ltm1sr 8002 Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.)
(𝐴 ∈ R → (𝐴 +R -1R) <R 𝐴)
 
20-Jan-2024mndinvmod 13551 Uniqueness of an inverse element in a monoid, if it exists. (Contributed by AV, 20-Jan-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝐴 ∈ 𝐵)    ⇒   (𝜑 → ∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))
 
20-Jan-2024ccats1val1g 11225 Value of a symbol in the left half of a word concatenated with a single symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by JJ, 20-Jan-2024.)
((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑌 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝑊 ++ ⟨“𝑆”⟩)‘𝐼) = (𝑊‘𝐼))
 
19-Jan-2024suplocsrlempr 8032 Lemma for suplocsr 8034. The set 𝐵 has a least upper bound. (Contributed by Jim Kingdon, 19-Jan-2024.)
𝐵 = {𝑤 ∈ P ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   (𝜑 → 𝐴 ⊆ R)    &   (𝜑 → 𝐶 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦)))    ⇒   (𝜑 → ∃𝑣 ∈ P (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢)))
 
18-Jan-2024ccatval1 11183 Value of a symbol in the left half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 30-Apr-2020.) (Revised by JJ, 18-Jan-2024.)
((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ 𝑇)‘𝐼) = (𝑆‘𝐼))
 
18-Jan-2024ccat0 11182 The concatenation of two words is empty iff the two words are empty. (Contributed by AV, 4-Mar-2022.) (Revised by JJ, 18-Jan-2024.)
((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) = ∅ ↔ (𝑆 = ∅ ∧ 𝑇 = ∅)))
 
18-Jan-2024suplocsrlemb 8031 Lemma for suplocsr 8034. The set 𝐵 is located. (Contributed by Jim Kingdon, 18-Jan-2024.)
𝐵 = {𝑤 ∈ P ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   (𝜑 → 𝐴 ⊆ R)    &   (𝜑 → 𝐶 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦)))    ⇒   (𝜑 → ∀𝑢 ∈ P ∀𝑣 ∈ P (𝑢<P 𝑣 → (∃𝑞 ∈ 𝐵 𝑢<P 𝑞 ∨ ∀𝑞 ∈ 𝐵 𝑞<P 𝑣)))
 
16-Jan-2024suplocsrlem 8033 Lemma for suplocsr 8034. The set 𝐴 has a least upper bound. (Contributed by Jim Kingdon, 16-Jan-2024.)
𝐵 = {𝑤 ∈ P ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   (𝜑 → 𝐴 ⊆ R)    &   (𝜑 → 𝐶 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦)))    ⇒   (𝜑 → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
 
15-Jan-2024eqg0el 13839 Equivalence class of a quotient group for a subgroup. (Contributed by Thierry Arnoux, 15-Jan-2024.)
∼ = (𝐺 ~QG 𝐻)    ⇒   ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([𝑋] ∼ = 𝐻 ↔ 𝑋 ∈ 𝐻))
 
14-Jan-2024wlklenvclwlk 16253 The number of vertices in a walk equals the length of the walk after it is "closed" (i.e. enhanced by an edge from its last vertex to its first vertex). (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 2-May-2021.) (Revised by JJ, 14-Jan-2024.)
(𝑊 ∈ Word (Vtx‘𝐺) → (⟨𝐹, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (Walks‘𝐺) → (♯‘𝐹) = (♯‘𝑊)))
 
14-Jan-2024suplocexprlemlub 7949 Lemma for suplocexpr 7950. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝜑 → (𝑦<P 𝐵 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
 
14-Jan-2024suplocexprlemub 7948 Lemma for suplocexpr 7950. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝜑 → ∀𝑦 ∈ 𝐴 ¬ 𝐵<P 𝑦)
 
10-Jan-2024nfcsbw 3163 Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3164 with a disjoint variable condition. (Contributed by Mario Carneiro, 12-Oct-2016.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐵    ⇒   Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵
 
10-Jan-2024nfsbcw 3161 Bound-variable hypothesis builder for class substitution. Version of nfsbc 3051 with a disjoint variable condition, which in the future may make it possible to reduce axiom usage. (Contributed by NM, 7-Sep-2014.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥[𝐴 / 𝑦]𝜑
 
10-Jan-2024nfsbcdw 3160 Version of nfsbcd 3050 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
 
10-Jan-2024cbvcsbw 3130 Version of cbvcsb 3131 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.)
Ⅎ𝑦𝐶    &   Ⅎ𝑥𝐷    &   (𝑥 = 𝑦 → 𝐶 = 𝐷)    ⇒   ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷
 
10-Jan-2024cbvsbcw 3058 Version of cbvsbc 3059 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
 
10-Jan-2024cbvrex2vw 2778 Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 2780 with a disjoint variable condition, which does not require ax-13 2203. (Contributed by FL, 2-Jul-2012.) (Revised by GG, 10-Jan-2024.)
(𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)
 
10-Jan-2024cbvral2vw 2777 Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 2779 with a disjoint variable condition, which does not require ax-13 2203. (Contributed by NM, 10-Aug-2004.) (Revised by GG, 10-Jan-2024.)
(𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)
 
10-Jan-2024cbvrexw 2760 Rule used to change bound variables, using implicit substitution. Version of cbvrexfw 2756 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 1555 and ax-bndl 1557 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024cbvralw 2759 Rule used to change bound variables, using implicit substitution. Version of cbvral 2762 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 1555 and ax-bndl 1557 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024cbvrexfw 2756 Rule used to change bound variables, using implicit substitution. Version of cbvrexf 2758 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 1555 and ax-bndl 1557 in the proof. (Contributed by FL, 27-Apr-2008.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑦𝐴    &   Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024nfralw 2568 Bound-variable hypothesis builder for restricted quantification. See nfralya 2571 for a version with 𝑦 and 𝐴 distinct instead of 𝑥 and 𝑦. (Contributed by NM, 1-Sep-1999.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑
 
10-Jan-2024nfraldw 2563 Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfraldya 2566 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by NM, 15-Feb-2013.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024nfabdw 2392 Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2393 with a disjoint variable condition. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
 
10-Jan-2024cbvex2vw 1981 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) (Revised by GG, 10-Jan-2024.)
((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)
 
10-Jan-2024cbval2vw 1980 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.) (Revised by GG, 10-Jan-2024.)
((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)
 
10-Jan-2024cbv2w 1797 Rule used to change bound variables, using implicit substitution. Version of cbv2 1796 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by GG, 10-Jan-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 
9-Jan-2024suplocexprlemloc 7946 Lemma for suplocexpr 7950. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝜑 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑟 ∈ (2nd ‘𝐵))))
 
9-Jan-2024suplocexprlemdisj 7945 Lemma for suplocexpr 7950. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝜑 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)))
 
9-Jan-2024suplocexprlemru 7944 Lemma for suplocexpr 7950. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝜑 → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))
 
9-Jan-2024suplocexprlemrl 7942 Lemma for suplocexpr 7950. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    ⇒   (𝜑 → ∀𝑞 ∈ Q (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))))
 
9-Jan-2024suplocexprlem2b 7939 Lemma for suplocexpr 7950. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝐴 ⊆ P → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
 
9-Jan-2024suplocexprlemell 7938 Lemma for suplocexpr 7950. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝐵 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (1st ‘𝑥))
 
7-Jan-2024suplocexpr 7950 An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    ⇒   (𝜑 → ∃𝑥 ∈ P (∀𝑦 ∈ 𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))
 
7-Jan-2024suplocexprlemex 7947 Lemma for suplocexpr 7950. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝜑 → 𝐵 ∈ P)
 
7-Jan-2024suplocexprlemmu 7943 Lemma for suplocexpr 7950. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩    ⇒   (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝐵))
 
7-Jan-2024suplocexprlemml 7941 Lemma for suplocexpr 7950. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    ⇒   (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ ∪ (1st “ 𝐴))
 
7-Jan-2024suplocexprlemss 7940 Lemma for suplocexpr 7950. 𝐴 is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))    ⇒   (𝜑 → 𝐴 ⊆ P)
 
5-Jan-2024dedekindicclemicc 15385 Lemma for dedekindicc 15386. Same as dedekindicc 15386, except that we merely show 𝑥 to be an element of (𝐴[,]𝐵). Later we will strengthen that to (𝐴(,)𝐵). (Contributed by Jim Kingdon, 5-Jan-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    &   (𝜑 → 𝐴 < 𝐵)    ⇒   (𝜑 → ∃!𝑥 ∈ (𝐴[,]𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟))
 
5-Jan-2024dedekindeu 15376 A Dedekind cut identifies a unique real number. Similar to df-inp 7691 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.)
(𝜑 → 𝐿 ⊆ ℝ)    &   (𝜑 → 𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿 ∩ 𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))    ⇒   (𝜑 → ∃!𝑥 ∈ ℝ (∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟))
 
1-Jan-2024ccatlen 11181 The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by JJ, 1-Jan-2024.)
((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑆) + (♯‘𝑇)))
 
31-Dec-2023dvmptsubcn 15476 Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)    &   ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ 𝑉)    &   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵))    &   ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐶 ∈ ℂ)    &   ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐷 ∈ 𝑊)    &   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐶)) = (𝑥 ∈ ℂ ↦ 𝐷))    ⇒   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 − 𝐶))) = (𝑥 ∈ ℂ ↦ (𝐵 − 𝐷)))
 
31-Dec-2023dvmptnegcn 15475 Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)    &   ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ 𝑉)    &   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵))    ⇒   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ -𝐴)) = (𝑥 ∈ ℂ ↦ -𝐵))
 
31-Dec-2023dvmptcmulcn 15474 Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)    &   ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ 𝑉)    &   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵))    &   (𝜑 → 𝐶 ∈ ℂ)    ⇒   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝐶 · 𝐴))) = (𝑥 ∈ ℂ ↦ (𝐶 · 𝐵)))
 
31-Dec-2023rinvmod 13919 Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6221. (Contributed by AV, 31-Dec-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝐴 ∈ 𝐵)    ⇒   (𝜑 → ∃*𝑤 ∈ 𝐵 (𝐴 + 𝑤) = 0 )
 
31-Dec-2023brm 4140 If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)
(𝐴𝑅𝐵 → ∃𝑥 𝑥 ∈ 𝑅)
 
30-Dec-2023dvmptccn 15468 Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
(𝜑 → 𝐴 ∈ ℂ)    ⇒   (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 0))
 
30-Dec-2023dvmptidcn 15467 Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
(ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1)
 
30-Dec-2023eqwrd 11163 Two words are equal iff they have the same length and the same symbol at each position. (Contributed by AV, 13-Apr-2018.) (Revised by JJ, 30-Dec-2023.)
((𝑈 ∈ Word 𝑆 ∧ 𝑊 ∈ Word 𝑇) → (𝑈 = 𝑊 ↔ ((♯‘𝑈) = (♯‘𝑊) ∧ ∀𝑖 ∈ (0..^(♯‘𝑈))(𝑈‘𝑖) = (𝑊‘𝑖))))
 
29-Dec-2023mndbn0 13537 The base set of a monoid is not empty. (It is also inhabited, as seen at mndidcl 13536). Statement in [Lang] p. 3. (Contributed by AV, 29-Dec-2023.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Mnd → 𝐵 ≠ ∅)
 
28-Dec-2023mulgnn0gsum 13738 Group multiple (exponentiation) operation at a nonnegative integer expressed by a group sum. This corresponds to the definition in [Lang] p. 6, second formula. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    &   𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋)    ⇒   ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))
 
28-Dec-2023mulgnngsum 13737 Group multiple (exponentiation) operation at a positive integer expressed by a group sum. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    &   𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋)    ⇒   ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))
 
26-Dec-2023gsumfzreidx 13947 Re-index a finite group sum using a bijection. Corresponds to the first equation in [Lang] p. 5 with 𝑀 = 1. (Contributed by AV, 26-Dec-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    &   (𝜑 → 𝐻:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    ⇒   (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))
 
26-Dec-2023gsumsplit1r 13504 Splitting off the rightmost summand of a group sum. This corresponds to the (inductive) definition of a (finite) product in [Lang] p. 4, first formula. (Contributed by AV, 26-Dec-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐹:(𝑀...(𝑁 + 1))⟶𝐵)    ⇒   (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ (𝑀...𝑁))) + (𝐹‘(𝑁 + 1))))
 
26-Dec-2023lidrididd 13488 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 13487) is equal to the two-sided identity element. (Contributed by AV, 26-Dec-2023.)
(𝜑 → 𝐿 ∈ 𝐵)    &   (𝜑 → 𝑅 ∈ 𝐵)    &   (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥)    &   (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥)    &   𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    ⇒   (𝜑 → 𝐿 = 0 )
 
26-Dec-2023lidrideqd 13487 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.)
(𝜑 → 𝐿 ∈ 𝐵)    &   (𝜑 → 𝑅 ∈ 𝐵)    &   (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥)    &   (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥)    ⇒   (𝜑 → 𝐿 = 𝑅)
 
25-Dec-2023ctfoex 7322 A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.)
(∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → 𝐴 ∈ V)
 
23-Dec-2023enct 13077 Countability is invariant relative to equinumerosity. (Contributed by Jim Kingdon, 23-Dec-2023.)
(𝐴 ≈ 𝐵 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)))
 
23-Dec-2023enctlem 13076 Lemma for enct 13077. One direction of the biconditional. (Contributed by Jim Kingdon, 23-Dec-2023.)
(𝐴 ≈ 𝐵 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)))
 
23-Dec-2023omct 7321 ω is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
∃𝑓 𝑓:ω–onto→(ω ⊔ 1o)
 
21-Dec-2023dvcoapbr 15460 The chain rule for derivatives at a point. The 𝑢 # 𝐶 → (𝐺‘𝑢) # (𝐺‘𝐶) hypothesis constrains what functions work for 𝐺. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 21-Dec-2023.)
(𝜑 → 𝐹:𝑋⟶ℂ)    &   (𝜑 → 𝑋 ⊆ 𝑆)    &   (𝜑 → 𝐺:𝑌⟶𝑋)    &   (𝜑 → 𝑌 ⊆ 𝑇)    &   (𝜑 → ∀𝑢 ∈ 𝑌 (𝑢 # 𝐶 → (𝐺‘𝑢) # (𝐺‘𝐶)))    &   (𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝑇 ⊆ ℂ)    &   (𝜑 → (𝐺‘𝐶)(𝑆 D 𝐹)𝐾)    &   (𝜑 → 𝐶(𝑇 D 𝐺)𝐿)    &   𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿))
 
19-Dec-2023apsscn 8832 The points apart from a given point are complex numbers. (Contributed by Jim Kingdon, 19-Dec-2023.)
{𝑥 ∈ 𝐴 ∣ 𝑥 # 𝐵} ⊆ ℂ
 
19-Dec-2023aprcl 8831 Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.)
(𝐴 # 𝐵 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
 
18-Dec-2023limccoap 15431 Composition of two limits. This theorem is only usable in the case where 𝑥 # 𝑋 implies R(x) # 𝐶 so it is less general than might appear at first. (Contributed by Mario Carneiro, 29-Dec-2016.) (Revised by Jim Kingdon, 18-Dec-2023.)
((𝜑 ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → 𝑅 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶})    &   ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶}) → 𝑆 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑅) limℂ 𝑋))    &   (𝜑 → 𝐷 ∈ ((𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ↦ 𝑆) limℂ 𝐶))    &   (𝑦 = 𝑅 → 𝑆 = 𝑇)    ⇒   (𝜑 → 𝐷 ∈ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑇) limℂ 𝑋))
 
16-Dec-2023cnreim 11561 Complex apartness in terms of real and imaginary parts. See also apreim 8788 which is similar but with different notation. (Contributed by Jim Kingdon, 16-Dec-2023.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 ↔ ((ℜ‘𝐴) # (ℜ‘𝐵) ∨ (ℑ‘𝐴) # (ℑ‘𝐵))))
 
14-Dec-2023cnopnap 15364 The complex numbers apart from a given complex number form an open set. (Contributed by Jim Kingdon, 14-Dec-2023.)
(𝐴 ∈ ℂ → {𝑤 ∈ ℂ ∣ 𝑤 # 𝐴} ∈ (MetOpen‘(abs ∘ − )))
 
14-Dec-2023cnovex 14949 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) → (𝐽 Cn 𝐾) ∈ V)
 
13-Dec-2023reopnap 15299 The real numbers apart from a given real number form an open set. (Contributed by Jim Kingdon, 13-Dec-2023.)
(𝐴 ∈ ℝ → {𝑤 ∈ ℝ ∣ 𝑤 # 𝐴} ∈ (topGen‘ran (,)))
 
12-Dec-2023cnopncntop 15297 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.)
ℂ ∈ (MetOpen‘(abs ∘ − ))
 
12-Dec-2023unicntopcntop 15295 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.)
ℂ = ∪ (MetOpen‘(abs ∘ − ))
 
4-Dec-2023bj-pm2.18st 16407 Clavius law for stable formulas. See pm2.18dc 862. (Contributed by BJ, 4-Dec-2023.)
(STAB 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))
 
4-Dec-2023bj-nnclavius 16394 Clavius law with doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)
((¬ 𝜑 → 𝜑) → ¬ ¬ 𝜑)
 
2-Dec-2023dvmulxx 15457 The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 15455. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 2-Dec-2023.)
(𝜑 → 𝐹:𝑋⟶ℂ)    &   (𝜑 → 𝑋 ⊆ 𝑆)    &   (𝜑 → 𝐺:𝑋⟶ℂ)    &   (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹))    &   (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐺))    ⇒   (𝜑 → ((𝑆 D (𝐹 ∘𝑓 · 𝐺))‘𝐶) = ((((𝑆 D 𝐹)‘𝐶) · (𝐺‘𝐶)) + (((𝑆 D 𝐺)‘𝐶) · (𝐹‘𝐶))))
 
1-Dec-2023dvmulxxbr 15455 The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmulxx 15457. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 1-Dec-2023.)
(𝜑 → 𝐹:𝑋⟶ℂ)    &   (𝜑 → 𝑋 ⊆ 𝑆)    &   (𝜑 → 𝐺:𝑋⟶ℂ)    &   (𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝐶(𝑆 D 𝐹)𝐾)    &   (𝜑 → 𝐶(𝑆 D 𝐺)𝐿)    &   𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   (𝜑 → 𝐶(𝑆 D (𝐹 ∘𝑓 · 𝐺))((𝐾 · (𝐺‘𝐶)) + (𝐿 · (𝐹‘𝐶))))
 
29-Nov-2023subctctexmid 16661 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.)
(𝜑 → ∀𝑥(∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝑥) → ∃𝑔 𝑔:ω–onto→(𝑥 ⊔ 1o)))    &   (𝜑 → ω ∈ Markov)    ⇒   (𝜑 → EXMID)
 
29-Nov-2023ismkvnex 7359 The predicate of being Markov stated in terms of double negation and comparison with 1o. (Contributed by Jim Kingdon, 29-Nov-2023.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
 
28-Nov-2023ccfunen 7488 Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)
(𝜑 → CCHOICE)    &   (𝜑 → 𝐴 ≈ ω)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥)    ⇒   (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))
 
28-Nov-2023exmid1stab 4300 If every proposition is stable, excluded middle follows. We are thinking of 𝑥 as a proposition and 𝑥 = {∅} as "𝑥 is true". (Contributed by Jim Kingdon, 28-Nov-2023.)
((𝜑 ∧ 𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})    ⇒   (𝜑 → EXMID)
 
27-Nov-2023df-cc 7487 The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7426 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.)
(CCHOICE ↔ ∀𝑥(dom 𝑥 ≈ ω → ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)))
 
26-Nov-2023offeq 6254 Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   (𝜑 → 𝐹:𝐴⟶𝑆)    &   (𝜑 → 𝐺:𝐵⟶𝑇)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝐴 ∩ 𝐵) = 𝐶    &   (𝜑 → 𝐻:𝐶⟶𝑈)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐷)    &   ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐸)    &   ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐷𝑅𝐸) = (𝐻‘𝑥))    ⇒   (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = 𝐻)
 
25-Nov-2023dvaddxx 15456 The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 15454. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
(𝜑 → 𝐹:𝑋⟶ℂ)    &   (𝜑 → 𝑋 ⊆ 𝑆)    &   (𝜑 → 𝐺:𝑋⟶ℂ)    &   (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹))    &   (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐺))    ⇒   (𝜑 → ((𝑆 D (𝐹 ∘𝑓 + 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶)))
 
25-Nov-2023dvaddxxbr 15454 The sum rule for derivatives at a point. That is, if the derivative of 𝐹 at 𝐶 is 𝐾 and the derivative of 𝐺 at 𝐶 is 𝐿, then the derivative of the pointwise sum of those two functions at 𝐶 is 𝐾 + 𝐿. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
(𝜑 → 𝐹:𝑋⟶ℂ)    &   (𝜑 → 𝑋 ⊆ 𝑆)    &   (𝜑 → 𝐺:𝑋⟶ℂ)    &   (𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝐶(𝑆 D 𝐹)𝐾)    &   (𝜑 → 𝐶(𝑆 D 𝐺)𝐿)    &   𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   (𝜑 → 𝐶(𝑆 D (𝐹 ∘𝑓 + 𝐺))(𝐾 + 𝐿))
 
25-Nov-2023dcnn 855 Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 849. The relation between dcn 849 and dcnn 855 is analogous to that between notnot 634 and notnotnot 639 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 855 means that a proposition is testable if and only if its negation is testable, and dcn 849 means that decidability implies testability. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof shortened by BJ, 25-Nov-2023.)
(DECID ¬ 𝜑 ↔ DECID ¬ ¬ 𝜑)
 
24-Nov-2023bj-dcst 16418 Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.)
(DECID STAB 𝜑 ↔ STAB 𝜑)
 
24-Nov-2023bj-nnbidc 16414 If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 16401. (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ 𝜑 → (DECID 𝜑 ↔ 𝜑))
 
24-Nov-2023bj-dcstab 16413 A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
(DECID 𝜑 → STAB 𝜑)
 
24-Nov-2023bj-fadc 16411 A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
(¬ 𝜑 → DECID 𝜑)
 
24-Nov-2023bj-trdc 16409 A provable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
(𝜑 → DECID 𝜑)
 
24-Nov-2023bj-stal 16406 The universal quantification of a stable formula is stable. See bj-stim 16403 for implication, stabnot 840 for negation, and bj-stan 16404 for conjunction. (Contributed by BJ, 24-Nov-2023.)
(∀𝑥STAB 𝜑 → STAB ∀𝑥𝜑)
 
24-Nov-2023bj-stand 16405 The conjunction of two stable formulas is stable. Deduction form of bj-stan 16404. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 16404 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
(𝜑 → STAB 𝜓)    &   (𝜑 → STAB 𝜒)    ⇒   (𝜑 → STAB (𝜓 ∧ 𝜒))
 
24-Nov-2023bj-stan 16404 The conjunction of two stable formulas is stable. See bj-stim 16403 for implication, stabnot 840 for negation, and bj-stal 16406 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
((STAB 𝜑 ∧ STAB 𝜓) → STAB (𝜑 ∧ 𝜓))
 
24-Nov-2023bj-stim 16403 A conjunction with a stable consequent is stable. See stabnot 840 for negation , bj-stan 16404 for conjunction , and bj-stal 16406 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
(STAB 𝜓 → STAB (𝜑 → 𝜓))
 
24-Nov-2023bj-nnbist 16401 If a formula is not refutable, then it is stable if and only if it is provable. By double-negation translation, if 𝜑 is a classical tautology, then ¬ ¬ 𝜑 is an intuitionistic tautology. Therefore, if 𝜑 is a classical tautology, then 𝜑 is intuitionistically equivalent to its stability (and to its decidability, see bj-nnbidc 16414). (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ 𝜑 → (STAB 𝜑 ↔ 𝜑))
 
24-Nov-2023bj-fast 16398 A refutable formula is stable. (Contributed by BJ, 24-Nov-2023.)
(¬ 𝜑 → STAB 𝜑)
 
24-Nov-2023bj-trst 16396 A provable formula is stable. (Contributed by BJ, 24-Nov-2023.)
(𝜑 → STAB 𝜑)
 
24-Nov-2023bj-nnan 16393 The double negation of a conjunction implies the conjunction of the double negations. (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ (𝜑 ∧ 𝜓) → (¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓))
 
24-Nov-2023bj-nnim 16392 The double negation of an implication implies the implication with the consequent doubly negated. (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ (𝜑 → 𝜓) → (𝜑 → ¬ ¬ 𝜓))
 
24-Nov-2023bj-nnsn 16390 As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.)
((𝜑 → ¬ 𝜓) ↔ (¬ ¬ 𝜑 → ¬ 𝜓))
 
24-Nov-2023nnal 1697 The double negation of a universal quantification implies the universal quantification of the double negation. The converse holds in classical but not in intuitionistic logic. (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)
 
22-Nov-2023ofvalg 6250 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)
(𝜑 → 𝐹 Fn 𝐴)    &   (𝜑 → 𝐺 Fn 𝐵)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝐴 ∩ 𝐵) = 𝑆    &   ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)    &   ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)    &   ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐶𝑅𝐷) ∈ 𝑈)    ⇒   ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
 
21-Nov-2023exmidac 7429 The axiom of choice implies excluded middle. See acexmid 6022 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 → EXMID)
 
21-Nov-2023exmidaclem 7428 Lemma for exmidac 7429. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})}    &   𝐶 = {𝐴, 𝐵}    ⇒   (CHOICE → EXMID)
 
21-Nov-2023exmid1dc 4292 A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4285 or ordtriexmid 4621. In this context 𝑥 = {∅} can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)
((𝜑 ∧ 𝑥 ⊆ {∅}) → DECID 𝑥 = {∅})    ⇒   (𝜑 → EXMID)
 
20-Nov-2023acfun 7427 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.)
(𝜑 → CHOICE)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥)    ⇒   (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))
 
18-Nov-2023rnrhmsubrg 14290 The range of a ring homomorphism is a subring. (Contributed by SN, 18-Nov-2023.)
(𝐹 ∈ (𝑀 RingHom 𝑁) → ran 𝐹 ∈ (SubRing‘𝑁))
 
18-Nov-2023condc 860 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 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
 
18-Nov-2023stdcn 854 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 849. (Contributed by BJ, 18-Nov-2023.)
(STAB 𝜑 ↔ (DECID ¬ 𝜑 → DECID 𝜑))
 
17-Nov-2023cnplimclemr 15422 Lemma for cnplimccntop 15423. The reverse direction. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐽 = (𝐾 ↾t 𝐴)    &   (𝜑 → 𝐴 ⊆ ℂ)    &   (𝜑 → 𝐹:𝐴⟶ℂ)    &   (𝜑 → 𝐵 ∈ 𝐴)    &   (𝜑 → (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵))    ⇒   (𝜑 → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵))
 
17-Nov-2023cnplimclemle 15421 Lemma for cnplimccntop 15423. Satisfying the epsilon condition for continuity. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐽 = (𝐾 ↾t 𝐴)    &   (𝜑 → 𝐴 ⊆ ℂ)    &   (𝜑 → 𝐹:𝐴⟶ℂ)    &   (𝜑 → 𝐵 ∈ 𝐴)    &   (𝜑 → (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵))    &   (𝜑 → 𝐸 ∈ ℝ+)    &   (𝜑 → 𝐷 ∈ ℝ+)    &   (𝜑 → 𝑍 ∈ 𝐴)    &   ((𝜑 ∧ 𝑍 # 𝐵 ∧ (abs‘(𝑍 − 𝐵)) < 𝐷) → (abs‘((𝐹‘𝑍) − (𝐹‘𝐵))) < (𝐸 / 2))    &   (𝜑 → (abs‘(𝑍 − 𝐵)) < 𝐷)    ⇒   (𝜑 → (abs‘((𝐹‘𝑍) − (𝐹‘𝐵))) < 𝐸)
 
14-Nov-2023limccnp2cntop 15430 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.)
((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝑋)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ 𝑌)    &   (𝜑 → 𝑋 ⊆ ℂ)    &   (𝜑 → 𝑌 ⊆ ℂ)    &   𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐽 = ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌))    &   (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝐵))    &   (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) limℂ 𝐵))    &   (𝜑 → 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))    ⇒   (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥 ∈ 𝐴 ↦ (𝑅𝐻𝑆)) limℂ 𝐵))
 
10-Nov-2023rpmaxcl 11806 The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → sup({𝐴, 𝐵}, ℝ, < ) ∈ ℝ+)
 
9-Nov-2023limccnp2lem 15429 Lemma for limccnp2cntop 15430. 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.)
((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝑋)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ 𝑌)    &   (𝜑 → 𝑋 ⊆ ℂ)    &   (𝜑 → 𝑌 ⊆ ℂ)    &   𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐽 = ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌))    &   (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝐵))    &   (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) limℂ 𝐵))    &   (𝜑 → 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))    &   Ⅎ𝑥𝜑    &   (𝜑 → 𝐸 ∈ ℝ+)    &   (𝜑 → 𝐿 ∈ ℝ+)    &   (𝜑 → ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝐿 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝐿) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝐸))    &   (𝜑 → 𝐹 ∈ ℝ+)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥 − 𝐵)) < 𝐹) → (abs‘(𝑅 − 𝐶)) < 𝐿))    &   (𝜑 → 𝐺 ∈ ℝ+)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥 − 𝐵)) < 𝐺) → (abs‘(𝑆 − 𝐷)) < 𝐿))    ⇒   (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥 − 𝐵)) < 𝑑) → (abs‘((𝑅𝐻𝑆) − (𝐶𝐻𝐷))) < 𝐸))
 
4-Nov-2023ellimc3apf 15413 Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 4-Nov-2023.)
(𝜑 → 𝐹:𝐴⟶ℂ)    &   (𝜑 → 𝐴 ⊆ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   Ⅎ𝑧𝐹    ⇒   (𝜑 → (𝐶 ∈ (𝐹 limℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))
 
3-Nov-2023limcmpted 15416 Express the limit operator for a function defined by a mapping, via epsilon-delta. (Contributed by Jim Kingdon, 3-Nov-2023.)
(𝜑 → 𝐴 ⊆ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐷 ∈ ℂ)    ⇒   (𝜑 → (𝐶 ∈ ((𝑧 ∈ 𝐴 ↦ 𝐷) limℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘(𝐷 − 𝐶)) < 𝑥))))
 
1-Nov-2023unct 13086 The union of two countable sets is countable. Corollary 8.1.20 of [AczelRathjen], p. 75. (Contributed by Jim Kingdon, 1-Nov-2023.)
((∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ∧ ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)) → ∃ℎ ℎ:ω–onto→((𝐴 ∪ 𝐵) ⊔ 1o))
 
31-Oct-2023ctiunct 13084 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 𝐵(𝑥): it refers to 𝐵(𝑥) together with the 𝐺(𝑥) which enumerates it. Theorem 8.1.19 of [AczelRathjen], p. 74.

For "countably many countable sets" the key hypothesis would be (𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑔𝑔:ω–onto→(𝐵 ⊔ 1o). This is almost omiunct 13088 (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 13086, 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 13039) and using the first number to map to an element 𝑥 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 ∪ 𝑥 ∈ 𝐴𝐵. Although it would be possible to work directly with countability expressed as 𝐹:ω–onto→(𝐴 ⊔ 1o), we instead use functions from subsets of the natural numbers via ctssdccl 7315 and ctssdc 7317.

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

(𝜑 → 𝐹:ω–onto→(𝐴 ⊔ 1o))    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:ω–onto→(𝐵 ⊔ 1o))    ⇒   (𝜑 → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o))
 
30-Oct-2023ctssdccl 7315 A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7317 but expressed in terms of classes rather than ∃. (Contributed by Jim Kingdon, 30-Oct-2023.)
(𝜑 → 𝐹:ω–onto→(𝐴 ⊔ 1o))    &   𝑆 = {𝑥 ∈ ω ∣ (𝐹‘𝑥) ∈ (inl “ 𝐴)}    &   𝐺 = (◡inl ∘ 𝐹)    ⇒   (𝜑 → (𝑆 ⊆ ω ∧ 𝐺:𝑆–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆))
 
28-Oct-2023ctiunctlemfo 13083 Lemma for ctiunct 13084. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑 → 𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)    &   (𝜑 → 𝐹:𝑆–onto→𝐴)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)    &   (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}    &   𝐻 = (𝑛 ∈ 𝑈 ↦ (⦋(𝐹‘(1st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2nd ‘(𝐽‘𝑛))))    &   Ⅎ𝑥𝐻    &   Ⅎ𝑥𝑈    ⇒   (𝜑 → 𝐻:𝑈–onto→∪ 𝑥 ∈ 𝐴 𝐵)
 
28-Oct-2023ctiunctlemf 13082 Lemma for ctiunct 13084. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑 → 𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)    &   (𝜑 → 𝐹:𝑆–onto→𝐴)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)    &   (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}    &   𝐻 = (𝑛 ∈ 𝑈 ↦ (⦋(𝐹‘(1st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2nd ‘(𝐽‘𝑛))))    ⇒   (𝜑 → 𝐻:𝑈⟶∪ 𝑥 ∈ 𝐴 𝐵)
 
28-Oct-2023ctiunctlemudc 13081 Lemma for ctiunct 13084. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑 → 𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)    &   (𝜑 → 𝐹:𝑆–onto→𝐴)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)    &   (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}    ⇒   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑈)
 
28-Oct-2023ctiunctlemuom 13080 Lemma for ctiunct 13084. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑 → 𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)    &   (𝜑 → 𝐹:𝑆–onto→𝐴)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)    &   (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}    ⇒   (𝜑 → 𝑈 ⊆ ω)
 
28-Oct-2023ctiunctlemu2nd 13079 Lemma for ctiunct 13084. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑 → 𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)    &   (𝜑 → 𝐹:𝑆–onto→𝐴)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)    &   (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}    &   (𝜑 → 𝑁 ∈ 𝑈)    ⇒   (𝜑 → (2nd ‘(𝐽‘𝑁)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇)
 
28-Oct-2023ctiunctlemu1st 13078 Lemma for ctiunct 13084. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑 → 𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)    &   (𝜑 → 𝐹:𝑆–onto→𝐴)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)    &   (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}    &   (𝜑 → 𝑁 ∈ 𝑈)    ⇒   (𝜑 → (1st ‘(𝐽‘𝑁)) ∈ 𝑆)
 
28-Oct-2023pm2.521gdc 875 A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107, under a decidability condition. (Contributed by BJ, 28-Oct-2023.)
(DECID 𝜑 → (¬ (𝜑 → 𝜓) → (𝜒 → 𝜑)))
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