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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 31-Jul-2026 at 7:19 AM ET.
Recent Additions to the Intuitionistic Logic Explorer
DateLabelDescription
Theorem
 
24-Jul-2026slotm 13398 A structure with an inhabited slot is inhabited. (Contributed by Jim Kingdon, 24-Jul-2026.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    ⇒   (𝐴 ∈ (𝐸‘𝐺) → ∃𝑗 𝑗 ∈ 𝐺)
 
22-Jul-2026mptmex 5939 If a function given by maps-to notation is inhabited, then the class it is defined on is inhabited. (Contributed by Jim Kingdon, 22-Jul-2026.)
(𝐶 ∈ (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑦 𝑦 ∈ 𝐴)
 
20-Jul-20262alsraln0idm 17133 Nested general "all some" quantifiers with class membership as their antecedents, for the same class 𝐴: 𝜑 holds for every 𝑥 and every 𝑦 in 𝐴, and 𝐴 is inhabited. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 20-Jul-2026.)
(∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐴 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴))
 
20-Jul-20262alsraln0m 17132 Nested general "all some" quantifiers with class membership as their antecedents: 𝜑 holds for every 𝑥 in 𝐴 and every 𝑦 in 𝐵, and both 𝐴 and 𝐵 are inhabited. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 20-Jul-2026.)
(∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐵 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵)))
 
20-Jul-2026n0alsm 17131 If 𝐴 is inhabited, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that 𝜑 holds for every 𝑥 in 𝐴. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)
(∃𝑥 𝑥 ∈ 𝐴 → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑))
 
20-Jul-2026alsraln0m 17128 The general "all some" quantifier with class membership as its antecedent holds if and only if 𝜑 holds for every 𝑥 in 𝐴 and 𝐴 is inhabited. This is the intuitionistic form of what set.mm states using 𝐴 ≠ ∅; see the section comment. (Contributed by Peter Mazsa, 28-Nov-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)
(∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴))
 
20-Jul-2026alsralrex 17127 The general "all some" quantifier with class membership as its antecedent holds if and only if 𝜑 holds for every 𝑥 in 𝐴 and some 𝑥 in 𝐴 satisfies 𝜑. (Contributed by Peter Mazsa, 27-Nov-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)
(∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑))
 
20-Jul-2026ralsanmo 17126 An "all some" statement restricted to a class, conjoined with the claim that at most one 𝑥 in 𝐴 satisfies its antecedent, is equivalent to the universal part conjoined with the claim that exactly one 𝑥 in 𝐴 satisfies the antecedent. This is the restricted counterpart of alsanmo 17125. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
((∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ∧ ∃*𝑥 ∈ 𝐴 𝜑) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑥 ∈ 𝐴 𝜑))
 
20-Jul-2026alsanmo 17125 An "all some" statement conjoined with the claim that at most one 𝑥 satisfies its antecedent is equivalent to the universal part conjoined with the claim that exactly one 𝑥 satisfies the antecedent. The "all some" quantifier supplies the existence of such an 𝑥 and ∃*𝑥𝜑 supplies the at-most-one part, so together they yield ∃!𝑥𝜑. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
((∀∃𝑥(𝜑 → 𝜓) ∧ ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))
 
20-Jul-2026rexrals 17124 If a member of 𝐴 satisfying the antecedent exists, then a restricted "all some" statement reduces to its universal part. This is the restricted counterpart of rexals 17130. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
(∃𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)))
 
20-Jul-2026ralrals 17123 If the universal part of a restricted "all some" statement holds, then the statement reduces to the existence of a member of 𝐴 satisfying its antecedent. This is the restricted counterpart of ralals 17129. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
(∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∃𝑥 ∈ 𝐴 𝜑))
 
20-Jul-2026als-no-surprise 17121 Demonstrate that there is never a "surprise" when using the allsome quantifier, that is, it is never possible for the consequent to be both always true and always false. This uses the definition of df-als 17102: the universal parts give ∀𝑥¬ 𝜑, which contradicts the witness that the allsome quantifier supplies. Ordinary "for all" with implication has no such property, since ∀𝑥(𝜑 → 𝜓) and ∀𝑥(𝜑 → ¬ 𝜓) can both hold when nothing satisfies 𝜑. (Contributed by David A. Wheeler, 27-Oct-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)
¬ (∀∃𝑥(𝜑 → 𝜓) ∧ ∀∃𝑥(𝜑 → ¬ 𝜓))
 
20-Jul-2026ralsmd 17112 Deduction rule: Given "all some" applied to a class, the class is inhabited. This is stronger than ralsn0d 17111, which only concludes that the class is nonempty; see n0r 3535. (Contributed by David A. Wheeler, 20-Jul-2026.)
(𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
 
19-Jul-2026disjdifg 3598 A class and its relative complement are disjoint. (Contributed by NM, 24-Mar-1998.) Generalize from disjdif 3599. (Revised by BJ, 19-Jul-2026.)
(𝐴 ⊆ 𝐵 → (𝐴 ∩ (𝐶 ∖ 𝐵)) = ∅)
 
19-Jul-2026sseq0b 3564 The only subclass of the empty class is itself. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Strengthen sseq0 3565 to a biconditional. (Revised by BJ, 19-Jul-2026.)
(𝐴 = ∅ → (𝐵 ⊆ 𝐴 ↔ 𝐵 = ∅))
 
18-Jul-2026sepab 4276 Separation Scheme (Aussonderung) in terms of a class abstraction. Prefer using the more natural statement rabexg 4277. (Contributed by NM, 8-Jun-1994.) Put in closed form. (Revised by BJ, 18-Jul-2026.)
(𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)
 
18-Jul-2026inssdif0im 3592 Intersection, subclass, and difference relationship. The converse holds in classical logic but not in intuitionistic logic. (Contributed by Jim Kingdon, 3-Aug-2018.) (Proof shortened by BJ, 18-Jul-2026.)
((𝐴 ∩ 𝐵) ⊆ 𝐶 → (𝐴 ∩ (𝐵 ∖ 𝐶)) = ∅)
 
15-Jul-2026rexals 17130 If some 𝑥 in 𝐴 satisfies 𝜑, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that 𝜑 holds for every 𝑥 in 𝐴. See rexrals 17124 for the restricted counterpart. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
(∃𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑))
 
15-Jul-2026ralals 17129 If 𝜑 holds for every 𝑥 in 𝐴, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that some 𝑥 in 𝐴 satisfies 𝜑. See ralrals 17123 for the restricted counterpart. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
(∀𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∃𝑥 ∈ 𝐴 𝜑))
 
14-Jul-2026uniex2 4579 The Axiom of Union using the standard abbreviation for union. Given any set 𝑥, its union 𝑦 exists. (Contributed by NM, 4-Jun-2006.) (Proof shortened by BJ, 14-Jul-2026.)
∃𝑦 𝑦 = ∪ 𝑥
 
14-Jul-2026sepgi 4250 Inference associated with sepg 4249. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 14-Jul-2026.)
𝐴 ∈ V    ⇒   ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
 
13-Jul-2026f1setfi 7311 The set of injections between two finite sets is finite. (Contributed by Jim Kingdon, 13-Jul-2026.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ∈ Fin)
 
13-Jul-2026fdcf1 7310 It is decidable whether a function from a finite set into another finite set is one-to-one. (Contributed by Jim Kingdon, 13-Jul-2026.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶𝐵) → DECID 𝐹:𝐴–1-1→𝐵)
 
12-Jul-2026rals-no-surprise 17122 Demonstrate that there is never a "surprise" when using the allsome quantifier restricted to a class, that is, it is never possible for the consequent to be both always true and always false of the members of 𝐴 that satisfy the antecedent. This is the restricted counterpart of als-no-surprise 17121, and follows from it by dfrals2 17104. Note that this holds without any assumption that 𝐴 is inhabited; that is the point of allsome, since the corresponding claim for the ordinary restricted "for all" fails when nothing in 𝐴 satisfies 𝜑. (Contributed by David A. Wheeler, 12-Jul-2026.)
¬ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ∧ ∀∃𝑥 ∈ 𝐴(𝜑 → ¬ 𝜓))
 
12-Jul-2026cbvals 17120 Rule used to change bound variables, using implicit substitution. (Contributed by David A. Wheeler, 12-Jul-2026.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    ⇒   (∀∃𝑥(𝜑 → 𝜓) ↔ ∀∃𝑦(𝜒 → 𝜃))
 
12-Jul-2026nfrals 17119 Bound-variable hypothesis builder for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    &   Ⅎ𝑥𝜓    ⇒   Ⅎ𝑥∀∃𝑦 ∈ 𝐴(𝜑 → 𝜓)
 
12-Jul-2026nfals 17118 Bound-variable hypothesis builder for "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝜓    ⇒   Ⅎ𝑥∀∃𝑦(𝜑 → 𝜓)
 
12-Jul-2026alsbid 17117 Deduction form of alsbii 17115. (Contributed by David A. Wheeler, 12-Jul-2026.)
Ⅎ𝑥𝜑    &   (𝜑 → (𝜓 ↔ 𝜃))    &   (𝜑 → (𝜒 ↔ 𝜏))    ⇒   (𝜑 → (∀∃𝑥(𝜓 → 𝜒) ↔ ∀∃𝑥(𝜃 → 𝜏)))
 
12-Jul-2026ralsbii 17116 Congruence for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026.)
(𝜑 ↔ 𝜒)    &   (𝜓 ↔ 𝜃)    ⇒   (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥 ∈ 𝐴(𝜒 → 𝜃))
 
12-Jul-2026alsbii 17115 Congruence: equivalents may be substituted inside an "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)
(𝜑 ↔ 𝜒)    &   (𝜓 ↔ 𝜃)    ⇒   (∀∃𝑥(𝜑 → 𝜓) ↔ ∀∃𝑥(𝜒 → 𝜃))
 
12-Jul-2026ralsex 17114 The consequent of an "all some" restricted to a class is witnessed: some member of 𝐴 satisfying 𝜑 also satisfies 𝜓. Restricted counterpart of alsex 17113. (Contributed by David A. Wheeler, 12-Jul-2026.)
(∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)
 
12-Jul-2026alsex 17113 The consequent of an "all some" is witnessed: if 𝜓 holds of every 𝑥 satisfying 𝜑, and some 𝑥 satisfies 𝜑, then some 𝑥 satisfies 𝜓. This is the positive counterpart of als-no-surprise 17121, and it is the property that ordinary "for all" with implication lacks: from ∀𝑥(𝜑 → 𝜓) alone nothing whatever follows about 𝜓, since nothing need satisfy 𝜑. It is the allsome quantifier says what a speaker of "all Martians are green" usually means. (Contributed by David A. Wheeler, 12-Jul-2026.)
(∀∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓)
 
12-Jul-2026ralsn0d 17111 Deduction rule: Given "all some" applied to a class, the class is not the empty set. (Contributed by David A. Wheeler, 23-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
(𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   (𝜑 → 𝐴 ≠ ∅)
 
12-Jul-2026rals2d 17110 Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. Note that the witness must satisfy the antecedent 𝜓, not merely be a member of 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
(𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
 
12-Jul-2026rals1d 17109 Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
(𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
 
12-Jul-2026ralsd 17106 Introduction rule for "all some" restricted to a class. This is the converse of rals1d 17109 and rals2d 17110 taken together. (Contributed by David A. Wheeler, 12-Jul-2026.)
(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))    &   (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))
 
12-Jul-2026alsd 17105 Introduction rule: "all some" holds if the "for all" part holds and the antecedent has a witness. This is the converse of als1d 17107 and als2d 17108 taken together, and is what lets an "all some" statement be proved rather than merely taken apart. (Contributed by David A. Wheeler, 12-Jul-2026.)
(𝜑 → ∀𝑥(𝜓 → 𝜒))    &   (𝜑 → ∃𝑥𝜓)    ⇒   (𝜑 → ∀∃𝑥(𝜓 → 𝜒))
 
12-Jul-2026dfrals2 17104 The bounded "all some" form is the general form with the class membership folded into the antecedent. (Contributed by David A. Wheeler, 22-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
(∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
 
12-Jul-2026df-rals 17103 Define "all some" applied to a class, which means 𝜓 is true whenever 𝜑 is true for 𝑥 in 𝐴, and there is at least one 𝑥 in 𝐴 where 𝜑 is true.

An older definition of the "all some" quantifier when scoped to a class, named df-alsc and now removed, instead applied a bare formula 𝜑 to the members of a class, asserting only (∀𝑥 ∈ 𝐴𝜑 ∧ ∃𝑥𝑥 ∈ 𝐴), that is, that the formula held throughout 𝐴 and that 𝐴 had at least one member. I've now decided that that was a mistake. Its older existence conjunct ∃𝑥𝑥 ∈ 𝐴 did not require any member of 𝐴 to satisfy the antecedent, so if the formula was itself an implication, that inner implication could still be vacuously true, which is precisely what the allsome quantifier exists to prevent. For example, the older definition meant that "among Martians, all tall ones are green" could be considered true if there are Martians, but no tall Martians. This version of the definition instead ensures that claims of the form "among Martians, all tall ones are green" can only be true if all tall Martians are green and that there is at least one tall Martian. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)

(∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑))
 
12-Jul-2026wrals 17101 Extend wff definition to include "all some" applied to a class, which means 𝜓 is true whenever 𝜑 is true for 𝑥 in 𝐴, and there is at least one 𝑥 in 𝐴 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
wff ∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓)
 
12-Jul-2026wals 17100 Extend wff definition to include "all some" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true, and there is at least one 𝑥 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
wff ∀∃𝑥(𝜑 → 𝜓)
 
12-Jul-2026vvin 3569 Two classes are both the universal class if and only if their intersection is the universal class. Dual of un00 3567. (Contributed by BJ, 12-Jul-2026.)
((𝐴 = V ∧ 𝐵 = V) ↔ (𝐴 ∩ 𝐵) = V)
 
7-Jul-2026cmnsubm 14095 A submonoid of a commutative monoid is commutative. (Contributed by Jim Kingdon, 7-Jul-2026.)
(𝜑 → 𝑆 ∈ (SubMnd‘𝐺))    &   (𝜑 → 𝐺 ∈ CMnd)    &   𝐻 = (𝐺 ↾s 𝑆)    ⇒   (𝜑 → 𝐻 ∈ CMnd)
 
29-Jun-2026dichmul0or 16743 Real number dichotomy is equivalent to the zero product principle for complex numbers: if a product is zero, one of its factors must be zero. (Contributed by Matthew House, 29-Jun-2026.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ∀𝑧 ∈ ℂ ∀𝑤 ∈ ℂ ((𝑧 · 𝑤) = 0 → (𝑧 = 0 ∨ 𝑤 = 0)))
 
29-Jun-2026dichmul0orlem5 16740 Lemma for dichmul0or 16743. (Contributed by Matthew House, 29-Jun-2026.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → ((abs‘𝐴) + 𝐴) = 0)    ⇒   (𝜑 → 𝐴 ≤ 0)
 
29-Jun-2026dichmul0orlem4 16739 Lemma for dichmul0or 16743. (Contributed by Matthew House, 29-Jun-2026.)
(𝜑 → 𝐴 ∈ ℝ)    ⇒   (𝜑 → (((abs‘𝐴) + 𝐴) · ((abs‘𝐴) − 𝐴)) = 0)
 
29-Jun-2026dichmul0orlem3 16738 Lemma for dichmul0or 16743. (Contributed by Matthew House, 29-Jun-2026.)
(𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))    &   (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 · 𝐵) = 0)    ⇒   (𝜑 → (𝐴 = 0 ∨ 𝐵 = 0))
 
29-Jun-2026dichmul0orlem2 16737 Lemma for dichmul0or 16743. (Contributed by Matthew House, 29-Jun-2026.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 · 𝐵) = 0)    &   (𝜑 → (abs‘𝐴) ≤ (abs‘𝐵))    ⇒   (𝜑 → 𝐴 = 0)
 
29-Jun-2026dichmul0orlem1 16736 Lemma for dichmul0or 16743. (Contributed by Matthew House, 29-Jun-2026.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → (𝐴 · 𝐵) = 0)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 𝐴 ≤ 𝐵)    ⇒   (𝜑 → 𝐴 = 0)
 
29-Jun-2026lealltlt2 16735 Alternative definition for ≤ on real numbers. (Contributed by Matthew House, 29-Jun-2026.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ (𝐵 < 𝑥 → 𝐴 < 𝑥)))
 
29-Jun-2026lealltlt1 16734 Alternative definition for ≤ on real numbers. (Contributed by Matthew House, 29-Jun-2026.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ (𝑥 < 𝐴 → 𝑥 < 𝐵)))
 
28-Jun-2026dichmul0orlem7 16742 Lemma for dichmul0or 16743. (Contributed by Matthew House, 28-Jun-2026.)
(𝜑 → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0)))    &   (𝜑 → 𝐴 ∈ ℝ)    ⇒   (𝜑 → (𝐴 ≤ 0 ∨ 0 ≤ 𝐴))
 
28-Jun-2026dichmul0orlem6 16741 Lemma for dichmul0or 16743. (Contributed by Matthew House, 28-Jun-2026.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → ((abs‘𝐴) − 𝐴) = 0)    ⇒   (𝜑 → 0 ≤ 𝐴)
 
28-Jun-2026msq0 8991 A number is zero iff its square is zero. (Contributed by Matthew House, 28-Jun-2026.)
(𝐴 ∈ ℂ → ((𝐴 · 𝐴) = 0 ↔ 𝐴 = 0))
 
28-Jun-2026msqap0 8990 A number is apart from zero iff its square is apart from zero. (Contributed by Matthew House, 28-Jun-2026.)
(𝐴 ∈ ℂ → ((𝐴 · 𝐴) # 0 ↔ 𝐴 # 0))
 
28-Jun-2026letrid 8436 Tightness of real apartness. (Contributed by Matthew House, 28-Jun-2026.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐴 ≤ 𝐵)    &   (𝜑 → 𝐵 ≤ 𝐴)    ⇒   (𝜑 → 𝐴 = 𝐵)
 
19-Jun-2026ringen1zr0 14605 The only unital ring with one element is the zero ring (at least if its operations are internal binary operations). This holds already for nonunital rings, see rngen1zr0 14244, and semirings, see srgen1zr0 14275. (Contributed by FL, 15-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 19-Jun-2026.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    ∗ = (.r‘𝑅)    &   𝑍 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
19-Jun-2026srg1zr 14274 The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 19-Jun-2026.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    ∗ = (.r‘𝑅)    ⇒   (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
18-Jun-2026rngen1zr0 14244 The only ring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 15-Feb-2010.) (Revised by AV, 18-Jun-2026.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    ∗ = (.r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨ 0 , 0 ⟩, 0 ⟩} ∧ ∗ = {⟨⟨ 0 , 0 ⟩, 0 ⟩})))
 
18-Jun-2026rngen1zr 14243 The only ring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 18-Jun-2026.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    ∗ = (.r‘𝑅)    ⇒   (((𝑅 ∈ Rng ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
18-Jun-2026rng1zr 14242 The only ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 18-Jun-2026.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    ∗ = (.r‘𝑅)    ⇒   (((𝑅 ∈ Rng ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
18-Jun-2026rng1zrlem 14241 Lemma for rng1zr 14242 and srg1zr 14274. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 18-Jun-2026.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    ∗ = (.r‘𝑅)    ⇒   (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
17-Jun-2026ballotfi 13265 Bertrand's ballot problem : the probability that A is ahead throughout the counting. The proof formalized here is a proof "by reflection", as opposed to other known proofs "by induction" or "by permutation". This is Metamath 100 proof #30. (Contributed by Thierry Arnoux, 7-Dec-2016.) (Revised by Jim Kingdon, 17-Jun-2026.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))    &   𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   𝑁 < 𝑀    ⇒   (𝑃‘𝐸) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))
 
17-Jun-2026ballotfilembfi 13222 The set of countings where B got the first vote is finite. (Contributed by Jim Kingdon, 17-Jun-2026.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))    &   𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    ⇒   {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin
 
17-Jun-2026ballotfilemafi 13221 The set of countings where A got the first vote, but does not stay strictly ahead throughout, is finite. (Contributed by Jim Kingdon, 17-Jun-2026.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))    &   𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    ⇒   {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∈ Fin
 
17-Jun-2026ballotfilemefi 13220 𝐸 is finite. (Contributed by Jim Kingdon, 17-Jun-2026.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))    &   𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    ⇒   𝐸 ∈ Fin
 
17-Jun-2026rabxmdc 3554 Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Jim Kingdon, 17-Jun-2026.)
(∀𝑥 ∈ 𝐴 DECID 𝜑 → 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))
 
15-Jun-2026ballotfilemgun 13251 A property of the defined ↑ operator. (Contributed by Thierry Arnoux, 26-Apr-2017.) (Revised by Jim Kingdon, 15-Jun-2026.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))    &   𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))    &    ↑ = (𝑢 ∈ 𝑂, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))    &   (𝜑 → 𝑈 ∈ 𝑂)    &   (𝜑 → 𝐿 ∈ (𝐽...𝐾))    ⇒   (𝜑 → (𝑈 ↑ (𝐽...𝐾)) = ((𝑈 ↑ (𝐽...(𝐿 − 1))) + (𝑈 ↑ (𝐿...𝐾))))
 
15-Jun-2026ballotfilemgval 13250 Expand the value of ↑. (Contributed by Thierry Arnoux, 21-Apr-2017.) (Revised by Jim Kingdon, 15-Jun-2026.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))    &   𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))    &    ↑ = (𝑢 ∈ 𝑂, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))    &   (𝜑 → 𝑈 ∈ 𝑂)    &   (𝜑 → 𝐽 ∈ ℤ)    &   (𝜑 → 𝐾 ∈ ℤ)    &   (𝜑 → 𝑉 = (𝐽...𝐾))    ⇒   (𝜑 → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))
 
15-Jun-2026ballotfilemdifcfz 13210 Lemma for ballotfi . The portion of a counting representing votes for B within a specified integer range is finite. (Contributed by Jim Kingdon, 15-Jun-2026.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}    &   (𝜑 → 𝐶 ∈ 𝑂)    &   (𝜑 → 𝐽 ∈ ℤ)    &   (𝜑 → 𝐾 ∈ ℤ)    ⇒   (𝜑 → ((𝐽...𝐾) ∖ 𝐶) ∈ Fin)
 
15-Jun-2026ballotfilemcinfz 13209 Lemma for ballotfi . The portion of a counting representing votes for A within a specified integer range is finite. (Contributed by Jim Kingdon, 15-Jun-2026.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}    &   (𝜑 → 𝐶 ∈ 𝑂)    &   (𝜑 → 𝐽 ∈ ℤ)    &   (𝜑 → 𝐾 ∈ ℤ)    ⇒   (𝜑 → ((𝐽...𝐾) ∩ 𝐶) ∈ Fin)
 
12-Jun-2026ballotfilemsle 13231 The infimum of the set of zeroes of 𝐹 is a lower bound. (Contributed by Jim Kingdon, 12-Jun-2026.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))    &   𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}    ⇒   ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑋 ∈ 𝑆) → inf(𝑆, ℝ, < ) ≤ 𝑋)
 
12-Jun-2026ballotfilemscl 13230 The set of zeroes of 𝐹 has an infimum. (Contributed by Jim Kingdon, 12-Jun-2026.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))    &   𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}    ⇒   (𝐶 ∈ (𝑂 ∖ 𝐸) → inf(𝑆, ℝ, < ) ∈ 𝑆)
 
12-Jun-2026infssfzledc 10653 The infimum of a decidable inhabited subset of an integer range is a lower bound for that set. (Contributed by Jim Kingdon, 12-Jun-2026.)
𝑆 = {𝑛 ∈ (𝑀...𝑁) ∣ 𝜓}    &   (𝜑 → 𝐴 ∈ 𝑆)    &   ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓)    ⇒   (𝜑 → inf(𝑆, ℝ, < ) ≤ 𝐴)
 
12-Jun-2026infssfzcldc 10652 The infimum of a decidable inhabited subset of an integer range is a member of the set. (Contributed by Jim Kingdon, 12-Jun-2026.)
𝑆 = {𝑛 ∈ (𝑀...𝑁) ∣ 𝜓}    &   (𝜑 → 𝐴 ∈ 𝑆)    &   ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓)    ⇒   (𝜑 → inf(𝑆, ℝ, < ) ∈ 𝑆)
 
8-Jun-2026ballotfilemdifcfi 13208 Lemma for ballotfi . The portion of a counting representing votes for B up to a specified integer is finite. (Contributed by Jim Kingdon, 8-Jun-2026.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}    &   (𝜑 → 𝐶 ∈ 𝑂)    &   (𝜑 → 𝐽 ∈ ℤ)    ⇒   (𝜑 → ((1...𝐽) ∖ 𝐶) ∈ Fin)
 
8-Jun-2026ballotfilemcinfi 13207 Lemma for ballotfi . The portion of a counting representing votes for A up to a specified integer is finite. (Contributed by Jim Kingdon, 8-Jun-2026.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}    &   (𝜑 → 𝐶 ∈ 𝑂)    &   (𝜑 → 𝐽 ∈ ℤ)    ⇒   (𝜑 → ((1...𝐽) ∩ 𝐶) ∈ Fin)
 
8-Jun-2026zfidc 9706 Whether an integer is an element of a finite set of integers is decidable. (Contributed by Jim Kingdon, 8-Jun-2026.)
((𝑆 ⊆ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝑆 ∈ Fin) → DECID 𝐴 ∈ 𝑆)
 
7-Jun-2026ballotfilemcdc 13206 Lemma for ballotfi . It is decidable whether a given integer is an element of a particular element of 𝑂. (Contributed by Jim Kingdon, 7-Jun-2026.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}    &   (𝜑 → 𝐶 ∈ 𝑂)    &   (𝜑 → 𝐾 ∈ ℤ)    ⇒   (𝜑 → DECID 𝐾 ∈ 𝐶)
 
5-Jun-2026hashpwfi 11252 The number of finite subsets of a finite set is two raised to the power of the size of the set. For a similar theorem with set size expressed using equinumerosity, see 2omapfi 7314. For the number of subsets (which need not be finite) of a set, see pw1mapen 17009. (Contributed by Jim Kingdon, 5-Jun-2026.)
(𝐴 ∈ Fin → (♯‘(𝒫 𝐴 ∩ Fin)) = (2↑(♯‘𝐴)))
 
4-Jun-2026ballotfilemonn 13204 The size of the universe is at least one. (Contributed by Jim Kingdon, 4-Jun-2026.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}    ⇒   (♯‘𝑂) ∈ ℕ
 
3-Jun-2026papeq2 7604 Equality theorem for apartness predicate. (Contributed by Jim Kingdon, 3-Jun-2026.)
(𝐴 = 𝐵 → (𝑅 Ap 𝐴 ↔ 𝑅 Ap 𝐵))
 
3-Jun-2026papeq1 7603 Equality theorem for apartness predicate. (Contributed by Jim Kingdon, 3-Jun-2026.)
(𝑅 = 𝑆 → (𝑅 Ap 𝐴 ↔ 𝑆 Ap 𝐴))
 
2-Jun-2026resq01 11078 If a real number equals its square, it must be 0 or 1. (Contributed by Jim Kingdon, 2-Jun-2026.)
(𝐴 ∈ ℝ → ((𝐴↑2) = 𝐴 ↔ (𝐴 = 0 ∨ 𝐴 = 1)))
 
31-May-2026aprprop 14584 If two structures have the same ring components (properties), df-apr 14573 generates the same relation for both of them. (Contributed by Jim Kingdon, 31-May-2026.)
(Base‘𝐾) = (Base‘𝐿)    &   (+g‘𝐾) = (+g‘𝐿)    &   (.r‘𝐾) = (.r‘𝐿)    ⇒   (𝐾 ∈ Ring → (#r‘𝐾) = (#r‘𝐿))
 
31-May-2026ringunitsap0 14577 The set of units of a ring. If 𝑅 is a local ring, # is an apartness and this theorem states that the units of a ring are those elements apart from zero (see aprlring 14583). Given the definition of #r this theorem holds even if # is not an apartness, however. (Contributed by Jim Kingdon, 31-May-2026.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    # = (#r‘𝑅)    ⇒   (𝑅 ∈ Ring → {𝑥 ∈ 𝐵 ∣ 𝑥 # 0 } = (Unit‘𝑅))
 
30-May-2026ringunitap 14576 Elementhood in the set of units. (Contributed by Jim Kingdon, 30-May-2026.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g‘𝑅)    &    # = (#r‘𝑅)    ⇒   (𝑅 ∈ Ring → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 # 0 )))
 
29-May-2026drnglring 14590 A division ring is a local ring. (Contributed by Jim Kingdon, 29-May-2026.)
(𝑅 ∈ DivRing → 𝑅 ∈ LRing)
 
29-May-2026isdrngtap 14589 The predicate "is a division ring". (Contributed by Jim Kingdon, 29-May-2026.)
𝐵 = (Base‘𝑅)    &    # = (#r‘𝑅)    ⇒   (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ # TAp 𝐵))
 
29-May-2026df-drngap 14587 Define class of all division rings. A division ring is a ring in which the relation given by df-apr 14573 is a tight apartness. (Contributed by Jim Kingdon, 29-May-2026.)
DivRing = {𝑟 ∈ Ring ∣ (#r‘𝑟) TAp (Base‘𝑟)}
 
29-May-2026aprunit 14575 The df-apr 14573 relation with zero expresses whether a ring element is a unit. That is, the difference of an element of a ring and zero is invertible iff the element is a unit. (Contributed by Jim Kingdon, 29-May-2026.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    # = (#r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 # 0 ↔ 𝑋 ∈ 𝑈))
 
29-May-2026tapap 7610 A tight apartness is an apartness. (Contributed by Jim Kingdon, 29-May-2026.)
(𝑅 TAp 𝐴 → 𝑅 Ap 𝐴)
 
28-May-2026aprlring 14583 A ring is a local ring if and only if the relation given by df-apr 14573 is an apartness relation. (Contributed by Jim Kingdon, 28-May-2026.)
(𝑅 ∈ Ring → (𝑅 ∈ LRing ↔ (#r‘𝑅) Ap (Base‘𝑅)))
 
28-May-2026papcotr 7607 An apartness is cotransitive. (Contributed by Jim Kingdon, 28-May-2026.)
(𝜑 → 𝑅 Ap 𝐴)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐴)    &   (𝜑 → 𝑋𝑅𝑌)    &   (𝜑 → 𝑍 ∈ 𝐴)    ⇒   (𝜑 → (𝑋𝑅𝑍 ∨ 𝑌𝑅𝑍))
 
27-May-2026aprnzr 14582 If the relation given by df-apr 14573 on a ring is an apartness relation, then the ring is a nonzero ring. (Contributed by Jim Kingdon, 27-May-2026.)
((𝑅 ∈ Ring ∧ (#r‘𝑅) Ap (Base‘𝑅)) → 𝑅 ∈ NzRing)
 
27-May-2026papsym 7606 An apartness is symmetric. (Contributed by Jim Kingdon, 27-May-2026.)
(𝜑 → 𝑅 Ap 𝐴)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐴)    &   (𝜑 → 𝑋𝑅𝑌)    ⇒   (𝜑 → 𝑌𝑅𝑋)
 
27-May-2026papirr 7605 An apartness is irreflexive. (Contributed by Jim Kingdon, 27-May-2026.)
((𝑅 Ap 𝐴 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋𝑅𝑋)
 
24-May-2026gsumzfi 14141 Value of a finite group sum over the zero element. (Contributed by Jim Kingdon, 24-May-2026.)
0 = (0g‘𝐺)    ⇒   ((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
 
22-May-2026sshashneg 11264 Subsets of a class of a negative size (a degenerate case). Together with ssenneg 11263 this shows that sseqn 11262 could not be extended beyond 𝑁 ∈ ℕ0. (Contributed by Jim Kingdon, 22-May-2026.)
((𝑁 ∈ ℤ ∧ 𝑁 < 0) → {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑁} = ∅)
 
22-May-2026ssenneg 11263 Subsets of a class of a negative size (a degenerate case). Together with sshashneg 11264 this shows that sseqn 11262 could not be extended beyond 𝑁 ∈ ℕ0. (Contributed by Jim Kingdon, 22-May-2026.)
((𝑁 ∈ ℤ ∧ 𝑁 < 0) → {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≈ (1...𝑁)} = {∅})
 
22-May-2026sseqn 11262 Two ways to express the subsets of a class of a given size. It might seem that {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁} would suffice, but that would require the converse of hashcl 11203 or something similar. Although each side of the equality would be well defined if we changed 𝑁 ∈ ℕ0 to 𝑁 ∈ ℤ, they would give different results for the (degenerate) case of a negative size, as shown at ssenneg 11263 and sshashneg 11264. (Contributed by Jim Kingdon, 22-May-2026.)
(𝑁 ∈ ℕ0 → {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≈ (1...𝑁)} = {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑁})
 
22-May-2026bilanri 389 Inference adding a conjunct to the right-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
(𝜑 ↔ 𝜓)    ⇒   ((𝜒 ∧ 𝜓) → 𝜑)
 
22-May-2026biranri 388 Inference adding a conjunct to the right-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
(𝜑 ↔ 𝜓)    ⇒   ((𝜓 ∧ 𝜒) → 𝜑)
 
22-May-2026bilani 387 Inference adding a conjunct to the left-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
(𝜑 ↔ 𝜓)    ⇒   ((𝜒 ∧ 𝜑) → 𝜓)
 
22-May-2026birani 386 Inference adding a conjunct to the left-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
(𝜑 ↔ 𝜓)    ⇒   ((𝜑 ∧ 𝜒) → 𝜓)
 
20-May-2026ballotfilemofi 13202 𝑂 is finite. (Contributed by Jim Kingdon, 20-May-2026.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}    ⇒   𝑂 ∈ Fin
 
19-May-2026fipwfi 7315 The set of finite subsets of a finite set is finite. (Contributed by Jim Kingdon, 19-May-2026.)
(𝐴 ∈ Fin → (𝒫 𝐴 ∩ Fin) ∈ Fin)
 
18-May-20262omapfi 7314 The number of finite subsets of a finite set. For a similar theorem with set size expressed using ♯ (df-ihash 11198), see hashpwfi 11252. (Contributed by Jim Kingdon, 18-May-2026.)
(𝐴 ∈ Fin → (2o ↑𝑚 𝐴) ≈ (𝒫 𝐴 ∩ Fin))
 
18-May-2026fissfi 7257 A finite subset of a finite set is a decidable subset. (Contributed by Jim Kingdon, 18-May-2026.)
((𝑆 ⊆ 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝑆 ∈ Fin) → ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝑆)
 
18-May-2026fresaunres1disj 5569 From the union of two functions with disjoint domains, either component can be recovered by restriction. (Contributed by Mario Carneiro, 16-Feb-2015.) (Revised by Jim Kingdon, 18-May-2026.)
((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹)
 
18-May-2026fresaunres2disj 5568 From the union of two functions with disjoint domains, either component can be recovered by restriction. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Jim Kingdon, 18-May-2026.)
((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)
 
15-May-2026fsuppcorn 7295 The composition of a 1-1 function with a finitely supported function is finitely supported. The purpose of the (𝐹 supp 𝑍) ⊆ ran 𝐺 condition is to ensure we don't subset the support of the function in such a way as to fun afoul of exmidssfi 7240. (Other alternative conditions might also be sufficient). (Contributed by AV, 28-May-2019.) (Revised by Jim Kingdon, 15-May-2026.)
(𝜑 → 𝐹 finSupp 𝑍)    &   (𝜑 → 𝐺:𝑋–1-1→𝑌)    &   (𝜑 → 𝑍 ∈ 𝑊)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝐺 ∈ 𝑈)    &   (𝜑 → (𝐹 supp 𝑍) ⊆ ran 𝐺)    ⇒   (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍)
 
13-May-2026lincmble 10389 A linear combination of two reals which lies in the interval between them. Like lincmb01cmp 10388 but generalized to require merely 𝐴 ≤ 𝐵 not 𝐴 < 𝐵. (Contributed by Jim Kingdon, 13-May-2026.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) ∧ 𝑇 ∈ (0[,]1)) → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐵)) ∈ (𝐴[,]𝐵))
 
5-May-2026fmelpw1o 7600 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 863, which translate to 1o and ∅ respectively by iftrue 3645 and iffalse 3648, giving pwtrufal 17010).

As proved in if0ab 3641, 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 4293 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 4625. (Contributed by BJ, 5-May-2026.)
(𝐴 ∈ 𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴)
 
5-May-2026if0ss 3642 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 17051 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 17049 and repiecege0 17050. 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. 17050 (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 17050 Piecewise definition on the reals agrees with the nonnegative part of the definition. See repiecef 17051 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 17049 Piecewise definition on the reals agrees with the nonpositive part of the definition. See repiecef 17051 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 17048 Lemma for repiecele0 17049, repiecege0 17050, and repiecef 17051. 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 17072 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 17071 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 17020 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 1499. (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 17019 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 1498. (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 17018 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 1497. (Contributed by Jim Kingdon, 22-Apr-2026.)
(EXMID ↔ ∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
 
18-Apr-2026hashtpglem 11281 Lemma for hashtpg 11282. 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 11280 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 16733 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 16732 Lemma for depind 16733. (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 16731 Lemma for depind 16733. (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 16730 Lemma for depind 16733. (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-2026gsumclfi 14142 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 Σg 𝐹) ∈ 𝐵)
 
4-Apr-2026gzsumsplit0 14131 Splitting off the rightmost summand of a group sum (even if it is the only summand). Similar to gzsumsplit1r 13698 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 Σgz 𝐹) = ((𝐺 Σgz (𝐹 ↾ (𝑀...𝑁))) + (𝐹‘(𝑁 + 1))))
 
4-Apr-2026fzf1o 12125 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-2026gsump1 14140 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 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝑌)) + (𝐹‘𝑍)))
 
2-Apr-2026gsumsncmn 14139 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 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)
 
31-Mar-2026sspw1or2 7538 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 7234 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-2026gsum0cmn 14137 An empty finite group sum is the identity. (Contributed by Jim Kingdon, 26-Mar-2026.)
(𝐺 ∈ CMnd → (𝐺 Σg ∅) = (0g‘𝐺))
 
26-Mar-2026gzsumshift 14132 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 Σgz 𝐹) = (𝐺 Σgz (𝐹 ∘ 𝑆)))
 
25-Mar-2026gzsumgsum 14138 On an integer range, Σgz and Σg 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 Σgz 𝐹) = (𝐺 Σg 𝐹))
 
25-Mar-2026gzsumgsum1 14136 On an integer range starting at one, Σgz and Σg agree. (Contributed by Jim Kingdon, 25-Mar-2026.)
𝐵 = (Base‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐹:(1...𝑁)⟶𝐵)    ⇒   (𝜑 → (𝐺 Σgz 𝐹) = (𝐺 Σg 𝐹))
 
24-Mar-2026gsumvalfi 14135 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 Σg 𝐹) = (𝑊 Σgz (𝐹 ∘ 𝐺)))
 
23-Mar-2026df-gsumfi 14134 Define the finite group sum (iterated sum) over an unordered finite set.

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. For this notation, 𝐴 is a finite set and 𝐺 is a commutative monoid, and the sum adds up these elements in some order (the sum does not depend on the order).

For a sum indexed by consecutive integers (and thus defining an order for the sum), see df-gzsum 13596. (Contributed by Jim Kingdon, 23-Mar-2026.)

Σg = (𝑤 ∈ CMnd, 𝑓 ∈ V ↦ (℩𝑥(dom 𝑓 ∈ Fin ∧ ∃𝑔(𝑔:(1...(♯‘dom 𝑓))–1-1-onto→dom 𝑓 ∧ 𝑥 = (𝑤 Σgz (𝑓 ∘ 𝑔))))))
 
20-Mar-2026exmidssfi 7240 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 16349 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 16348 A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 16345 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 16611 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 16670 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 16531 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 11222 A set equinumerous to the ordinal one has size 1 . (Contributed by Jim Kingdon, 11-Mar-2026.)
(𝐴 ≈ 1o → (♯‘𝐴) = 1)
 
4-Mar-2026elmpom 6468 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 16631 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 16622 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 16517 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 16514 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 16513 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 7204 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 17004 The powerset of 1o is not infinite. Since we cannot prove it is finite (see pw1fin 7211), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7207. (Contributed by Jim Kingdon, 14-Feb-2026.)
¬ ω ≼ 𝒫 1o
 
14-Feb-2026pw1ndom3 17003 The powerset of 1o does not dominate 3o. This is another way of saying that 𝒫 1o does not have three elements (like pwntru 4334). (Contributed by Steven Nguyen and Jim Kingdon, 14-Feb-2026.)
¬ 3o ≼ 𝒫 1o
 
14-Feb-2026pw1ndom3lem 17002 Lemma for pw1ndom3 17003. (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 17017 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 17001 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 7203 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 16515 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 7197 Equinumerosity of finite sets is decidable. (Contributed by Jim Kingdon, 10-Feb-2026.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → DECID 𝐴 ≈ 𝐵)
 
8-Feb-2026wlkvtxm 16564 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 16608 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 16549 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 7111 A set with an element dominates one. (Contributed by Jim Kingdon, 3-Feb-2026.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → 1o ≼ 𝐴)
 
2-Feb-2026edginwlkd 16579 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 16573 A walk is an ordered pair. (Contributed by Jim Kingdon, 2-Feb-2026.)
(𝑊 ∈ (Walks‘𝐺) → 𝑊 ∈ (V × V))
 
1-Feb-2026wlkcprim 16574 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 16543 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 5728 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 5727 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 4998 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 993 Forward direction of dfifp2dc 994. This direction does not require decidability. (Contributed by Jim Kingdon, 25-Jan-2026.)
(if-(𝜑, 𝜓, 𝜒) → ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
 
25-Jan-2026ifpdc 992 The conditional operator for propositions implies decidability. (Contributed by Jim Kingdon, 25-Jan-2026.)
(if-(𝜑, 𝜓, 𝜒) → DECID 𝜑)
 
24-Jan-2026eqabcbw 2376 Version of eqabcb 2377 using implicit substitution, which requires fewer axioms. (Contributed by TM, 24-Jan-2026.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑦(𝜓 ↔ 𝑦 ∈ 𝐴))
 
20-Jan-2026cats1fvd 11521 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 11520 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 11524 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 11523 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 11522 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 2656 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 16379 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 7533 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 17009 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 7578 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 7577 A truth value which is inhabited is equal to true. This is a variation of pwntru 4334 and pwtrufal 17010. (Contributed by Jim Kingdon, 10-Jan-2026.)
((𝐴 ∈ 𝒫 1o ∧ ∃𝑥 𝑥 ∈ 𝐴) → 𝐴 = 1o)
 
10-Jan-20261ndom2 7160 Two is not dominated by one. (Contributed by Jim Kingdon, 10-Jan-2026.)
¬ 2o ≼ 1o
 
9-Jan-2026pw1map 17008 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 7576 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 11434 Closure of the prefix extractor. This extends pfxclg 11433 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 11432 The domain of the prefix extractor. (Contributed by Jim Kingdon, 8-Jan-2026.)
prefix Fn (V × ℕ0)
 
7-Jan-2026pr1or2 7534 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 16344 Lemma for upgr1edc 16345. (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 16318 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 16317 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 16318). (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 7110 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 7107 A set with two elements is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
(𝐴 ≈ 2o → ∃𝑥 𝑥 ∈ 𝐴)
 
3-Jan-2026en1m 7086 A set with one element is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
(𝐴 ≈ 1o → ∃𝑥 𝑥 ∈ 𝐴)
 
31-Dec-2025pw0ss 16307 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 16294 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 16293 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 16243 Applying the indexed edge function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
(𝐺 ∈ 𝑉 → (iEdg‘𝐺) ∈ V)
 
29-Dec-2025vtxex 16242 Applying the vertex function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
(𝐺 ∈ 𝑉 → (Vtx‘𝐺) ∈ V)
 
29-Dec-2025snmb 3832 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 11339 Existence of the last symbol. The last symbol of a word is a set. See lsw0g 11336 or lswcl 11338 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 11402 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 11345 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 11334 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 13455 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 16256 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 16255 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 16254 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 16253 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 11278 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 17040 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 9950 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 15050 Simpler form of psrelbas 15049 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 15074 Lemma for mplsubgfi 15075. 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 15073 Lemma for mplsubgfi 15075. 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 7514 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 7467). (Contributed by Jim Kingdon, 25-Nov-2025.)
(∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ ω ∈ WOmni)
 
23-Nov-2025psrbagfi 15042 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 7519 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 15072 Lemma for mplsubgfi 15075. 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 9948 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 7313 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 7312 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 17014 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 14157 Lemma for prdsbas 14159 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 15067 mPoly has universal domain. (Contributed by Jim Kingdon, 5-Nov-2025.)
mPoly Fn (V × V)
 
4-Nov-2025mplelbascoe 15066 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 15065 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 15064 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 7528 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 12697 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 11974 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 10664 Rational ≤ is decidable. (Contributed by Jim Kingdon, 28-Oct-2025.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 ≤ 𝐵)
 
17-Oct-2025plycoeid3 15841 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 7231 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 7229 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 3637 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 13000 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 15031 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 15782 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 1499 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 926), 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 1498 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 865), 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 1497 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 855), 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 15781 Real derivative of the identity function. (Contributed by Jim Kingdon, 3-Oct-2025.)
(ℝ D ( I ↾ ℝ)) = (ℝ × {1})
 
3-Oct-2025dvconstre 15780 Real derivative of a constant function. (Contributed by Jim Kingdon, 3-Oct-2025.)
(𝐴 ∈ ℂ → (ℝ D (ℝ × {𝐴})) = (ℝ × {0}))
 
3-Oct-2025dvidsslem 15777 Lemma for dvconstss 15782. Analogue of dvidlemap 15775 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 15776 Lemma for dvidre 15781 and dvconstre 15780. Analogue of dvidlemap 15775 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 14875 Applying metUnif yields a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
(𝐴 ∈ 𝑉 → (metUnif‘𝐴) ∈ V)
 
28-Sep-2025cndsex 14873 The standard distance function on the complex numbers is a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
(abs ∘ − ) ∈ V
 
25-Sep-2025cntopex 14874 The standard topology on the complex numbers is a set. (Contributed by Jim Kingdon, 25-Sep-2025.)
(MetOpen‘(abs ∘ − )) ∈ V
 
24-Sep-2025mopnset 14872 Getting a set by applying MetOpen. (Contributed by Jim Kingdon, 24-Sep-2025.)
(𝐷 ∈ 𝑉 → (MetOpen‘𝐷) ∈ V)
 
24-Sep-2025blfn 14871 The ball function has universal domain. (Contributed by Jim Kingdon, 24-Sep-2025.)
ball Fn V
 
23-Sep-2025elfzoext 10593 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 15844 Lemma for plycj 15845. (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))))
 
22-Sep-2025asclelbas 15009 Lifted scalars are in the base set of the algebra. (Contributed by Zhi Wang, 11-Sep-2025.) (Proof shortened by Thierry Arnoux, 22-Sep-2025.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &   (𝜑 → 𝑊 ∈ AssAlg)    &   (𝜑 → 𝐶 ∈ 𝐵)    ⇒   (𝜑 → (𝐴‘𝐶) ∈ (Base‘𝑊))
 
20-Sep-2025plycolemc 15842 Lemma for plyco 15843. 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 10592 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 16178 Lemma for lgsquad 16182. 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 16177 Lemma for lgsquad 16182. 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 7241 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 6365 Principle of unique choice. This is also called non-choice. The name choice results in its similarity to something like acfun 7557 (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-2025assa2ass2 14993 Left- and right-associative property of an associative algebra. Notice that the scalars are not commuted! (Contributed by Zhi Wang, 11-Sep-2025.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    ∗ = (.r‘𝐹)    &    · = ( ·𝑠 ‘𝑊)    &    × = (.r‘𝑊)    ⇒   ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴 · 𝑋) × (𝐶 · 𝑌)) = ((𝐴 ∗ 𝐶) · (𝑋 × 𝑌)))
 
11-Sep-2025assa2ass 14992 Left- and right-associative property of an associative algebra. Notice that the scalars are commuted! (Contributed by AV, 14-Aug-2019.) (Proof shortened by Zhi Wang, 11-Sep-2025.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    ∗ = (.r‘𝐹)    &    · = ( ·𝑠 ‘𝑊)    &    × = (.r‘𝑊)    ⇒   ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴 · 𝑋) × (𝐶 · 𝑌)) = ((𝐶 ∗ 𝐴) · (𝑋 × 𝑌)))
 
11-Sep-2025expghmap 14925 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 14907 The invertible complex numbers are exactly those apart from zero. This is recapb 8995 but expressed in terms of ℂfld. (Contributed by Jim Kingdon, 11-Sep-2025.)
{𝑧 ∈ ℂ ∣ 𝑧 # 0} = (Unit‘ℂfld)
 
9-Sep-2025gzsummhm2 14129 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 Σgz (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) → 𝐶 = 𝐸)    ⇒   (𝜑 → (𝐻 Σgz (𝑘 ∈ (𝑀...𝑁) ↦ 𝐷)) = 𝐸)
 
8-Sep-2025gzsummhm 14128 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 Σgz (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σgz 𝐹)))
 
8-Sep-20255ndvds6 12685 5 does not divide 6. (Contributed by AV, 8-Sep-2025.)
¬ 5 ∥ 6
 
8-Sep-20255ndvds3 12684 5 does not divide 3. (Contributed by AV, 8-Sep-2025.)
¬ 5 ∥ 3
 
7-Sep-20255eluz3 9944 5 is an integer greater than or equal to 3. (Contributed by AV, 7-Sep-2025.)
5 ∈ (ℤ≥‘3)
 
6-Sep-2025gzsumconst 14126 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 Σgz (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) = (((𝑁 − 𝑀) + 1) · 𝑋))
 
5-Sep-2025uzuzle34 9947 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))
 
30-Aug-2025gzsumsubmcl 14125 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 Σgz 𝐹) ∈ 𝑆)
 
30-Aug-2025seqm1g 10894 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 10941 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 10031 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 10030. (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 10886 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 10883 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 11294 A zero-based sequence is a word. In iswrdinn0 11292 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-2025gzsumcl 13787 Closure of an ordered 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 Σgz 𝐹) ∈ 𝐵)
 
16-Aug-2025iswrdinn0 11292 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)
 
14-Aug-2025gzsumfzval 13694 An expression for Σgz 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 Σgz 𝐹) = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)))
 
13-Aug-2025znidom 14975 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 14552 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 16161 Lemma for gausslemma2dlem1 16163. 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 16162 Lemma for gausslemma2dlem1 16163. (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 10663 Rational < is decidable. (Contributed by Jim Kingdon, 7-Aug-2025.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 < 𝐵)
 
22-Jul-2025ivthdich 15737 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 15727 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 15730, hovera 15731, and hoverb 15732, 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 8387, 𝑐 < 1 or 0 < 𝑐, and that leads to 𝑧 ≤ 0 by hoverlt1 15733 or 0 ≤ 𝑧 by hovergt0 15734. (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 15736 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 15735 Lemma for ivthdich 15737. 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 15734 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 15733 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 15732 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 15731 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 2760 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 2759 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 2758 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 2757 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 15730 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 15700 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 15699 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 10857 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 7457 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 12800 Lemma for nninfct 12801. (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 17038 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 12801 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 10863 ℕ∞ is infinte. (Contributed by Jim Kingdon, 8-Jul-2025.)
ω ≼ ℕ∞
 
7-Jul-2025ivthreinc 15729 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 15727). 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-2025fngzsum 13691 Iterated sum has a universal domain. (Contributed by Jim Kingdon, 28-Jun-2025.)
Σgz Fn (V × V)
 
28-Jun-2025iotaexel 6037 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-gzsum 13596 Define a finite group sum (also called "iterated sum") of a structure.

Given 𝐺 Σgz 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. But see df-gsumfi 14134 for the case where 𝐴 is a finite set (which need not specify an order) and 𝐺 is a commutative monoid.

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

Σgz = (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 14475 The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr 14476. (Contributed by SN, 20-Jun-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing))
 
16-Jun-2025fnpsr 15034 The multivariate power series constructor has a universal domain. (Contributed by Jim Kingdon, 16-Jun-2025.)
mPwSer Fn (V × V)
 
14-Jun-2025basm 13397 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 5726 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 13074 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 10685 "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 2862 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 2895 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 13161 Exercise which may help in understanding the proof of 4sqlemsdc 13162. (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 5354 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 13162 Lemma for 4sq 13172. 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 13160 and 4sqexercise2 13161 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 13160 Exercise which may help in understanding the proof of 4sqlemsdc 13162. (Contributed by Jim Kingdon, 25-May-2025.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ 𝑛 = (𝑥↑2)}    ⇒   (𝐴 ∈ ℕ0 → DECID 𝐴 ∈ 𝑆)
 
24-May-20254sqleminfi 13159 Lemma for 4sq 13172. 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 13158 Lemma for 4sq 13172. 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 13157 Lemma for 4sq 13172. 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 7242 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 14939 Set existence for ℤRHom. (Contributed by Jim Kingdon, 19-May-2025.)
𝐿 = (ℤRHom‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → 𝐿 ∈ V)
 
16-May-2025rhmex 14447 Set existence for ring homomorphism. (Contributed by Jim Kingdon, 16-May-2025.)
((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 RingHom 𝑆) ∈ V)
 
15-May-2025ghmex 14041 The set of group homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)
((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) ∈ V)
 
15-May-2025mhmex 13752 The set of monoid homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)
((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 MndHom 𝑇) ∈ V)
 
14-May-2025idomcringd 14570 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 14560 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 14236 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 13408. (Contributed by Jim Kingdon, 5-May-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Rng → (𝐺 ↾s 𝐵) ∈ Rng)
 
5-May-2025ablressid 14122 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 13408. (Contributed by Jim Kingdon, 5-May-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Abel → (𝐺 ↾s 𝐵) ∈ Abel)
 
30-Apr-2025dvply2g 15850 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 14780 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 13407 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 14675 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 14884 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 14882 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 14793 Existence of the set of left ideals. (Contributed by Jim Kingdon, 27-Apr-2025.)
(𝑊 ∈ 𝑉 → (LIdeal‘𝑊) ∈ V)
 
27-Apr-2025lssex 14674 Existence of a linear subspace. (Contributed by Jim Kingdon, 27-Apr-2025.)
(𝑊 ∈ 𝑉 → (LSubSp‘𝑊) ∈ V)
 
25-Apr-2025rspex 14794 Existence of the ring span. (Contributed by Jim Kingdon, 25-Apr-2025.)
(𝑊 ∈ 𝑉 → (RSpan‘𝑊) ∈ V)
 
25-Apr-2025lspex 14715 Existence of the span of a set of vectors. (Contributed by Jim Kingdon, 25-Apr-2025.)
(𝑊 ∈ 𝑋 → (LSpan‘𝑊) ∈ V)
 
25-Apr-2025eqgex 14007 The left coset equivalence relation exists. (Contributed by Jim Kingdon, 25-Apr-2025.)
((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐺 ~QG 𝑆) ∈ V)
 
25-Apr-2025qusex 13629 Existence of a quotient structure. (Contributed by Jim Kingdon, 25-Apr-2025.)
((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) ∈ V)
 
23-Apr-20251dom1el 7101 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 13909 Existence of the group multiple operation. (Contributed by Jim Kingdon, 22-Apr-2025.)
(𝐺 ∈ 𝑉 → (.g‘𝐺) ∈ V)
 
21-Apr-2025uspgruhgr 16411 An undirected simple pseudograph is an undirected hypergraph. (Contributed by AV, 21-Apr-2025.)
(𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
 
20-Apr-2025uspgriedgedg 16403 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 16402 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 6281 Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 6282 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 5743 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 16088 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 8296. (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 16087 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 14829 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 14821 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 14808 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 14795 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 14749 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 14766 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 13814 A group is a magma, deduction form. (Contributed by SN, 14-Apr-2025.)
(𝜑 → 𝐺 ∈ Grp)    ⇒   (𝜑 → 𝐺 ∈ Mgm)
 
12-Apr-2025psraddcl 15054 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 17083 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 14059 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 14853 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 14888 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 14877. (Revised by GG, 31-Mar-2025.)
(abs ∘ − ) = (dist‘ℂfld)
 
31-Mar-2025cnfldle 14887 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 14877. (Revised by GG, 31-Mar-2025.)
≤ = (le‘ℂfld)
 
31-Mar-2025cnfldtset 14886 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 14883 The multiplication operation of the field of complex numbers. Version of cnfldmul 14884 using maps-to notation, which does not require ax-mulf 8296. (Contributed by GG, 31-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (.r‘ℂfld)
 
31-Mar-2025mpocnfldadd 14881 The addition operation of the field of complex numbers. Version of cnfldadd 14882 using maps-to notation, which does not require ax-addf 8295. (Contributed by GG, 31-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (+g‘ℂfld)
 
31-Mar-2025df-cnfld 14877 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 14879, cnfldadd 14882, cnfldmul 14884, cnfldcj 14885, cnfldtset 14886, cnfldle 14887, cnfldds 14888, and cnfldbas 14880. 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 14844 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 14571 An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
(𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → 𝑅 ∈ Ring)
 
22-Mar-2025idomdomd 14569 An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
(𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → 𝑅 ∈ Domn)
 
21-Mar-2025df2idl2rng 14828 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 14802 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 14801 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 13575 Slot property of comp. (Contributed by Jim Kingdon, 20-Mar-2025.)
(comp = Slot (comp‘ndx) ∧ (comp‘ndx) ∈ ℕ)
 
20-Mar-2025homslid 13572 Slot property of Hom. (Contributed by Jim Kingdon, 20-Mar-2025.)
(Hom = Slot (Hom ‘ndx) ∧ (Hom ‘ndx) ∈ ℕ)
 
19-Mar-2025ptex 13601 Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.)
(𝐹 ∈ 𝑉 → (∏t‘𝐹) ∈ V)
 
18-Mar-2025prdsex 14155 Existence of the structure product. (Contributed by Jim Kingdon, 18-Mar-2025.)
((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑆Xs𝑅) ∈ V)
 
16-Mar-2025plycn 15846 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8296. (Revised by GG, 16-Mar-2025.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))
 
16-Mar-2025expcn 15653 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 8296. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpen‘ℂfld)    ⇒   (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽))
 
16-Mar-2025mpomulcn 15650 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 8310 Multiplication is an operation on complex numbers. Version of ax-mulf 8296 using maps-to notation, proved from the axioms of set theory and ax-mulcl 8271. (Contributed by GG, 16-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
 
15-Mar-2025lmodvscld 14625 Closure of scalar product for a left module. (Contributed by SN, 15-Mar-2025.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠 ‘𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑 → 𝑊 ∈ LMod)    &   (𝜑 → 𝑅 ∈ 𝐾)    &   (𝜑 → 𝑋 ∈ 𝑉)    ⇒   (𝜑 → (𝑅 · 𝑋) ∈ 𝑉)
 
13-Mar-20252idlss 14834 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 13609 Existence of the image structure. (Contributed by Jim Kingdon, 13-Mar-2025.)
((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐹 “s 𝑅) ∈ V)
 
11-Mar-2025rng2idlsubgsubrng 14840 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 14837 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 14818 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 14817 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 14816 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 13610 Value of an image structure. The is a lemma for the theorems imasbas 13611, imasplusg 13612, and imasmulr 13613 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 14827 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 14826 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 14019 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 6188 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 13962 The class of subgroups of a group is a set. (Contributed by Jim Kingdon, 8-Mar-2025.)
(𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ V)
 
8-Mar-2025fsuppfund 7288 A finitely supported function is a function. (Contributed by SN, 8-Mar-2025.)
(𝜑 → 𝐹 finSupp 𝑍)    ⇒   (𝜑 → Fun 𝐹)
 
7-Mar-2025ringrzd 14334 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 14333 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 14118 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))
 
2-Mar-2025isassad 14994 Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by SN, 2-Mar-2025.)
(𝜑 → 𝑉 = (Base‘𝑊))    &   (𝜑 → 𝐹 = (Scalar‘𝑊))    &   (𝜑 → 𝐵 = (Base‘𝐹))    &   (𝜑 → · = ( ·𝑠 ‘𝑊))    &   (𝜑 → × = (.r‘𝑊))    &   (𝜑 → 𝑊 ∈ LMod)    &   (𝜑 → 𝑊 ∈ Ring)    &   ((𝜑 ∧ (𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)))    &   ((𝜑 ∧ (𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))    ⇒   (𝜑 → 𝑊 ∈ AssAlg)
 
2-Mar-2025assasca 14991 The scalars of an associative algebra form a ring. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by SN, 2-Mar-2025.)
𝐹 = (Scalar‘𝑊)    ⇒   (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)
 
2-Mar-2025isassa 14985 The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by SN, 2-Mar-2025.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    · = ( ·𝑠 ‘𝑊)    &    × = (.r‘𝑊)    ⇒   (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
 
2-Mar-2025df-assa 14982 Definition of an associative algebra. An associative algebra is a set equipped with a left-module structure on a ring, coupled with a multiplicative internal operation on the vectors of the module that is associative and distributive for the additive structure of the left-module (so giving the vectors a ring structure) and that is also bilinear under the scalar product. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by SN, 2-Mar-2025.)
AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣ [(Scalar‘𝑤) / 𝑓]∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))}
 
1-Mar-2025quselbasg 14016 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 14852 Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 14853. Similar to qusmul2 14849. (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 14351 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 13408. (Contributed by Jim Kingdon, 28-Feb-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Ring)
 
28-Feb-2025grpressid 13849 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 13408. (Contributed by Jim Kingdon, 28-Feb-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp)
 
27-Feb-2025imasringf1 14353 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 13442 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 14496 A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.)
(𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅))
 
25-Feb-2025rngansg 14232 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 14024 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 14025 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 14017 Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) Generalization of quseccl 14019 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 17094 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 17092). (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 17093 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 14845 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 14847 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 14843 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 14486 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 14485 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 14484 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 14483 A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
(𝑅 ∈ LRing → 𝑅 ∈ NzRing)
 
23-Feb-2025islring 14482 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 14481 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 14479 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 14473 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 14240 The quotient structure of a non-unital ring is a non-unital ring (qusring2 14354 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 14234 Ring multiplication distributes over subtraction. (subdir 8707 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 14345. (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 14233 Ring multiplication distributes over subtraction. (subdi 8706 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdi 14344. (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)))
 
23-Feb-2025abbib 2356 Equal class abstractions require equivalent formulas, and conversely. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-8 1557 and df-clel 2234 (by avoiding use of cleqh 2338). (Revised by BJ, 23-Jun-2019.) Definitial form. (Revised by Wolf Lammen, 23-Feb-2025.)
({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 ↔ 𝜓))
 
22-Feb-2025imasrngf1 14239 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 14238 The image structure of a non-unital ring is a non-unital ring (imasring 14352 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 14219 Restricted functionality of the multiplicative group on non-unital rings (mgpf 14298 analog). (Contributed by AV, 22-Feb-2025.)
(mulGrp ↾ Rng):Rng⟶Smgrp
 
22-Feb-2025imasabl 14123 The image structure of an abelian group is an abelian group (imasgrp 13897 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 14174 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 14173 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 7611 Tight apartness with the apartness properties from df-pap 7602 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 14842 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 14841 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 14839 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 14838 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 14836 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 14835 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 14825 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 14581 The relation given by df-apr 14573 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 13376 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 3686 The converse of ifbi 3661 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 3685 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 3684 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 14800 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 14580 The apartness relation given by df-apr 14573 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 14579 The apartness relation given by df-apr 14573 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 14574 Expand Definition df-apr 14573. (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 14507 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 14231 Double negation of a product in a non-unital ring (mul2neg 8719 analog). (Contributed by Mario Carneiro, 4-Dec-2014.) Generalization of ringm2neg 14343. (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 14230 Negation of a product in a non-unital ring (mulneg2 8717 analog). In contrast to ringmneg2 14342, 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 14229 Negation of a product in a non-unital ring (mulneg1 8716 analog). In contrast to ringmneg1 14341, 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 14578 The apartness relation given by df-apr 14573 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 14228 The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 14332. (Revised by AV, 16-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )
 
16-Feb-2025rng0cl 14225 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 14224 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 14220 A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)
(𝑅 ∈ Rng → 𝑅 ∈ Grp)
 
16-Feb-2025aptap 8972 Complex apartness (as defined at df-ap 8904) is a tight apartness (as defined at df-tap 7609). (Contributed by Jim Kingdon, 16-Feb-2025.)
# TAp ℂ
 
15-Feb-2025subsubrng2 14506 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 14505 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 14504 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 14503 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 14502 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 14501 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 14366 A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 14365. (Contributed by AV, 15-Feb-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → (𝑅 ∈ Rng ↔ 𝑂 ∈ Rng))
 
15-Feb-2025opprrng 14365 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 14237 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 13711 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 13707 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 7613 Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.)
(𝐴 = 𝐵 → (𝑅 TAp 𝐴 ↔ 𝑅 TAp 𝐵))
 
14-Feb-2025subrngmcl 14500 A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 14524. (Revised by AV, 14-Feb-2025.)
· = (.r‘𝑅)    ⇒   ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴)
 
14-Feb-2025subrngacl 14499 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 14498 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 14497 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 14495 A subring is a subgroup. (Contributed by AV, 14-Feb-2025.)
(𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
 
14-Feb-2025subrngrcl 14494 Reverse closure for a subring predicate. (Contributed by AV, 14-Feb-2025.)
(𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
 
14-Feb-2025subrngrng 14493 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 14492 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 14491 A subring is a subset. (Contributed by AV, 14-Feb-2025.)
𝐵 = (Base‘𝑅)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵)
 
14-Feb-2025issubrng 14490 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 14489 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 14235 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 14222 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 7621 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 7620 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 7602 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 14833 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 14832 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 14831 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 14830 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 14824 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 14573 The relation between elements whose difference is invertible, which for a local ring is an apartness relation by aprap 14581. (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 14221 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 13710 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)
 
13-Feb-2025eqab 2373 One direction of eqabb 2374. (Contributed by Wolf Lammen, 13-Feb-2025.)
(∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑) → 𝐴 = {𝑥 ∣ 𝜑})
 
12-Feb-2025eqabb 2374 Equality of a class variable and a class abstraction (also called a class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the relationship between expressions with class abstractions and expressions with class variables. Note that abbib 2356 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting a class abstraction for each class variable.

Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable 𝜑 (that has a free variable 𝑥) to a theorem with a class variable 𝐴, we substitute 𝑥 ∈ 𝐴 for 𝜑 throughout and simplify, where 𝐴 is a new class variable not already in the wff. An example is the conversion of sepgi 4250 to inex1 4265 (look at the instance of sepgi 4250 that occurs in the proof of inex1 4265). Conversely, to convert a theorem with a class variable 𝐴 to one with 𝜑, we substitute {𝑥 ∣ 𝜑} for 𝐴 throughout and simplify, where 𝑥 and 𝜑 are new setvar and wff variables not already in the wff. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13.

(Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2025.)

(𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))
 
8-Feb-20252oneel 7616 ∅ 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 7612 Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.)
(𝑅 = 𝑆 → (𝑅 TAp 𝐴 ↔ 𝑆 TAp 𝐴))
 
7-Feb-2025psrgrp 15059 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 14540 Restriction of the codomain of a (ring) homomorphism. resghm2b 14048 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 11129 An integer is less than or equal to its square. (Contributed by BJ, 6-Feb-2025.)
(𝑁 ∈ ℤ → 𝑁 ≤ (𝑁↑2))
 
6-Feb-20252omotap 7619 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 7618 Lemma for 2omotap 7619. (Contributed by Jim Kingdon, 6-Feb-2025.)
((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → 𝜑)
 
6-Feb-20252omotaplemap 7617 Lemma for 2omotap 7619. (Contributed by Jim Kingdon, 6-Feb-2025.)
(¬ ¬ 𝜑 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} TAp 2o)
 
6-Feb-20252onetap 7615 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 7614 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 7609 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 13929 Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 13924. (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 13985 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 13844 The right inverse of a group element. Deduction associated with grprinv 13839. (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 13843 The left inverse of a group element. Deduction associated with grplinv 13838. (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 13837 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 13822 The identity element of a group is a right identity. Deduction associated with grprid 13820. (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 13821 The identity element of a group is a left identity. Deduction associated with grplid 13819. (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 13800 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 14388 Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)
(𝑅 ∈ SRing → (∥r‘𝑅) ∈ V)
 
24-Jan-2025reldvdsrsrg 14382 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 9167 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 8995 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-2025flddrngd 14598 A field is a division ring. (Contributed by SN, 17-Jan-2025.)
(𝜑 → 𝑅 ∈ Field)    ⇒   (𝜑 → 𝑅 ∈ DivRing)
 
17-Jan-2025ressval3d 13409 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 13408 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 4348 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 4257. (Revised by BJ, 17-Jan-2025.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))
 
16-Jan-2025ressex 13402 Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) ∈ V)
 
16-Jan-2025ressvalsets 13401 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))⟩))
 
15-Jan-2025vsnex 4346 A singleton built on a setvar is a set. (Contributed by BJ, 15-Jan-2025.)
{𝑥} ∈ V
 
12-Jan-2025isrim 14459 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 14461 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 14451 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 14361 Existence of the opposite ring. If you know that 𝑅 is a ring, see opprring 14367. (Contributed by Jim Kingdon, 10-Jan-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
 
10-Jan-2025mgpex 14206 Existence of the multiplication group. If 𝑅 is known to be a semiring, see srgmgp 14255. (Contributed by Jim Kingdon, 10-Jan-2025.)
𝑀 = (mulGrp‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → 𝑀 ∈ V)
 
6-Jan-2025ord3 6693 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 3848 The singleton of an element of a class is a subset of the class (inference form of snssg 3847). 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 3847 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 3846 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 7104 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 7100 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 4423 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 2781 Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 2784 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 7513 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 7512 Lemma for nninfinfwlpo 7514. (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 7511 Lemma for nninfwlpoim 7513. (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 7510 Lemma for nninfwlpoim 7513. (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 7509 Lemma for nninfwlpoim 7513. (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 7508 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 7507 Lemma for nninfwlpor 7508. 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 7058 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 7033 The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 7038 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 7032 The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 7037 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 7506 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 7515 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 7505 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 7025 Dominance relation. This variation of brdomg 7026 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 7026. (Revised by BTernaryTau, 29-Nov-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
 
28-Nov-2024basmexd 13396 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)
 
23-Nov-2024fldcrngd 14599 A field is a commutative ring. (Contributed by SN, 23-Nov-2024.)
(𝜑 → 𝑅 ∈ Field)    ⇒   (𝜑 → 𝑅 ∈ CRing)
 
22-Nov-2024eliotaeu 5364 An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.)
(𝐴 ∈ (℩𝑥𝜑) → ∃!𝑥𝜑)
 
22-Nov-2024eliota 5363 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 13395 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 945 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 14763 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 14762 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 13512 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 16762 Contraposition when the antecedent is a negated stable proposition. See con1dc 868. (Contributed by BJ, 11-Nov-2024.)
(STAB 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))
 
11-Nov-2024slotsdifdsndx 13562 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 13551 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 13550 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 13547 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 13531 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 6298 Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)
(𝜑 → 𝑅 = 𝑆)    ⇒   (𝜑 → ∘𝑓 𝑅 = ∘𝑓 𝑆)
 
11-Nov-2024const 864 Contraposition when the antecedent is a negated stable proposition. See comment of condc 865. (Contributed by BJ, 18-Nov-2023.) (Proof shortened by BJ, 11-Nov-2024.)
(STAB 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
 
10-Nov-2024slotsdifunifndx 13569 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 13404 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 14364 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 14363 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 14362 Lemma for opprbasg 14363 and oppraddg 14364. (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 16107 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 14960 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 14959 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 14958 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 14957 Lemma for znbas 14962. (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 14949 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 14948 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 14947 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 14946 Lemma for zlmbasg 14947 and zlmplusgg 14948. (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 14950 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 13546 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 13545 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 13544 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 11070 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 13559 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 13530 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 13528 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 13527 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 13526 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 16234 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 13542 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 13541 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 13540 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 14768 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 14765 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 14761 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 14760 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 14759 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 14758 Lemma for srabaseg 14759 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 13561 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 13560 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 13532 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 13511 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 13510 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 13498 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 13493 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 11256 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 11255 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 11254 Lemma for fiubz 11255 and fiubnn 11256. 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 16233 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 13568 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 13566 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 13565 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 13557 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 13556 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 13555 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 16754 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 16999 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 16749 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 16748 as its last step. (Contributed by BJ, 27-Oct-2024.)
(𝜑 → 𝜓)    &   (¬ 𝜑 → 𝜓)    ⇒    ¬ ¬ 𝜓
 
25-Oct-2024nnwosdc 12799 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 12796 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 12797 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 16999 Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 862, 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 13567 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 13509 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 13491 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 12902 Lemma for isprm5 12903. 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 13410 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 14671 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 14608 of a left module, see also islmod 14610. (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 14213 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 13558 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 13543 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 13529 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 13499 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 13497 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 13496 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 13492 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 13481 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 13480 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 13479 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 13450 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 13448 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 9995 Membership of an integer in ℕ is decidable. (Contributed by Jim Kingdon, 17-Oct-2024.)
(𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ)
 
14-Oct-20242zinfmin 11992 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 11991 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 16232 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 16231 Index value of the df-edgf 16229 slot. (Contributed by AV, 13-Oct-2024.) (New usage is discouraged.)
(.ef‘ndx) = 18
 
13-Oct-2024prdsvallem 13604 Lemma for prdsval 14156. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 14156, dependency on df-hom 13438 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 13072 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 10188 Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
 
13-Oct-2024xnn0dcle 10187 Decidability of ≤ for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → DECID 𝐴 ≤ 𝐵)
 
9-Oct-2024nn0leexp2 11131 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 13050 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 9994 Membership of an integer in ℕ0 is decidable. (Contributed by Jim Kingdon, 8-Oct-2024.)
(𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ0)
 
7-Oct-2024pclemub 13049 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 13048 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 11130 Special case of ltexp2 16026 which we use here because we haven't yet defined df-rpcxp 15943 which is used in the current proof of ltexp2 16026. (Contributed by Jim Kingdon, 7-Oct-2024.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁)))
 
6-Oct-2024suprzcl2dc 10657 The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 8294.) (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 10656 An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 8294.) (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 10654 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, < ))
 
3-Oct-2024ab0w 3550 The class of sets verifying a property is the empty class if and only if that property is a contradiction. (Contributed by GG, 3-Oct-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝜓)
 
1-Oct-2024infex2g 7368 Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
(𝐴 ∈ 𝐶 → inf(𝐵, 𝐴, 𝑅) ∈ V)
 
30-Sep-2024unbendc 13328 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 12891 Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.)
(𝑁 ∈ ℕ → DECID 𝑁 ∈ ℙ)
 
30-Sep-2024dcfi 7309 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 6043 Change bound variable in a restricted description binder. Version of cbvriotav 6045 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 5341 Change bound variables in a description binder. Version of cbviotav 5342 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 13321 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 13320 Lemma for ssnnct 13321. 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 10655 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 9981 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 13324 Lemma for nninfdc 13327. 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 12795 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 12794. (Contributed by Jim Kingdon, 26-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝐵)
 
25-Sep-2024nninfdclemcl 13322 Lemma for nninfdc 13327. (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 13325 Lemma for nninfdc 13327. The function from nninfdclemf 13323 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 13327 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 13326 Lemma for nninfdc 13327. The function from nninfdclemf 13323 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 13323 Lemma for nninfdc 13327. 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 12794 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 7023 Equinumerosity relation. This variation of bren 7024 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 7024. (Revised by BTernaryTau, 23-Sep-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
 
23-Sep-2024dfnul3 3524 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.) (Proof shortened by BJ, 23-Sep-2024.)
∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}
 
23-Sep-2024dfnul2 3523 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Reduce axiom dependencies. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)
∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
 
23-Sep-2024dfnul4 3522 Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-13 2211, df-clel 2234. (Revised by GG, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 3523. (Revised by BJ, 23-Sep-2024.)
∅ = {𝑥 ∣ ⊥}
 
20-Sep-2024ineqcomi 3423 Two ways of expressing that two classes have a given intersection. Inference form of ineqcom 3422. Disjointness inference when 𝐶 = ∅. (Contributed by Peter Mazsa, 26-Mar-2017.) (Proof shortened by SN, 20-Sep-2024.)
(𝐴 ∩ 𝐵) = 𝐶    ⇒   (𝐵 ∩ 𝐴) = 𝐶
 
19-Sep-2024ssomct 13319 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 6698 2o is a set. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by Zhi Wang, 19-Sep-2024.)
2o ∈ V
 
19-Sep-2024ecase2d 1392 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 5888 Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 5550. (Contributed by AV, 18-Sep-2024.)
((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
 
18-Sep-2024eqabbw 2375 Version of eqabb 2374 using implicit substitution. (Contributed by GG and AV, 18-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝜓))
 
17-Sep-2024fncofn 5887 Composition of a function with domain and a function as a function with domain. Generalization of fnco 5489. (Contributed by AV, 17-Sep-2024.)
((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))
 
14-Sep-2024nnpredlt 4769 The predecessor (see nnpredcl 4768) of a nonzero natural number is less than (see df-iord 4509) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴)
 
13-Sep-2024nninfisollemeq 7466 Lemma for nninfisol 7467. 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 7465 Lemma for nninfisol 7467. 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 7464 Lemma for nninfisol 7467. 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)
 
13-Sep-2024fset0 6943 The set of functions from the empty set is the singleton containing the empty set. (Contributed by AV, 13-Sep-2024.)
{𝑓 ∣ 𝑓:∅⟶𝐵} = {∅}
 
13-Sep-2024fsetsspwxp 6942 The class of all functions from 𝐴 into 𝐵 is a subclass of the power class of the cartesion product of 𝐴 and 𝐵. (Contributed by AV, 13-Sep-2024.)
{𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
 
12-Sep-2024nninfisol 7467 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 7514). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.)

((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
 
8-Sep-2024relopabv 4902 A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopab 4904. (Contributed by SN, 8-Sep-2024.)
Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
7-Sep-2024eulerthlemfi 12989 Lemma for eulerth 12994. 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 11087 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 12992 Lemma for eulerth 12994. 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 12993 Lemma for eulerth 12994. 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 12991 Lemma for eulerth 12994. (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 12990 Lemma for eulerth 12994. 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 14849 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 12386 A finite product of terms apart from zero is apart from zero. A version of fprodap0 12371 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 12379 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 7596 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 7592 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 4717 Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4716. (Contributed by Jim Kingdon, 26-Aug-2024.)
(∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → DECID 𝜑)
 
26-Aug-2024ontriexmidim 4667 Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4666. (Contributed by Jim Kingdon, 26-Aug-2024.)
(∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → DECID 𝜑)
 
25-Aug-2024onntri2or 7599 Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
(¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
 
25-Aug-2024onntri3or 7598 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 3158 Composition law for chained substitutions into a class. Version of csbco 3157 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 2792 Version of cbvreuv 2788 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 2791 Version of cbvrexv 2787 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 2790 Version of cbvralv 2786 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 2363 Version of cbvab 2364 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 2007 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 2006 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 1976 Change bound variable. See cbvexv 1974 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1501. (Revised by GG, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥𝜑 ↔ ∃𝑦𝜓)
 
25-Aug-2024cbvalvw 1975 Change bound variable. See cbvalv 1973 for a version with fewer disjoint variable conditions. (Contributed by NM, 9-Apr-2017.) Avoid ax-7 1501. (Revised by GG, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 
25-Aug-2024nfal 1629 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 1563. (Revised by GG, 25-Aug-2024.)
Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∀𝑦𝜑
 
24-Aug-2024gcdcomd 12734 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))
 
22-Aug-2024fsetdmprc0 6944 The set of functions with a proper class as domain is empty. (Contributed by AV, 22-Aug-2024.)
(𝐴 ∉ V → {𝑓 ∣ 𝑓 Fn 𝐴} = ∅)
 
22-Aug-2024abanssr 3502 A class abstraction with a conjunction is a subset of the class abstraction with the right conjunct only. (Contributed by AV, 7-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
{𝑓 ∣ (𝜑 ∧ 𝜓)} ⊆ {𝑓 ∣ 𝜓}
 
22-Aug-2024abanssl 3501 A class abstraction with a conjunction is a subset of the class abstraction with the left conjunct only. (Contributed by AV, 7-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
{𝑓 ∣ (𝜑 ∧ 𝜓)} ⊆ {𝑓 ∣ 𝜑}
 
21-Aug-2024dvds2addd 12579 Deduction form of dvds2add 12575. (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))
 
20-Aug-2024rspcedvdw 2936 Version of rspcedvd 2935 where the implicit substitution hypothesis does not have an antecedent, which also avoids a disjoint variable condition on 𝜑, 𝑥. (Contributed by SN, 20-Aug-2024.)
(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   (𝜑 → 𝐴 ∈ 𝐵)    &   (𝜑 → 𝜒)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
 
18-Aug-2024prdsmulr 14161 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 14160 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 14159 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 14158 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 14156 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 14153 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 12355 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 12354 Multiply in an additional term in a finite product. See also fprodsplitsn 12383 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 3641 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 16818 Alternate proof of bj-charfundc 16817. It was expected to be much shorter since it uses bj-charfun 16816 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 16816 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 8967 Equality of complex numbers is stable. Stability here means ¬ ¬ 𝐴 = 𝐵 → 𝐴 = 𝐵 as defined at df-stab 843. 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 8966 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 4628 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 4627 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 4626 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 4625 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 3656 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 3655 A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.)
if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)
 
14-Aug-2024hashf1lem1 11268 Lemma for hashf1 11270. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 14-Aug-2024.)
(𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐵 ∈ Fin)    &   (𝜑 → ¬ 𝑧 ∈ 𝐴)    &   (𝜑 → ((♯‘𝐴) + 1) ≤ (♯‘𝐵))    &   (𝜑 → 𝐹:𝐴–1-1→𝐵)    ⇒   (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝐹))
 
14-Aug-2024f1setexg 6945 The set of injections between two sets exists. (Contributed by AV, 14-Aug-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ∈ V)
 
12-Aug-2024exmidontriimlem2 7572 Lemma for exmidontriim 7575. (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 7571 Lemma for exmidontriim 7575. A variation of r19.30dc 2698. (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 863 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 862 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 1702 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 4330): then, we can prove ∀𝑥 ∈ 𝒫 1o¬ ¬ DECID 𝑥 = 1o but we cannot prove ¬ ¬ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o because the converse of nnral 2540 does not hold.

Actually, ¬ ¬ EXMID is not provable in intuitionistic logic since intuitionistic logic has models satisfying ¬ EXMID and noncontradiction holds (pm3.24 705). (Contributed by BJ, 9-Oct-2019.) Add explanation on non-provability of ¬ ¬ EXMID. (Revised by BJ, 11-Aug-2024.)

¬ ¬ DECID 𝜑
 
10-Aug-2024exmidontriim 7575 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 7574 Lemma for exmidontriim 7575. 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 7573 Lemma for exmidontriim 7575. 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 7461 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 7458 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 4820 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
(𝑖 ∈ ω ↦ 1o) ∈ ℕ∞
 
10-Aug-2024mapssfsetg 6940 The value of the set exponentiation (𝐵 ↑𝑚 𝐴) is a subset of the class of functions from 𝐴 to 𝐵. (Contributed by AV, 10-Aug-2024.)
(𝐵 ∈ 𝑉 → (𝐵 ↑𝑚 𝐴) ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵})
 
9-Aug-2024ss1o0el1o 7214 Reformulation of ss1o0el1 4332 using 1o instead of {∅}. (Contributed by BJ, 9-Aug-2024.)
(𝐴 ⊆ 1o → (∅ ∈ 𝐴 ↔ 𝐴 = 1o))
 
9-Aug-2024pw1dc0el 7212 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 4332 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 7215 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-2024mapfset 6939 If 𝐵 is a set, the value of the set exponentiation (𝐵 ↑𝑚 𝐴) is the class of all functions from 𝐴 to 𝐵. Generalisation of mapvalg 6926 to arbitrary domains. Note that the class {𝑓 ∣ 𝑓:𝐴⟶𝐵} can only contain set-functions, as opposed to arbitrary class-functions. When 𝐴 is a proper class, there can be no set-functions on it, so the above class is empty (see also fsetdmprc0 6944), hence a set. In this case, both sides of the equality in this theorem are the empty set. (Contributed by AV, 8-Aug-2024.)
(𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = (𝐵 ↑𝑚 𝐴))
 
8-Aug-2024fdmexb 5933 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 5932 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-2024psrbagaddclfi 15044 The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.) Shorten proof and remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝐼 ∈ Fin) → (𝐹 ∘𝑓 + 𝐺) ∈ 𝐷)
 
7-Aug-2024psrbagfsupp 15038 Finite bags have finite support. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   (𝐹 ∈ 𝐷 → 𝐹 finSupp 0)
 
7-Aug-2024pw1fin 7211 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-2024mapfoss 6941 The value of the set exponentiation (𝐵 ↑𝑚 𝐴) is a superset of the set of all functions from 𝐴 onto 𝐵. (Contributed by AV, 7-Aug-2024.)
{𝑓 ∣ 𝑓:𝐴–onto→𝐵} ⊆ (𝐵 ↑𝑚 𝐴)
 
7-Aug-2024elomssom 4750 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 4751. (Revised by BJ, 7-Aug-2024.)
(𝐴 ∈ ω → 𝐴 ⊆ ω)
 
6-Aug-2024bj-charfunbi 16820 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 16819 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 16817 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 15047 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 15046 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 12339 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 16815 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 16814 The maps-to notation defines a function (deduction form).

Note: one should similarly prove a deduction form of funopab4 5412, then prove funmptd 16814 from it, and then prove funmpt 5413 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 16766 The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID ⊥
 
5-Aug-2024bj-dctru 16764 The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID ⊤
 
5-Aug-2024bj-stfal 16753 The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)
STAB ⊥
 
5-Aug-2024bj-sttru 16751 The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)
STAB ⊤
 
5-Aug-2024psrbagcon 15045 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 15043 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 15041 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 12336 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-2024fcdmnn0fsuppg 9601 Version of fcdmnn0fsupp 9599 avoiding ax-coll 4244 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 5-Aug-2024.)
((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (◡𝐹 “ ℕ) ∈ Fin))
 
5-Aug-2024fcdmnn0suppg 9600 Version of fcdmnn0supp 9598 avoiding ax-coll 4244 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 5-Aug-2024.)
((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
 
5-Aug-20242ssom 6791 The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
2o ⊆ ω
 
5-Aug-2024suppssdc 6494 Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.) (Proof shortened by SN, 5-Aug-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍)    &   (𝜑 → ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝑊)    ⇒   (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
 
5-Aug-2024elsuppfng 6476 An element of the support of a function with a given domain. This version of elsuppfn 6477 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-coll 4244. (Contributed by SN, 5-Aug-2024.)
((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆 ∈ 𝑋 ∧ (𝐹‘𝑆) ≠ 𝑍)))
 
5-Aug-2024suppvalfng 6474 The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 6475 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-coll 4244. (Contributed by SN, 5-Aug-2024.)
((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍})
 
4-Aug-2024en3d 7049 Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by AV, 4-Aug-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))    &   (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴))    &   (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)))    ⇒   (𝜑 → 𝐴 ≈ 𝐵)
 
4-Aug-2024en2d 7048 Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by AV, 4-Aug-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝑋))    &   (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝑌))    &   (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)))    ⇒   (𝜑 → 𝐴 ≈ 𝐵)
 
2-Aug-2024onntri52 7597 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 7595 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 7594 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 7593 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 7591 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 7590 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 7591), (3) implies (5) (onntri35 7590), (5) implies (1) (onntri51 7593), (2) implies (4) (onntri24 7595), (4) implies (5) (onntri45 7594), and (5) implies (2) (onntri52 7597).

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 7592 and exmidontri2or 7596, respectively. Thus ¬ ¬ EXMID is equivalent to (1) or (2). In addition, ¬ ¬ EXMID is equivalent to (3) by onntri3or 7598 and (4) by onntri2or 7599.

(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 2540 The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1702. (Contributed by Jim Kingdon, 1-Aug-2024.)
(¬ ¬ ∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑)
 
31-Jul-20243nsssucpw1 7589 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 7588 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 15037 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 7587 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 7586 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 7585 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 7584 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 7583 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 7582 The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
𝒫 1o ≠ 1o
 
30-Jul-2024pw1ne0 7581 The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
𝒫 1o ≠ ∅
 
30-Jul-2024fsuppeqg 6482 Version of fsuppeq 6481 avoiding ax-coll 4244 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 30-Jul-2024.)
((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍}))))
 
29-Jul-2024ringcld 14305 Closure of the multiplication operation of a ring. (Contributed by SN, 29-Jul-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)
 
29-Jul-2024grpcld 13802 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 7579 The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
𝒫 1o ∈ On
 
29-Jul-2024isfsuppd 7284 Deduction form of isfsupp 7283. (Contributed by SN, 29-Jul-2024.)
(𝜑 → 𝑅 ∈ 𝑉)    &   (𝜑 → 𝑍 ∈ 𝑊)    &   (𝜑 → Fun 𝑅)    &   (𝜑 → (𝑅 supp 𝑍) ∈ Fin)    ⇒   (𝜑 → 𝑅 finSupp 𝑍)
 
28-Jul-2024exmidpweq 7210 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 17086 Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 17085 by means of dceqnconst 17084. (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 17082 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 17088 Lemma for nconstwlpo 17090. 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 17087 Lemma for nconstwlpo 17090. 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 17080 Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 17066 and redcwlpo 17079). Thus, this is an analytic analogue to lpowlpo 7502. (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 17075 Weak omniscience stated in terms of equality with 0. Like iswomninn 17074 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-2024asclmulg 15027 Apply group multiplication to the algebra scalars. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    ↑ = (.g‘𝑊)    &    ∗ = (.g‘𝐹)    ⇒   ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → (𝐴‘(𝑁 ∗ 𝑋)) = (𝑁 ↑ (𝐴‘𝑋)))
 
24-Jul-2024lpowlpo 7502 LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7501. There is an analogue in terms of analytic omniscience principles at tridceq 17080. (Contributed by Jim Kingdon, 24-Jul-2024.)
(ω ∈ Omni → ω ∈ WOmni)
 
23-Jul-2024nconstwlpolem 17089 Lemma for nconstwlpo 17090. (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 17084 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 17079 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 17081 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 6030 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 1553 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 17090 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)
 
16-Jul-2024unexd 4890 The union of two sets is a set. (Contributed by SN, 16-Jul-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
 
16-Jul-2024difexd 4275 Existence of a difference. (Contributed by SN, 16-Jul-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (𝐴 ∖ 𝐵) ∈ V)
 
15-Jul-2024fprodseq 12333 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 2555 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 2554 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 16063 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 12941, 2logb9irrap 16062 and sqrt2cxp2logb9e3 16060. 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 16062 Example for logbgcd1irrap 16055. 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 13636 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 13635 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 13634 Lemma for ercpbl 13635. (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 13633 Value of the function in qusval 13627. (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 13632 Value of the function in qusval 13627. (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 16053 Lemma for logbgcd1irrap 16055. 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 15862 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 16024 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 16054 Lemma for logbgcd1irrap 16055. 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 11139 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 15961 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 16030 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 15962 Deduction form of logrpap0 15961. (Contributed by Jim Kingdon, 3-Jul-2024.)
(𝜑 → 𝐴 ∈ ℝ+)    &   (𝜑 → 𝐴 # 1)    ⇒   (𝜑 → (log‘𝐴) # 0)
 
3-Jul-2024logrpap0b 15960 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 17007 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 17006 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 17058 An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.)
(0(,)1) ≈ ℝ
 
27-Jun-2024iooref1o 17057 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 17091 Lemma for neapmkv 17092. 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 17077 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 17076 Lemma for ismkvnn 17077. 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 7495 Lemma for enmkv 7496. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ Markov → 𝐵 ∈ Markov))
 
24-Jun-2024neapmkv 17092 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 17085 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 17066 for more discussion of decidability of real number apartness.

This is a weaker form of dceqnconst 17084 and in fact this theorem can be proved using dceqnconst 17084 as shown at dcapnconstALT 17086. (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 7496 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 6696 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 17078 Lemma for redcwlpo 17079. A biconditionalized version of trilpolemeq1 17063. (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 17079 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 17078). 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 10662 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)

(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ω ∈ WOmni)
 
20-Jun-2024iswomninn 17074 Weak omniscience stated in terms of natural numbers. Similar to iswomnimap 7500 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 17073 Lemma for iswomnimap 7500. 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 7504 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 6696 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 7503 Lemma for enwomni 7504. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni → 𝐵 ∈ WOmni))
 
19-Jun-2024rpabscxpbnd 16025 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 16007 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 622 Deduction associated with biadani 620. 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 15990 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 15989 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 15987 Closure of the complex power function. (Contributed by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) ∈ ℂ)
 
12-Jun-2024rpcxp0 15983 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 15981 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 15980 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 15979 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 15943 Define the power function on complex numbers. Because df-relog 15942 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 17068 Version of trirec0 17067 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 17067 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 17066). (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 7501 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 7500 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 7499 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 7498 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 14323 A ring is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → 𝑅 ∈ CMnd)
 
1-Jun-2024ringabld 14322 A ring is an Abelian group. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → 𝑅 ∈ Abel)
 
1-Jun-2024cmnmndd 14094 A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝐺 ∈ CMnd)    ⇒   (𝜑 → 𝐺 ∈ Mnd)
 
1-Jun-2024ablcmnd 14078 An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝐺 ∈ Abel)    ⇒   (𝜑 → 𝐺 ∈ CMnd)
 
1-Jun-2024grpmndd 13801 A group is a monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝐺 ∈ Grp)    ⇒   (𝜑 → 𝐺 ∈ Mnd)
 
1-Jun-2024fndmi 5479 The domain of a function. (Contributed by Wolf Lammen, 1-Jun-2024.)
𝐹 Fn 𝐴    ⇒   dom 𝐹 = 𝐴
 
29-May-2024pw1nct 17016 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 17015 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)
 
28-May-2024fvun2d 5769 The value of a union when the argument is in the second domain, a deduction version. (Contributed by metakunt, 28-May-2024.)
(𝜑 → 𝐹 Fn 𝐴)    &   (𝜑 → 𝐺 Fn 𝐵)    &   (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))
 
28-May-2024fvun1d 5768 The value of a union when the argument is in the first domain, a deduction version. (Contributed by metakunt, 28-May-2024.)
(𝜑 → 𝐹 Fn 𝐴)    &   (𝜑 → 𝐺 Fn 𝐵)    &   (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   (𝜑 → 𝑋 ∈ 𝐴)    ⇒   (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))
 
26-May-2024elpwi2 4292 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 15079 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 15808 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 2775 Rule used to change bound variables, using implicit substitution. Version of cbvralf 2777 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 1560 and ax-bndl 1562 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 2735 Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo 2785 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 2127 Rule used to change bound variables, using implicit substitution. Version of cbvmo 2126 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 15858 Lemma for eflt 15859. The converse of efltim 12448 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 15859 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 659 A negated syllogism inference. (Contributed by Wolf Lammen, 20-May-2024.)
(𝜑 → 𝜓)    &   (¬ 𝜑 → 𝜒)    ⇒   (¬ 𝜓 → 𝜒)
 
19-May-2024apdifflemr 17070 Lemma for apdiff 17071. (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 17069 Lemma for apdiff 17071. 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 17071 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 14616 A left module is a group. (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑊 ∈ LMod)    ⇒   (𝜑 → 𝑊 ∈ Grp)
 
16-May-2024drnggrpd 14595 A division ring is a group (deduction form). (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑅 ∈ DivRing)    ⇒   (𝜑 → 𝑅 ∈ Grp)
 
16-May-2024drngringd 14594 A division ring is a ring. (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑅 ∈ DivRing)    ⇒   (𝜑 → 𝑅 ∈ Ring)
 
16-May-2024crnggrpd 14297 A commutative ring is a group. (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑅 ∈ CRing)    ⇒   (𝜑 → 𝑅 ∈ Grp)
 
16-May-2024crngringd 14296 A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑅 ∈ CRing)    ⇒   (𝜑 → 𝑅 ∈ Ring)
 
16-May-2024ringgrpd 14292 A ring is a group. (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → 𝑅 ∈ Grp)
 
15-May-2024reeff1oleme 15856 Lemma for reeff1o 15857. (Contributed by Jim Kingdon, 15-May-2024.)
(𝑈 ∈ (0(,)e) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈)
 
14-May-2024df-relog 15942 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 6219 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 5579 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 12580 The divides relation is transitive, a deduction version of dvdstr 12578. (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 15938 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 15936 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 15937 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 13318 The union of a countably infinite collection of countable sets is countable. Theorem 8.1.28 of [AczelRathjen], p. 78. Compare with ctiunct 13314 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 13315 Variation of ctiunct 13314 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-2024suppssrgst 6496 A function is zero outside its support. Version of suppssrst 6495 avoiding ax-coll 4244 by assuming 𝐹 is a set rather than its domain 𝐴. (Contributed by SN, 5-May-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 STAB 𝑢 = 𝑣)    ⇒   ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍)
 
5-May-2024ifpnst 1001 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 7631 Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7630, 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 7629 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 7630 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 998 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 7628 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 1006 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 7627 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 7626 Lemma for cc2 7627. (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 7625 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 14752 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 13317 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 1005 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 12327 Lemma for prodmodc 12328. (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))
 
13-Apr-2024sspwd 3703 The powerclass preserves inclusion (deduction form). (Contributed by BJ, 13-Apr-2024.)
(𝜑 → 𝐴 ⊆ 𝐵)    ⇒   (𝜑 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 
13-Apr-2024sspwi 3702 The powerclass preserves inclusion (inference form). (Contributed by BJ, 13-Apr-2024.)
𝐴 ⊆ 𝐵    ⇒   𝒫 𝐴 ⊆ 𝒫 𝐵
 
13-Apr-2024sspw 3701 The powerclass preserves inclusion. See sspwb 4354 for the biconditional version. (Contributed by NM, 13-Oct-1996.) Extract forward implication of sspwb 4354 since it requires fewer axioms. (Revised by BJ, 13-Apr-2024.)
(𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 
11-Apr-2024prodmodclem2a 12326 Lemma for prodmodc 12328. (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 12325 Lemma for prodmodc 12328. (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 662 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 12321 Lemma for prodrbdc 12324. (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 12297 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 12296 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 12295 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 12291 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 12301 Define the product of a series with an index set of integers 𝐴. This definition takes most of the aspects of df-sumdc 12103 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 15935 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 15934 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 15918 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 15917 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 15867 Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.)
(π ∈ (2(,)4) ∧ (sin‘π) = 0)
 
9-Mar-2024exmidonfin 7540 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 7168 and nnon 4755. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
(ω = (On ∩ Fin) → EXMID)
 
9-Mar-2024exmidonfinlem 7539 Lemma for exmidonfin 7540. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
𝐴 = {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}    ⇒   (ω = (On ∩ Fin) → DECID 𝜑)
 
8-Mar-2024sin0pilem2 15866 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 15865 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 15864 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 12518 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 16652 (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 14626 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 14423 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 13665 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 13775 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 10148 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 10147 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 8970 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 8953 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 8971 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 15728 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 15726 Lemma for ivthinc 15727. 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 15722 Lemma for ivthinc 15727. 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 15717 A Dedekind cut identifies a unique real number. Similar to df-inp 7827 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 13833 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 15725 Lemma for ivthinc 15727. 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)))
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