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

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

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Last updated on 9-Sep-2026 at 7:08 AM ET.
Recent Additions to the Intuitionistic Logic Explorer
DateLabelDescription
Theorem
 
27-Aug-2026prmdcz 12925 Primality is decidable. (Contributed by Jim Kingdon, 27-Aug-2026.)
(𝑁 ∈ ℤ → DECID 𝑁 ∈ ℙ)
 
27-Aug-2026zmincl 12020 The minumum of two integers is an integer. (Contributed by Jim Kingdon, 27-Aug-2026.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → inf({𝐴, 𝐵}, ℝ, < ) ∈ ℤ)
 
25-Aug-2026nn0sqdcq 13004 A nonnegative integer is a perfect square or not. This is similar to nn0sqdc 11160 but expresses the idea of being a perfect square as having a rational number which, when squared, gives the original number. (Contributed by Jim Kingdon, 25-Aug-2026.)
(𝑁 ∈ ℕ0 → DECID ∃𝑞 ∈ ℚ 𝑁 = (𝑞↑2))
 
25-Aug-2026qabscl 11857 The absolute value of a rational number is a rational number. (Contributed by Jim Kingdon, 25-Aug-2026.)
(𝐴 ∈ ℚ → (abs‘𝐴) ∈ ℚ)
 
25-Aug-2026nn0sqdc 11160 A nonnegative integer is a perfect square or not. (Contributed by Jim Kingdon, 25-Aug-2026.)
(𝑁 ∈ ℕ0 → DECID ∃𝑞 ∈ ℕ0 𝑁 = (𝑞↑2))
 
24-Aug-2026sqrtrirr 13005 The square root of a nonnegative integer is either rational or irrational. (Contributed by Jim Kingdon, 24-Aug-2026.)
(𝐴 ∈ ℕ0 → ((√‘𝐴) ∈ ℚ ∨ ((√‘𝐴) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (√‘𝐴) # 𝑞)))
 
21-Aug-2026prmefexple 16227 Convert a bound on a power of a prime to a bound on the exponent. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Jim Kingdon, 21-Aug-2026.)
((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐴↑𝑁) ≤ 𝐵 ↔ 𝑁 ≤ (⌊‘((log‘𝐵) / (log‘𝐴)))))
 
20-Aug-2026zprmlogbap 16137 The logarithm of a natural number to a prime base is either rational or irrational.

The proof decomposes 𝑋 into 𝑚 ∈ ℕ and 𝑎 ∈ ℕ0 such that 𝑋 = ((𝐵↑𝑎) · 𝑚) (using nnmaxpw 12969). If 𝑚 = 1 the logarithm is 𝑎, which is rational. If 1 < 𝑚 then we can apply logbgcd1irrap 16125 to show that the logarithm is irrational. (Contributed by Jim Kingdon and Taylor Barrella, 20-Aug-2026.)

((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → ((𝐵 logb 𝑋) ∈ ℚ ∨ ((𝐵 logb 𝑋) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐵 logb 𝑋) # 𝑞)))
 
20-Aug-2026zprmlogbaplem3 16136 Lemma for zprmlogbap 16137. Decomposing a natural number into a power of a prime base and a factor not divisible by that prime. (Contributed by Jim Kingdon, 20-Aug-2026.)
𝐽 = {𝑧 ∈ ℕ ∣ ¬ 𝐵 ∥ 𝑧}    &   𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((𝐵↑𝑦) · 𝑥))    ⇒   ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → ∃𝑚 ∈ ℕ ∃𝑎 ∈ ℕ0 (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚)))
 
20-Aug-2026zprmlogbaplem2 16135 Lemma for zprmlogbap 16137. The logarithm is either rational or irrational. (Contributed by Jim Kingdon, 20-Aug-2026.)
(𝜑 → 𝐵 ∈ ℙ)    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → ¬ 𝐵 ∥ 𝑀)    &   (𝜑 → 𝐴 ∈ ℕ0)    &   𝑋 = ((𝐵↑𝐴) · 𝑀)    ⇒   (𝜑 → ((𝐵 logb 𝑋) ∈ ℚ ∨ ((𝐵 logb 𝑋) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐵 logb 𝑋) # 𝑞)))
 
20-Aug-2026zprmlogbaplem1 16134 Lemma for zprmlogbap 16137. Rearranging an expression involving logarithms. (Contributed by Jim Kingdon, 20-Aug-2026.)
(𝜑 → 𝐵 ∈ ℙ)    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → ¬ 𝐵 ∥ 𝑀)    &   (𝜑 → 𝐴 ∈ ℕ0)    ⇒   (𝜑 → (𝐵 logb ((𝐵↑𝐴) · 𝑀)) = (𝐴 + (𝐵 logb 𝑀)))
 
20-Aug-2026flaplelt 10723 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 20-Aug-2026.)
((𝐴 ∈ ℚ ∨ (𝐴 ∈ ℝ ∧ ∀𝑞 ∈ ℚ 𝐴 # 𝑞)) → ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1)))
 
20-Aug-2026flapcl 10721 The floor (greatest integer) function yields an integer when applied to a number which is either rational or irrational. (Contributed by Jim Kingdon, 20-Aug-2026.)
((𝐴 ∈ ℚ ∨ (𝐴 ∈ ℝ ∧ ∀𝑞 ∈ ℚ 𝐴 # 𝑞)) → (⌊‘𝐴) ∈ ℤ)
 
20-Aug-2026irraddap 10056 The sum of an irrational number and a rational number is irrational. (Contributed by Jim Kingdon, 20-Aug-2026.)
(((𝐴 ∈ ℝ ∧ ∀𝑞 ∈ ℚ 𝐴 # 𝑞) ∧ 𝐵 ∈ ℚ) → ((𝐴 + 𝐵) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐴 + 𝐵) # 𝑞))
 
19-Aug-2026nnmaxpw 12969 The function 𝐹 that decomposes a number into its "odd" and "even" parts, which is to say the largest power of a base and largest divisor of the number not divisible by that base, is a bijection from pairs of a nonnegative integer and a number not divisible by that base to positive integers. (Contributed by Thierry Arnoux, 15-Aug-2017.) (Revised by Jim Kingdon, 19-Aug-2026.)
𝐽 = {𝑧 ∈ ℕ ∣ ¬ 𝐵 ∥ 𝑧}    &   𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((𝐵↑𝑦) · 𝑥))    ⇒   (𝐵 ∈ (ℤ≥‘2) → 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ)
 
19-Aug-2026nnmaxpwlemparts 12968 Lemma for nnmaxpw 12969. Decomposing a number into parts. (Contributed by Jim Kingdon, 16-Nov-2021.) (Revised by Jim Kingdon, 19-Aug-2026.)
(𝐵 ∈ (ℤ≥‘2) → ((((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋)) ↔ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))))
 
19-Aug-2026nnmaxpwlemnfac 12967 Lemma for nnmaxpw 12969. Removing the powers of a base from a natural number produces a number not divisible by that base. (Contributed by Jim Kingdon, 16-Nov-2021.) (Revised by Jim Kingdon, 19-Aug-2026.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → ¬ 𝐵 ∥ (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))))
 
19-Aug-2026nnmaxpwlemndvds 12966 Lemma for nnmaxpw 12969. A natural number is not divisible by one more than the highest power of a base which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.) (Revised by Jim Kingdon, 19-Aug-2026.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → ¬ (𝐵↑((℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)) + 1)) ∥ 𝐴)
 
19-Aug-2026nnmaxpwlemdvds 12965 Lemma for nnmaxpw 12969. A natural number is divisible by the highest power of a base which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.) (Revised by Jim Kingdon, 19-Aug-2026.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))) ∥ 𝐴)
 
18-Aug-2026nnmaxpwlemxy 12964 Lemma for nnmaxpw 12969. Another way of stating that decomposing a natural number into a power of a base and a number not divisible by that base is unique. (Contributed by Jim Kingdon, 16-Nov-2021.) (Revised by Jim Kingdon, 18-Aug-2026.)
(𝜑 → 𝑋 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ (ℤ≥‘2))    &   (𝜑 → ¬ 𝐵 ∥ 𝑋)    &   (𝜑 → 𝑌 ∈ ℕ0)    &   (𝜑 → 𝐴 = ((𝐵↑𝑌) · 𝑋))    ⇒   (𝜑 → (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))
 
18-Aug-2026pwbdvdseu 12963 A natural number has a unique highest power of a base which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.) (Revised by Jim Kingdon, 18-Aug-2026.)
((𝑁 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → ∃!𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))
 
18-Aug-2026pwbdvdseulemle 12962 Lemma for pwbdvdseu 12963. Powers of a base which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.) (Revised by Jim Kingdon, 18-Aug-2026.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝐴 ∈ ℕ0)    &   (𝜑 → 𝐵 ∈ ℕ0)    &   (𝜑 → 𝑃 ∈ ℕ)    &   (𝜑 → (𝑃↑𝐴) ∥ 𝑁)    &   (𝜑 → ¬ (𝑃↑(𝐵 + 1)) ∥ 𝑁)    ⇒   (𝜑 → 𝐴 ≤ 𝐵)
 
18-Aug-2026pwbdvds 12961 A natural number has a highest power of a base which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.) (Revised by Jim Kingdon, 18-Aug-2026.)
((𝑁 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))
 
17-Aug-2026pwbdvdslemn 12960 Lemma for pwbdvds 12961. If a natural number has some power of a base which does not divide it, there is a highest power of the base which does divide it. (Contributed by Jim Kingdon, 14-Nov-2021.) (Revised by Jim Kingdon, 17-Aug-2026.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → ¬ (𝐵↑𝐴) ∥ 𝑁)    ⇒   (𝜑 → ∃𝑚 ∈ ℕ0 ((𝐵↑𝑚) ∥ 𝑁 ∧ ¬ (𝐵↑(𝑚 + 1)) ∥ 𝑁))
 
14-Aug-2026reaplog 16019 Apartness and the real natural logarithm. (Contributed by Jim Kingdon, 14-Aug-2026.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 # 𝐵 ↔ (log‘𝐴) # (log‘𝐵)))
 
13-Aug-2026efap1p 15929 If the exponential of a number is apart from one plus that number, the number is apart from zero. To some extent can be thought of as the converse of efgt1p 12479. (Contributed by Jim Kingdon, 13-Aug-2026.)
((𝐴 ∈ ℝ ∧ (1 + 𝐴) # (exp‘𝐴)) → 𝐴 # 0)
 
6-Aug-2026relndmfv 5728 The value of a relation outside its domain is the empty set. (Contributed by Jim Kingdon, 6-Aug-2026.)
((Rel 𝐹 ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∅)
 
1-Aug-2026wexmiddifxy 17165 Being able to subtract an arbitrary finite set from a finite set and get a finite set is equivalent to weak excluded middle. By adding additional conditions we can get a theorem which does not need weak excluded middle, at diffifi 7198. (Contributed by Jim Kingdon, 1-Aug-2026.)
(WEXMID ↔ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin))
 
1-Aug-2026wexmiddifxylem 17164 Lemma for wexmiddifxylem 17164. Showing weak excluded middle given a suitable finite set. (Contributed by Jim Kingdon, 1-Aug-2026.)
(({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) ∈ Fin → DECID ¬ 𝜑)
 
31-Jul-2026rabid1o 17153 Converting between propositions and corresponding subsets of a singleton. (Contributed by Jim Kingdon, 31-Jul-2026.)
({𝑥 ∈ 1o ∣ 𝜑} = 1o ↔ 𝜑)
 
30-Jul-2026wexmiddc 17161 Weak excluded middle expressed using WEXMID implies decidability of a negated proposition. (Contributed by Jim Kingdon, 30-Jul-2026.)
(WEXMID → DECID ¬ 𝜑)
 
30-Jul-2026df-wexmid 17160 Weak excluded middle is the principle that any negated proposition is decidable. (Contributed by Jim Kingdon, 30-Jul-2026.)
(WEXMID ↔ ∀𝑝 ∈ 𝒫 1o(¬ 𝑝 = 1o ∨ ¬ ¬ 𝑝 = 1o))
 
29-Jul-2026wexmiddiffi 17163 Being able to subtract an arbitrary set from a finite set and get a finite set is equivalent to weak excluded middle. By adding additional conditions we can get a theorem which does not need weak excluded middle, at diffifi 7198. (Contributed by Jim Kingdon, 29-Jul-2026.)
(WEXMID ↔ ∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin))
 
29-Jul-2026wexmiddiffilem 17162 Lemma for wexmiddiffi 17163. The reverse direction, using different notation. (Contributed by Jim Kingdon, 29-Jul-2026.)
(∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
 
24-Jul-2026stnot 17158 A proposition is double negation stable if and only if it is equivalent to a negated proposition. Here by "proposition" we mean a subset of a singleton (which is a choice which allows us to quantify over them). Posed as an exercise online by Yannick Forster. (Contributed by Jim Kingdon, 24-Jul-2026.)
(𝐴 ∈ 𝒫 1o → ((¬ ¬ 𝐴 = 1o → 𝐴 = 1o) ↔ ∃𝑦 ∈ 𝒫 1o(𝐴 = 1o ↔ ¬ 𝑦 = 1o)))
 
24-Jul-2026slotm 13464 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-2026alseu-no-surprise 17298 Demonstrate that there is never a "surprise" when using the "all some one" quantifier, that is, it is never possible for the consequent to be both always true and always false. This follows from als-no-surprise 17266 by alseuals 17284. See als-no-surprise 17266 for why ordinary "for all" with implication has no such property. (Contributed by David A. Wheeler, 22-Jul-2026.)
¬ (∀∃!𝑥(𝜑 → 𝜓) ∧ ∀∃!𝑥(𝜑 → ¬ 𝜓))
 
22-Jul-2026alseueu 17297 "The 𝜑 is 𝜓 " implies that exactly one thing is both 𝜑 and 𝜓. This is the half of dfalseu2 17296 that drops the universal conjunct; it does not reverse, so ∃!𝑥(𝜑 ∧ 𝜓) cannot be used in place of an "all some one" statement. (Contributed by David A. Wheeler, 22-Jul-2026.)
(∀∃!𝑥(𝜑 → 𝜓) → ∃!𝑥(𝜑 ∧ 𝜓))
 
22-Jul-2026dfalseu2 17296 An "all some one" statement is equivalent to its universal part conjoined with the claim that exactly one 𝑥 satisfies both 𝜑 and 𝜓. In other words, given ∀𝑥(𝜑 → 𝜓), requiring exactly one 𝑥 to satisfy 𝜑, which is what df-alseu 17281 requires, and requiring exactly one 𝑥 to satisfy (𝜑 ∧ 𝜓) come to the same thing. Read 𝜑 as "is a king" and 𝜓 as "is hungry": if every king is hungry, then "there is exactly one king" and "there is exactly one hungry king" say the same thing, so either of them, together with "every king is hungry", gives "the king is hungry".

The universal conjunct is what makes that work, and it cannot be dropped. ∃!𝑥(𝜑 ∧ 𝜓) on its own is strictly weaker than ∀∃!𝑥(𝜑 → 𝜓), since it is satisfied when many things are 𝜑 and just one of those is 𝜓, as in a region with five kings exactly one of whom is hungry; see alseueu 17297 for the one direction that does hold without it. Uniqueness attaches to the antecedent, not to the conjunction. Russell's analysis of a definite description is built the same way: its uniqueness clause constrains the description predicate alone, while the predication is a separate conjunct. See his worked example of "the father of Charles II was executed", [Russell1905] p. 482. (Contributed by David A. Wheeler, 22-Jul-2026.)

(∀∃!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥(𝜑 ∧ 𝜓)))
 
22-Jul-2026nfralseu 17295 Bound-variable hypothesis builder for "all some one" restricted to a class. This is the "all some one" counterpart of nfrals 17264. (Contributed by David A. Wheeler, 22-Jul-2026.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    &   Ⅎ𝑥𝜓    ⇒   Ⅎ𝑥∀∃!𝑦 ∈ 𝐴(𝜑 → 𝜓)
 
22-Jul-2026nfalseu 17294 Bound-variable hypothesis builder for "all some one". This is the "all some one" counterpart of nfals 17263. Unlike the set.mm version of this theorem, no disjoint variable condition is needed, because nfeu 2105 here does not require one. (Contributed by David A. Wheeler, 22-Jul-2026.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝜓    ⇒   Ⅎ𝑥∀∃!𝑦(𝜑 → 𝜓)
 
22-Jul-2026ralseubii 17293 Congruence for "all some one" restricted to a class. This is the "all some one" counterpart of ralsbii 17261. (Contributed by David A. Wheeler, 22-Jul-2026.)
(𝜑 ↔ 𝜒)    &   (𝜓 ↔ 𝜃)    ⇒   (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃!𝑥 ∈ 𝐴(𝜒 → 𝜃))
 
22-Jul-2026alseubii 17292 Congruence: equivalents may be substituted inside an "all some one". This is the "all some one" counterpart of alsbii 17260. (Contributed by David A. Wheeler, 22-Jul-2026.)
(𝜑 ↔ 𝜒)    &   (𝜓 ↔ 𝜃)    ⇒   (∀∃!𝑥(𝜑 → 𝜓) ↔ ∀∃!𝑥(𝜒 → 𝜃))
 
22-Jul-2026ralseu2d 17291 Deduction rule: Given "all some one" applied to a class, you can extract the "exactly one" part. Note that the witness must satisfy the antecedent 𝜓, not merely be a member of 𝐴. (Contributed by David A. Wheeler, 22-Jul-2026.)
(𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)
 
22-Jul-2026ralseu1d 17290 Deduction rule: Given "all some one" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 22-Jul-2026.)
(𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
 
22-Jul-2026alseu2d 17289 Deduction rule: Given "all some one" applied to a top-level inference, you can extract the "exactly one" part. (Contributed by David A. Wheeler, 22-Jul-2026.)
(𝜑 → ∀∃!𝑥(𝜓 → 𝜒))    ⇒   (𝜑 → ∃!𝑥𝜓)
 
22-Jul-2026alseu1d 17288 Deduction rule: Given "all some one" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 22-Jul-2026.)
(𝜑 → ∀∃!𝑥(𝜓 → 𝜒))    ⇒   (𝜑 → ∀𝑥(𝜓 → 𝜒))
 
22-Jul-2026ralseud 17287 Introduction rule for "all some one" restricted to a class. This is the converse of ralseu1d 17290 and ralseu2d 17291 taken together. (Contributed by David A. Wheeler, 22-Jul-2026.)
(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))    &   (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)    ⇒   (𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))
 
22-Jul-2026alseud 17286 Introduction rule: "all some one" holds if the "for all" part holds and the antecedent has exactly one witness. This is the converse of alseu1d 17288 and alseu2d 17289 taken together. (Contributed by David A. Wheeler, 22-Jul-2026.)
(𝜑 → ∀𝑥(𝜓 → 𝜒))    &   (𝜑 → ∃!𝑥𝜓)    ⇒   (𝜑 → ∀∃!𝑥(𝜓 → 𝜒))
 
22-Jul-2026ralseurals 17285 "All some one" restricted to a class implies "all some" restricted to that class. Restricted counterpart of alseuals 17284. (Contributed by David A. Wheeler, 22-Jul-2026.)
(∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓))
 
22-Jul-2026alseuals 17284 "All some one" implies "all some": requiring exactly one witness is stronger than requiring at least one. Any consequence of an allsome statement is therefore a consequence of the corresponding "all some one" statement, which is how alseu-no-surprise 17298 is proved. (Contributed by David A. Wheeler, 22-Jul-2026.)
(∀∃!𝑥(𝜑 → 𝜓) → ∀∃𝑥(𝜑 → 𝜓))
 
22-Jul-2026dfralseu2 17283 The bounded "all some one" form is the general form with the class membership folded into the antecedent. This is the "all some one" counterpart of dfrals2 17249. (Contributed by David A. Wheeler, 22-Jul-2026.)
(∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃!𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
 
22-Jul-2026df-ralseu 17282 Define "all some one" applied to a class, which means 𝜓 is true whenever 𝜑 is true for 𝑥 in 𝐴, and exactly one 𝑥 in 𝐴 satisfies 𝜑. (Contributed by David A. Wheeler, 22-Jul-2026.)
(∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑥 ∈ 𝐴 𝜑))
 
22-Jul-2026df-alseu 17281 Define "all some one" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true and exactly one 𝑥 satisfies 𝜑. (Contributed by David A. Wheeler, 22-Jul-2026.)
(∀∃!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))
 
22-Jul-2026wralseu 17280 Extend wff definition to include "all some one" applied to a class, which means 𝜓 is true whenever 𝜑 is true for 𝑥 in 𝐴, and exactly one 𝑥 in 𝐴 satisfies 𝜑. (Contributed by David A. Wheeler, 22-Jul-2026.)
wff ∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓)
 
22-Jul-2026walseu 17279 Extend wff definition to include "all some one" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true, and exactly one 𝑥 satisfies 𝜑. (Contributed by David A. Wheeler, 22-Jul-2026.)
wff ∀∃!𝑥(𝜑 → 𝜓)
 
22-Jul-2026mptmex 5945 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 17278 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 17277 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 17276 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 17273 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 17272 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 17271 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 17270. (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 17270 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 17269 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 17275. (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 17268 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 17274. (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 17266 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 17247: 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 17257 Deduction rule: Given "all some" applied to a class, the class is inhabited. This is stronger than ralsn0d 17256, 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 4278 Separation Scheme (Aussonderung) in terms of a class abstraction. Prefer using the more natural statement rabexg 4279. (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 17275 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 17269 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 17274 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 17268 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 4581 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 4252 Inference associated with sepg 4251. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 14-Jul-2026.)
𝐴 ∈ V    ⇒   ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
 
13-Jul-2026f1setfi 7317 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 7316 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 17267 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 17266, and follows from it by dfrals2 17249. 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 17265 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 17264 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 17263 Bound-variable hypothesis builder for "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝜓    ⇒   Ⅎ𝑥∀∃𝑦(𝜑 → 𝜓)
 
12-Jul-2026alsbid 17262 Deduction form of alsbii 17260. (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 17261 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 17260 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 17259 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 17258. (Contributed by David A. Wheeler, 12-Jul-2026.)
(∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)
 
12-Jul-2026alsex 17258 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 17266, 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 17256 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 17255 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 17254 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 17251 Introduction rule for "all some" restricted to a class. This is the converse of rals1d 17254 and rals2d 17255 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 17250 Introduction rule: "all some" holds if the "for all" part holds and the antecedent has a witness. This is the converse of als1d 17252 and als2d 17253 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 17249 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 17248 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 17246 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 17245 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 14161 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 16879 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 16876 Lemma for dichmul0or 16879. (Contributed by Matthew House, 29-Jun-2026.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → ((abs‘𝐴) + 𝐴) = 0)    ⇒   (𝜑 → 𝐴 ≤ 0)
 
29-Jun-2026dichmul0orlem4 16875 Lemma for dichmul0or 16879. (Contributed by Matthew House, 29-Jun-2026.)
(𝜑 → 𝐴 ∈ ℝ)    ⇒   (𝜑 → (((abs‘𝐴) + 𝐴) · ((abs‘𝐴) − 𝐴)) = 0)
 
29-Jun-2026dichmul0orlem3 16874 Lemma for dichmul0or 16879. (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 16873 Lemma for dichmul0or 16879. (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 16872 Lemma for dichmul0or 16879. (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 16871 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 16870 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 16878 Lemma for dichmul0or 16879. (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 16877 Lemma for dichmul0or 16879. (Contributed by Matthew House, 28-Jun-2026.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → ((abs‘𝐴) − 𝐴) = 0)    ⇒   (𝜑 → 0 ≤ 𝐴)
 
28-Jun-2026msq0 8999 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 8998 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 8443 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 14671 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 14310, and semirings, see srgen1zr0 14341. (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 14340 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 14310 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 14309 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 14308 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 14307 Lemma for rng1zr 14308 and srg1zr 14340. (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 13331 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 13288 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 13287 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 13286 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 13317 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 13316 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 13276 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 13275 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 13297 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 13296 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 10680 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 10679 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)
 
10-Jun-202610nprm 13248 10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
¬ 10 ∈ ℙ
 
10-Jun-202611nn 9805 11 is a positive integer. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
11 ∈ ℕ
 
10-Jun-202625nn0 9799 25 is a nonnegative integer. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
25 ∈ ℕ0
 
10-Jun-202616nn0 9798 16 is a nonnegative integer. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
16 ∈ ℕ0
 
10-Jun-202612nn0 9797 12 is a nonnegative integer. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
12 ∈ ℕ0
 
10-Jun-202611nn0 9796 11 is a nonnegative integer. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
11 ∈ ℕ0
 
10-Jun-20262t4e8 9467 2 times 4 equals 8. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
(2 · 4) = 8
 
10-Jun-20262t3e6 9464 2 times 3 equals 6. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
(2 · 3) = 6
 
8-Jun-2026ballotfilemdifcfi 13274 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 13273 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 9727 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 13272 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 11283 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 7320. For the number of subsets (which need not be finite) of a set, see pw1mapen 17145. (Contributed by Jim Kingdon, 5-Jun-2026.)
(𝐴 ∈ Fin → (♯‘(𝒫 𝐴 ∩ Fin)) = (2↑(♯‘𝐴)))
 
4-Jun-2026ballotfilemonn 13270 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 7610 Equality theorem for apartness predicate. (Contributed by Jim Kingdon, 3-Jun-2026.)
(𝐴 = 𝐵 → (𝑅 Ap 𝐴 ↔ 𝑅 Ap 𝐵))
 
3-Jun-2026papeq1 7609 Equality theorem for apartness predicate. (Contributed by Jim Kingdon, 3-Jun-2026.)
(𝑅 = 𝑆 → (𝑅 Ap 𝐴 ↔ 𝑆 Ap 𝐴))
 
2-Jun-2026resq01 11108 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 14650 If two structures have the same ring components (properties), df-apr 14639 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 14643 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 14649). 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 14642 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 14656 A division ring is a local ring. (Contributed by Jim Kingdon, 29-May-2026.)
(𝑅 ∈ DivRing → 𝑅 ∈ LRing)
 
29-May-2026isdrngtap 14655 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 14653 Define class of all division rings. A division ring is a ring in which the relation given by df-apr 14639 is a tight apartness. (Contributed by Jim Kingdon, 29-May-2026.)
DivRing = {𝑟 ∈ Ring ∣ (#r‘𝑟) TAp (Base‘𝑟)}
 
29-May-2026aprunit 14641 The df-apr 14639 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 7616 A tight apartness is an apartness. (Contributed by Jim Kingdon, 29-May-2026.)
(𝑅 TAp 𝐴 → 𝑅 Ap 𝐴)
 
28-May-2026aprlring 14649 A ring is a local ring if and only if the relation given by df-apr 14639 is an apartness relation. (Contributed by Jim Kingdon, 28-May-2026.)
(𝑅 ∈ Ring → (𝑅 ∈ LRing ↔ (#r‘𝑅) Ap (Base‘𝑅)))
 
28-May-2026papcotr 7613 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 14648 If the relation given by df-apr 14639 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 7612 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 7611 An apartness is irreflexive. (Contributed by Jim Kingdon, 27-May-2026.)
((𝑅 Ap 𝐴 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋𝑅𝑋)
 
24-May-2026gsumzfi 14207 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 11295 Subsets of a class of a negative size (a degenerate case). Together with ssenneg 11294 this shows that sseqn 11293 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 11294 Subsets of a class of a negative size (a degenerate case). Together with sshashneg 11295 this shows that sseqn 11293 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 11293 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 11234 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 11294 and sshashneg 11295. (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 13268 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 7321 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 7320 The number of finite subsets of a finite set. For a similar theorem with set size expressed using ♯ (df-ihash 11229), see hashpwfi 11283. (Contributed by Jim Kingdon, 18-May-2026.)
(𝐴 ∈ Fin → (2o ↑𝑚 𝐴) ≈ (𝒫 𝐴 ∩ Fin))
 
18-May-2026fissfi 7263 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 5571 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 5570 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 7301 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 7246. (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 10416 A linear combination of two reals which lies in the interval between them. Like lincmb01cmp 10415 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 7606 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 17146).

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 4295 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 4627. (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 17196 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 17194 and repiecege0 17195. 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. 17195 (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 17195 Piecewise definition on the reals agrees with the nonnegative part of the definition. See repiecef 17196 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 17194 Piecewise definition on the reals agrees with the nonpositive part of the definition. See repiecef 17196 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 17193 Lemma for repiecele0 17194, repiecege0 17195, and repiecef 17196. 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 17217 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 17216 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 17157 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 17156 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 17155 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 11312 Lemma for hashtpg 11313. 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 11311 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 16869 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 16868 Lemma for depind 16869. (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 16867 Lemma for depind 16869. (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 16866 Lemma for depind 16869. (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))))
 
11-Apr-2026indval0 9299 The indicator function generator does not generate a (meaningful) indicator function for a class which is not a subset of the domain. (Contributed by AV, 11-Apr-2026.)
(¬ 𝐴 ⊆ 𝑂 → ((𝟭‘𝑂)‘𝐴) = ∅)
 
8-Apr-2026gsumclfi 14208 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 14197 Splitting off the rightmost summand of a group sum (even if it is the only summand). Similar to gzsumsplit1r 13764 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 12158 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 14206 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 14205 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 7544 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 7240 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 14203 An empty finite group sum is the identity. (Contributed by Jim Kingdon, 26-Mar-2026.)
(𝐺 ∈ CMnd → (𝐺 Σg ∅) = (0g‘𝐺))
 
26-Mar-2026gzsumshift 14198 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 14204 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 14202 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 14201 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 14200 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 13662. (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 7246 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 16485 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 16484 A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 16481 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 16747 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 16806 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 16667 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 11253 A set equinumerous to the ordinal one has size 1 . (Contributed by Jim Kingdon, 11-Mar-2026.)
(𝐴 ≈ 1o → (♯‘𝐴) = 1)
 
4-Mar-2026elmpom 6474 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 16767 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 16758 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 16653 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 16650 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 16649 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 7210 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 17140 The powerset of 1o is not infinite. Since we cannot prove it is finite (see pw1fin 7217), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7213. (Contributed by Jim Kingdon, 14-Feb-2026.)
¬ ω ≼ 𝒫 1o
 
14-Feb-2026pw1ndom3 17139 The powerset of 1o does not dominate 3o. This is another way of saying that 𝒫 1o does not have three elements (like pwntru 4336). (Contributed by Steven Nguyen and Jim Kingdon, 14-Feb-2026.)
¬ 3o ≼ 𝒫 1o
 
14-Feb-2026pw1ndom3lem 17138 Lemma for pw1ndom3 17139. (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 17154 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 17137 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 7209 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 16651 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 7203 Equinumerosity of finite sets is decidable. (Contributed by Jim Kingdon, 10-Feb-2026.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → DECID 𝐴 ≈ 𝐵)
 
8-Feb-2026wlkvtxm 16700 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 16744 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 16685 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 7117 A set with an element dominates one. (Contributed by Jim Kingdon, 3-Feb-2026.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → 1o ≼ 𝐴)
 
2-Feb-2026edginwlkd 16715 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 16709 A walk is an ordered pair. (Contributed by Jim Kingdon, 2-Feb-2026.)
(𝑊 ∈ (Walks‘𝐺) → 𝑊 ∈ (V × V))
 
1-Feb-2026wlkcprim 16710 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 16679 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 5731 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 5730 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 5000 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-2026indconst1 9305 Indicator of the whole set. (Contributed by Thierry Arnoux, 25-Jan-2026.)
(𝑂 ∈ 𝑉 → ((𝟭‘𝑂)‘𝑂) = (𝑂 × {1}))
 
25-Jan-2026indconst0 9304 Indicator of the empty set. (Contributed by Thierry Arnoux, 25-Jan-2026.)
(𝑂 ∈ 𝑉 → ((𝟭‘𝑂)‘∅) = (𝑂 × {0}))
 
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 11552 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 11551 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 11555 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 11554 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 11553 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 16515 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 7539 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 17145 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 7584 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 7583 A truth value which is inhabited is equal to true. This is a variation of pwntru 4336 and pwtrufal 17146. (Contributed by Jim Kingdon, 10-Jan-2026.)
((𝐴 ∈ 𝒫 1o ∧ ∃𝑥 𝑥 ∈ 𝐴) → 𝐴 = 1o)
 
10-Jan-20261ndom2 7166 Two is not dominated by one. (Contributed by Jim Kingdon, 10-Jan-2026.)
¬ 2o ≼ 1o
 
9-Jan-2026pw1map 17144 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 7582 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 11465 Closure of the prefix extractor. This extends pfxclg 11464 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 11463 The domain of the prefix extractor. (Contributed by Jim Kingdon, 8-Jan-2026.)
prefix Fn (V × ℕ0)
 
7-Jan-2026pr1or2 7540 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 16480 Lemma for upgr1edc 16481. (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 16454 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 16453 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 16454). (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 7116 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 7113 A set with two elements is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
(𝐴 ≈ 2o → ∃𝑥 𝑥 ∈ 𝐴)
 
3-Jan-2026en1m 7092 A set with one element is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
(𝐴 ≈ 1o → ∃𝑥 𝑥 ∈ 𝐴)
 
31-Dec-2025pw0ss 16443 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 16430 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 16429 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 16379 Applying the indexed edge function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
(𝐺 ∈ 𝑉 → (iEdg‘𝐺) ∈ V)
 
29-Dec-2025vtxex 16378 Applying the vertex function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
(𝐺 ∈ 𝑉 → (Vtx‘𝐺) ∈ V)
 
29-Dec-2025snmb 3834 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 11370 Existence of the last symbol. The last symbol of a word is a set. See lsw0g 11367 or lswcl 11369 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 11433 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 11376 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 11365 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 13521 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 16392 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 16391 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 16390 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 16389 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 11309 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 17185 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 9976 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 15116 Simpler form of psrelbas 15115 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 15140 Lemma for mplsubgfi 15141. 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 15139 Lemma for mplsubgfi 15141. 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 7520 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 7473). (Contributed by Jim Kingdon, 25-Nov-2025.)
(∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ ω ∈ WOmni)
 
23-Nov-2025psrbagfi 15108 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 7525 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 15138 Lemma for mplsubgfi 15141. 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 9974 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 7319 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 7318 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 17150 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 14223 Lemma for prdsbas 14225 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 15133 mPoly has universal domain. (Contributed by Jim Kingdon, 5-Nov-2025.)
mPoly Fn (V × V)
 
4-Nov-2025mplelbascoe 15132 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 15131 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 15130 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 7534 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 12730 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 12006 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 10691 Rational ≤ is decidable. (Contributed by Jim Kingdon, 28-Oct-2025.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 ≤ 𝐵)
 
17-Oct-2025plycoeid3 15907 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 7237 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 7235 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 13037 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 15097 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 15848 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 15847 Real derivative of the identity function. (Contributed by Jim Kingdon, 3-Oct-2025.)
(ℝ D ( I ↾ ℝ)) = (ℝ × {1})
 
3-Oct-2025dvconstre 15846 Real derivative of a constant function. (Contributed by Jim Kingdon, 3-Oct-2025.)
(𝐴 ∈ ℂ → (ℝ D (ℝ × {𝐴})) = (ℝ × {0}))
 
3-Oct-2025dvidsslem 15843 Lemma for dvconstss 15848. Analogue of dvidlemap 15841 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 15842 Lemma for dvidre 15847 and dvconstre 15846. Analogue of dvidlemap 15841 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 14941 Applying metUnif yields a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
(𝐴 ∈ 𝑉 → (metUnif‘𝐴) ∈ V)
 
28-Sep-2025cndsex 14939 The standard distance function on the complex numbers is a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
(abs ∘ − ) ∈ V
 
25-Sep-2025cntopex 14940 The standard topology on the complex numbers is a set. (Contributed by Jim Kingdon, 25-Sep-2025.)
(MetOpen‘(abs ∘ − )) ∈ V
 
24-Sep-2025mopnset 14938 Getting a set by applying MetOpen. (Contributed by Jim Kingdon, 24-Sep-2025.)
(𝐷 ∈ 𝑉 → (MetOpen‘𝐷) ∈ V)
 
24-Sep-2025blfn 14937 The ball function has universal domain. (Contributed by Jim Kingdon, 24-Sep-2025.)
ball Fn V
 
23-Sep-2025elfzoext 10620 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 15910 Lemma for plycj 15911. (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 15075 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 15908 Lemma for plyco 15909. 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 10619 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 16314 Lemma for lgsquad 16318. 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 16313 Lemma for lgsquad 16318. 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 7247 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 6371 Principle of unique choice. This is also called non-choice. The name choice results in its similarity to something like acfun 7563 (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 15059 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 15058 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 14991 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 14973 The invertible complex numbers are exactly those apart from zero. This is recapb 9003 but expressed in terms of ℂfld. (Contributed by Jim Kingdon, 11-Sep-2025.)
{𝑧 ∈ ℂ ∣ 𝑧 # 0} = (Unit‘ℂfld)
 
9-Sep-2025gzsummhm2 14195 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 14194 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 12718 5 does not divide 6. (Contributed by AV, 8-Sep-2025.)
¬ 5 ∥ 6
 
8-Sep-20255ndvds3 12717 5 does not divide 3. (Contributed by AV, 8-Sep-2025.)
¬ 5 ∥ 3
 
7-Sep-20255eluz3 9970 5 is an integer greater than or equal to 3. (Contributed by AV, 7-Sep-2025.)
5 ∈ (ℤ≥‘3)
 
6-Sep-2025gzsumconst 14192 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 9973 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 14191 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 10924 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 10971 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 10058 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 10057. (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 10916 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 10913 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 11325 A zero-based sequence is a word. In iswrdinn0 11323 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 13853 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 11323 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 13760 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 15041 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 14618 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 16297 Lemma for gausslemma2dlem1 16299. 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 16298 Lemma for gausslemma2dlem1 16299. (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 10690 Rational < is decidable. (Contributed by Jim Kingdon, 7-Aug-2025.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 < 𝐵)
 
22-Jul-2025ivthdich 15803 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 15793 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 15796, hovera 15797, and hoverb 15798, 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 8393, 𝑐 < 1 or 0 < 𝑐, and that leads to 𝑧 ≤ 0 by hoverlt1 15799 or 0 ≤ 𝑧 by hovergt0 15800. (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 15802 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 15801 Lemma for ivthdich 15803. 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 15800 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 15799 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 15798 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 15797 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 15796 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 15766 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 15765 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 10887 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 7463 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 12833 Lemma for nninfct 12834. (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 17183 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 12834 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 10893 ℕ∞ is infinte. (Contributed by Jim Kingdon, 8-Jul-2025.)
ω ≼ ℕ∞
 
7-Jul-2025ivthreinc 15795 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 15793). 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 13757 Iterated sum has a universal domain. (Contributed by Jim Kingdon, 28-Jun-2025.)
Σgz Fn (V × V)
 
28-Jun-2025iotaexel 6043 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 13662 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 14200 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 14541 The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr 14542. (Contributed by SN, 20-Jun-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing))
 
16-Jun-2025fnpsr 15100 The multivariate power series constructor has a universal domain. (Contributed by Jim Kingdon, 16-Jun-2025.)
mPwSer Fn (V × V)
 
14-Jun-2025basm 13463 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 5729 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 13111 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 10712 "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 13198 Exercise which may help in understanding the proof of 4sqlemsdc 13199. (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 5356 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 13199 Lemma for 4sq 13209. 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 13197 and 4sqexercise2 13198 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 13197 Exercise which may help in understanding the proof of 4sqlemsdc 13199. (Contributed by Jim Kingdon, 25-May-2025.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ 𝑛 = (𝑥↑2)}    ⇒   (𝐴 ∈ ℕ0 → DECID 𝐴 ∈ 𝑆)
 
24-May-20254sqleminfi 13196 Lemma for 4sq 13209. 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 13195 Lemma for 4sq 13209. 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 13194 Lemma for 4sq 13209. 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 7248 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 15005 Set existence for ℤRHom. (Contributed by Jim Kingdon, 19-May-2025.)
𝐿 = (ℤRHom‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → 𝐿 ∈ V)
 
16-May-2025rhmex 14513 Set existence for ring homomorphism. (Contributed by Jim Kingdon, 16-May-2025.)
((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 RingHom 𝑆) ∈ V)
 
15-May-2025ghmex 14107 The set of group homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)
((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) ∈ V)
 
15-May-2025mhmex 13818 The set of monoid homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)
((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 MndHom 𝑇) ∈ V)
 
14-May-2025idomcringd 14636 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 14626 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 14302 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 13474. (Contributed by Jim Kingdon, 5-May-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Rng → (𝐺 ↾s 𝐵) ∈ Rng)
 
5-May-2025ablressid 14188 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 13474. (Contributed by Jim Kingdon, 5-May-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Abel → (𝐺 ↾s 𝐵) ∈ Abel)
 
30-Apr-2025dvply2g 15916 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 14846 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 13473 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 14741 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 14950 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 14948 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 14859 Existence of the set of left ideals. (Contributed by Jim Kingdon, 27-Apr-2025.)
(𝑊 ∈ 𝑉 → (LIdeal‘𝑊) ∈ V)
 
27-Apr-2025lssex 14740 Existence of a linear subspace. (Contributed by Jim Kingdon, 27-Apr-2025.)
(𝑊 ∈ 𝑉 → (LSubSp‘𝑊) ∈ V)
 
25-Apr-2025rspex 14860 Existence of the ring span. (Contributed by Jim Kingdon, 25-Apr-2025.)
(𝑊 ∈ 𝑉 → (RSpan‘𝑊) ∈ V)
 
25-Apr-2025lspex 14781 Existence of the span of a set of vectors. (Contributed by Jim Kingdon, 25-Apr-2025.)
(𝑊 ∈ 𝑋 → (LSpan‘𝑊) ∈ V)
 
25-Apr-2025eqgex 14073 The left coset equivalence relation exists. (Contributed by Jim Kingdon, 25-Apr-2025.)
((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐺 ~QG 𝑆) ∈ V)
 
25-Apr-2025qusex 13695 Existence of a quotient structure. (Contributed by Jim Kingdon, 25-Apr-2025.)
((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) ∈ V)
 
23-Apr-20251dom1el 7107 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 13975 Existence of the group multiple operation. (Contributed by Jim Kingdon, 22-Apr-2025.)
(𝐺 ∈ 𝑉 → (.g‘𝐺) ∈ V)
 
21-Apr-2025uspgruhgr 16547 An undirected simple pseudograph is an undirected hypergraph. (Contributed by AV, 21-Apr-2025.)
(𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
 
20-Apr-2025uspgriedgedg 16539 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 16538 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 6287 Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 6288 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 5746 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 16204 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 8302. (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 16203 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 14895 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 14887 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 14874 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 14861 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 14815 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 14832 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 13880 A group is a magma, deduction form. (Contributed by SN, 14-Apr-2025.)
(𝜑 → 𝐺 ∈ Grp)    ⇒   (𝜑 → 𝐺 ∈ Mgm)
 
12-Apr-2025psraddcl 15120 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 17228 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 14125 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 14919 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 14954 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 14943. (Revised by GG, 31-Mar-2025.)
(abs ∘ − ) = (dist‘ℂfld)
 
31-Mar-2025cnfldle 14953 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 14943. (Revised by GG, 31-Mar-2025.)
≤ = (le‘ℂfld)
 
31-Mar-2025cnfldtset 14952 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 14949 The multiplication operation of the field of complex numbers. Version of cnfldmul 14950 using maps-to notation, which does not require ax-mulf 8302. (Contributed by GG, 31-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (.r‘ℂfld)
 
31-Mar-2025mpocnfldadd 14947 The addition operation of the field of complex numbers. Version of cnfldadd 14948 using maps-to notation, which does not require ax-addf 8301. (Contributed by GG, 31-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (+g‘ℂfld)
 
31-Mar-2025df-cnfld 14943 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 14945, cnfldadd 14948, cnfldmul 14950, cnfldcj 14951, cnfldtset 14952, cnfldle 14953, cnfldds 14954, and cnfldbas 14946. 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 14910 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 14637 An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
(𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → 𝑅 ∈ Ring)
 
22-Mar-2025idomdomd 14635 An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
(𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → 𝑅 ∈ Domn)
 
21-Mar-2025df2idl2rng 14894 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 14868 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 14867 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 13641 Slot property of comp. (Contributed by Jim Kingdon, 20-Mar-2025.)
(comp = Slot (comp‘ndx) ∧ (comp‘ndx) ∈ ℕ)
 
20-Mar-2025homslid 13638 Slot property of Hom. (Contributed by Jim Kingdon, 20-Mar-2025.)
(Hom = Slot (Hom ‘ndx) ∧ (Hom ‘ndx) ∈ ℕ)
 
19-Mar-2025ptex 13667 Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.)
(𝐹 ∈ 𝑉 → (∏t‘𝐹) ∈ V)
 
18-Mar-2025prdsex 14221 Existence of the structure product. (Contributed by Jim Kingdon, 18-Mar-2025.)
((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑆Xs𝑅) ∈ V)
 
16-Mar-2025plycn 15912 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8302. (Revised by GG, 16-Mar-2025.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))
 
16-Mar-2025expcn 15719 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 8302. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpen‘ℂfld)    ⇒   (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽))
 
16-Mar-2025mpomulcn 15716 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 8316 Multiplication is an operation on complex numbers. Version of ax-mulf 8302 using maps-to notation, proved from the axioms of set theory and ax-mulcl 8277. (Contributed by GG, 16-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
 
15-Mar-2025lmodvscld 14691 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 14900 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 13675 Existence of the image structure. (Contributed by Jim Kingdon, 13-Mar-2025.)
((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐹 “s 𝑅) ∈ V)
 
11-Mar-2025rng2idlsubgsubrng 14906 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 14903 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 14884 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 14883 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 14882 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 13676 Value of an image structure. The is a lemma for the theorems imasbas 13677, imasplusg 13678, and imasmulr 13679 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 14893 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 14892 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 14085 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 6194 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 14028 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 7294 A finitely supported function is a function. (Contributed by SN, 8-Mar-2025.)
(𝜑 → 𝐹 finSupp 𝑍)    ⇒   (𝜑 → Fun 𝐹)
 
7-Mar-2025ringrzd 14400 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 14399 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 14184 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 15060 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 15057 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 15051 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 15048 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 14082 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 14918 Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 14919. Similar to qusmul2 14915. (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 14417 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 13474. (Contributed by Jim Kingdon, 28-Feb-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Ring)
 
28-Feb-2025grpressid 13915 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 13474. (Contributed by Jim Kingdon, 28-Feb-2025.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp)
 
27-Feb-2025imasringf1 14419 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 13508 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 14562 A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.)
(𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅))
 
25-Feb-2025rngansg 14298 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 14090 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 14091 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 14083 Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) Generalization of quseccl 14085 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 17239 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 17237). (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 17238 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 14911 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 14913 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 14909 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 14552 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 14551 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 14550 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 14549 A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
(𝑅 ∈ LRing → 𝑅 ∈ NzRing)
 
23-Feb-2025islring 14548 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 14547 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 14545 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 14539 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 14306 The quotient structure of a non-unital ring is a non-unital ring (qusring2 14420 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 14300 Ring multiplication distributes over subtraction. (subdir 8714 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 14411. (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 14299 Ring multiplication distributes over subtraction. (subdi 8713 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdi 14410. (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 14305 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 14304 The image structure of a non-unital ring is a non-unital ring (imasring 14418 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 14285 Restricted functionality of the multiplicative group on non-unital rings (mgpf 14364 analog). (Contributed by AV, 22-Feb-2025.)
(mulGrp ↾ Rng):Rng⟶Smgrp
 
22-Feb-2025imasabl 14189 The image structure of an abelian group is an abelian group (imasgrp 13963 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 14240 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 14239 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 7617 Tight apartness with the apartness properties from df-pap 7608 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 14908 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 14907 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 14905 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 14904 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 14902 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 14901 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 14891 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 14647 The relation given by df-apr 14639 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 13442 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 14866 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 14646 The apartness relation given by df-apr 14639 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 14645 The apartness relation given by df-apr 14639 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 14640 Expand Definition df-apr 14639. (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 14573 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 14297 Double negation of a product in a non-unital ring (mul2neg 8726 analog). (Contributed by Mario Carneiro, 4-Dec-2014.) Generalization of ringm2neg 14409. (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 14296 Negation of a product in a non-unital ring (mulneg2 8724 analog). In contrast to ringmneg2 14408, 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 14295 Negation of a product in a non-unital ring (mulneg1 8723 analog). In contrast to ringmneg1 14407, 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 14644 The apartness relation given by df-apr 14639 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 14294 The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 14398. (Revised by AV, 16-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )
 
16-Feb-2025rng0cl 14291 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 14290 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 14286 A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)
(𝑅 ∈ Rng → 𝑅 ∈ Grp)
 
16-Feb-2025aptap 8980 Complex apartness (as defined at df-ap 8912) is a tight apartness (as defined at df-tap 7615). (Contributed by Jim Kingdon, 16-Feb-2025.)
# TAp ℂ
 
15-Feb-2025subsubrng2 14572 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 14571 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 14570 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 14569 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 14568 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 14567 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 14432 A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 14431. (Contributed by AV, 15-Feb-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → (𝑅 ∈ Rng ↔ 𝑂 ∈ Rng))
 
15-Feb-2025opprrng 14431 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 14303 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 13777 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 13773 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 7619 Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.)
(𝐴 = 𝐵 → (𝑅 TAp 𝐴 ↔ 𝑅 TAp 𝐵))
 
14-Feb-2025subrngmcl 14566 A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 14590. (Revised by AV, 14-Feb-2025.)
· = (.r‘𝑅)    ⇒   ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴)
 
14-Feb-2025subrngacl 14565 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 14564 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 14563 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 14561 A subring is a subgroup. (Contributed by AV, 14-Feb-2025.)
(𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
 
14-Feb-2025subrngrcl 14560 Reverse closure for a subring predicate. (Contributed by AV, 14-Feb-2025.)
(𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
 
14-Feb-2025subrngrng 14559 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 14558 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 14557 A subring is a subset. (Contributed by AV, 14-Feb-2025.)
𝐵 = (Base‘𝑅)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵)
 
14-Feb-2025issubrng 14556 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 14555 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 14301 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 14288 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 7627 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 7626 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 7608 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 14899 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 14898 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 14897 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 14896 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 14890 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 14639 The relation between elements whose difference is invertible, which for a local ring is an apartness relation by aprap 14647. (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 14287 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 13776 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 4252 to inex1 4267 (look at the instance of sepgi 4252 that occurs in the proof of inex1 4267). 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 7622 ∅ 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 7618 Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.)
(𝑅 = 𝑆 → (𝑅 TAp 𝐴 ↔ 𝑆 TAp 𝐴))
 
7-Feb-2025psrgrp 15125 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 14606 Restriction of the codomain of a (ring) homomorphism. resghm2b 14114 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 11159 An integer is less than or equal to its square. (Contributed by BJ, 6-Feb-2025.)
(𝑁 ∈ ℤ → 𝑁 ≤ (𝑁↑2))
 
6-Feb-20252omotap 7625 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 7624 Lemma for 2omotap 7625. (Contributed by Jim Kingdon, 6-Feb-2025.)
((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → 𝜑)
 
6-Feb-20252omotaplemap 7623 Lemma for 2omotap 7625. (Contributed by Jim Kingdon, 6-Feb-2025.)
(¬ ¬ 𝜑 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} TAp 2o)
 
6-Feb-20252onetap 7621 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 7620 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 7615 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 13995 Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 13990. (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 14051 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 13910 The right inverse of a group element. Deduction associated with grprinv 13905. (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 13909 The left inverse of a group element. Deduction associated with grplinv 13904. (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 13903 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 13888 The identity element of a group is a right identity. Deduction associated with grprid 13886. (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 13887 The identity element of a group is a left identity. Deduction associated with grplid 13885. (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 13866 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 14454 Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)
(𝑅 ∈ SRing → (∥r‘𝑅) ∈ V)
 
24-Jan-2025reldvdsrsrg 14448 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 9175 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 9003 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 14664 A field is a division ring. (Contributed by SN, 17-Jan-2025.)
(𝜑 → 𝑅 ∈ Field)    ⇒   (𝜑 → 𝑅 ∈ DivRing)
 
17-Jan-2025ressval3d 13475 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 13474 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 4350 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 4259. (Revised by BJ, 17-Jan-2025.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))
 
16-Jan-2025ressex 13468 Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) ∈ V)
 
16-Jan-2025ressvalsets 13467 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 4348 A singleton built on a setvar is a set. (Contributed by BJ, 15-Jan-2025.)
{𝑥} ∈ V
 
12-Jan-2025isrim 14525 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 14527 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 14517 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 14427 Existence of the opposite ring. If you know that 𝑅 is a ring, see opprring 14433. (Contributed by Jim Kingdon, 10-Jan-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
 
10-Jan-2025mgpex 14272 Existence of the multiplication group. If 𝑅 is known to be a semiring, see srgmgp 14321. (Contributed by Jim Kingdon, 10-Jan-2025.)
𝑀 = (mulGrp‘𝑅)    ⇒   (𝑅 ∈ 𝑉 → 𝑀 ∈ V)
 
6-Jan-2025ord3 6699 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 3850 The singleton of an element of a class is a subset of the class (inference form of snssg 3849). 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 3849 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 3848 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 7110 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 7106 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 4425 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 7519 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 7518 Lemma for nninfinfwlpo 7520. (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 7517 Lemma for nninfwlpoim 7519. (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 7516 Lemma for nninfwlpoim 7519. (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 7515 Lemma for nninfwlpoim 7519. (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 7514 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 7513 Lemma for nninfwlpor 7514. 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 7064 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 7039 The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 7044 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 7038 The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 7043 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 7512 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 7521 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 7511 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 7031 Dominance relation. This variation of brdomg 7032 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 7032. (Revised by BTernaryTau, 29-Nov-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
 
28-Nov-2024basmexd 13462 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 14665 A field is a commutative ring. (Contributed by SN, 23-Nov-2024.)
(𝜑 → 𝑅 ∈ Field)    ⇒   (𝜑 → 𝑅 ∈ CRing)
 
22-Nov-2024eliotaeu 5366 An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.)
(𝐴 ∈ (℩𝑥𝜑) → ∃!𝑥𝜑)
 
22-Nov-2024eliota 5365 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 13461 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 14829 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 14828 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 13578 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 16898 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 13628 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 13617 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 13616 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 13613 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 13597 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 6304 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 13635 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 13470 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 14430 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 14429 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 14428 Lemma for opprbasg 14429 and oppraddg 14430. (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 16243 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 15026 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 15025 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 15024 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 15023 Lemma for znbas 15028. (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 15015 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 15014 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 15013 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 15012 Lemma for zlmbasg 15013 and zlmplusgg 15014. (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 15016 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 13612 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 13611 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 13610 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 11100 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 13625 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 13596 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 13594 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 13593 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 13592 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 16370 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 13608 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 13607 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 13606 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 14834 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 14831 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 14827 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 14826 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 14825 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 14824 Lemma for srabaseg 14825 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 13627 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 13626 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 13598 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 13577 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 13576 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 13564 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 13559 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 11287 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 11286 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 11285 Lemma for fiubz 11286 and fiubnn 11287. 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 16369 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 13634 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 13632 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 13631 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 13623 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 13622 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 13621 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 16890 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 17135 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 16885 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 16884 as its last step. (Contributed by BJ, 27-Oct-2024.)
(𝜑 → 𝜓)    &   (¬ 𝜑 → 𝜓)    ⇒    ¬ ¬ 𝜓
 
25-Oct-2024nnwosdc 12832 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 12829 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 12830 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 17135 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 13633 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 13575 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 13557 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 12936 Lemma for isprm5 12937. 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 13476 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 14737 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 14674 of a left module, see also islmod 14676. (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 14279 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 13624 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 13609 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 13595 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 13565 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 13563 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 13562 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 13558 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 13547 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 13546 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 13545 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 13516 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 13514 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 10021 Membership of an integer in ℕ is decidable. (Contributed by Jim Kingdon, 17-Oct-2024.)
(𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ)
 
14-Oct-20242zinfmin 12025 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 12024 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 16368 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 16367 Index value of the df-edgf 16365 slot. (Contributed by AV, 13-Oct-2024.) (New usage is discouraged.)
(.ef‘ndx) = 18
 
13-Oct-2024prdsvallem 13670 Lemma for prdsval 14222. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 14222, dependency on df-hom 13504 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 13109 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 10215 Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
 
13-Oct-2024xnn0dcle 10214 Decidability of ≤ for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → DECID 𝐴 ≤ 𝐵)
 
9-Oct-2024nn0leexp2 11162 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 13087 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 10020 Membership of an integer in ℕ0 is decidable. (Contributed by Jim Kingdon, 8-Oct-2024.)
(𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ0)
 
7-Oct-2024pclemub 13086 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 13085 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 11161 Special case of ltexp2 16096 which we use here because we haven't yet defined df-rpcxp 16010 which is used in the current proof of ltexp2 16096. (Contributed by Jim Kingdon, 7-Oct-2024.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁)))
 
6-Oct-2024suprzcl2dc 10684 The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 8300.) (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 10683 An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 8300.) (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 10681 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 7374 Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
(𝐴 ∈ 𝐶 → inf(𝐵, 𝐴, 𝑅) ∈ V)
 
30-Sep-2024unbendc 13394 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 12924 Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.)
(𝑁 ∈ ℕ → DECID 𝑁 ∈ ℙ)
 
30-Sep-2024dcfi 7315 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 6049 Change bound variable in a restricted description binder. Version of cbvriotav 6051 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 5343 Change bound variables in a description binder. Version of cbviotav 5344 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 13387 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 13386 Lemma for ssnnct 13387. 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 10682 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 10007 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 13390 Lemma for nninfdc 13393. 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 12828 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 12827. (Contributed by Jim Kingdon, 26-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝐵)
 
25-Sep-2024nninfdclemcl 13388 Lemma for nninfdc 13393. (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 13391 Lemma for nninfdc 13393. The function from nninfdclemf 13389 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 13393 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 13392 Lemma for nninfdc 13393. The function from nninfdclemf 13389 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 13389 Lemma for nninfdc 13393. 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 12827 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 7029 Equinumerosity relation. This variation of bren 7030 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 7030. (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 13385 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 6704 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 5894 Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 5552. (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 5893 Composition of a function with domain and a function as a function with domain. Generalization of fnco 5491. (Contributed by AV, 17-Sep-2024.)
((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))
 
14-Sep-2024nnpredlt 4771 The predecessor (see nnpredcl 4770) of a nonzero natural number is less than (see df-iord 4511) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴)
 
13-Sep-2024nninfisollemeq 7472 Lemma for nninfisol 7473. 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 7471 Lemma for nninfisol 7473. 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 7470 Lemma for nninfisol 7473. 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 6949 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 6948 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 7473 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 7520). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.)

((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
 
8-Sep-2024relopabv 4904 A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopab 4906. (Contributed by SN, 8-Sep-2024.)
Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
7-Sep-2024eulerthlemfi 13026 Lemma for eulerth 13031. 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 11117 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 13029 Lemma for eulerth 13031. 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 13030 Lemma for eulerth 13031. 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 13028 Lemma for eulerth 13031. (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 13027 Lemma for eulerth 13031. 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 14915 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 12419 A finite product of terms apart from zero is apart from zero. A version of fprodap0 12404 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 12412 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 7602 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 7598 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 4719 Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4718. (Contributed by Jim Kingdon, 26-Aug-2024.)
(∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → DECID 𝜑)
 
26-Aug-2024ontriexmidim 4669 Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4668. (Contributed by Jim Kingdon, 26-Aug-2024.)
(∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → DECID 𝜑)
 
25-Aug-2024onntri2or 7605 Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
(¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
 
25-Aug-2024onntri3or 7604 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 12767 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 6950 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 12612 Deduction form of dvds2add 12608. (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 14227 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 14226 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 14225 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 14224 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 14222 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 14219 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 12388 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 12387 Multiply in an additional term in a finite product. See also fprodsplitsn 12416 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 16954 Alternate proof of bj-charfundc 16953. It was expected to be much shorter since it uses bj-charfun 16952 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 16952 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 8975 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 8974 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 4630 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 4629 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 4628 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 4627 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 11299 Lemma for hashf1 11301. (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 6951 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 7578 Lemma for exmidontriim 7581. (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 7577 Lemma for exmidontriim 7581. 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 4332): 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 7581 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 7580 Lemma for exmidontriim 7581. 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 7579 Lemma for exmidontriim 7581. 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 7467 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 7464 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 4822 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 6946 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 7220 Reformulation of ss1o0el1 4334 using 1o instead of {∅}. (Contributed by BJ, 9-Aug-2024.)
(𝐴 ⊆ 1o → (∅ ∈ 𝐴 ↔ 𝐴 = 1o))
 
9-Aug-2024pw1dc0el 7218 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 4334 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 7221 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 6945 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 6932 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 6950), 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 5939 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 5938 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 15110 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 15104 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 7217 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 6947 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 4752 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 4753. (Revised by BJ, 7-Aug-2024.)
(𝐴 ∈ ω → 𝐴 ⊆ ω)
 
6-Aug-2024bj-charfunbi 16956 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 16955 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 16953 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 15113 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 15112 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 12372 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 16951 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 16950 The maps-to notation defines a function (deduction form).

Note: one should similarly prove a deduction form of funopab4 5414, then prove funmptd 16950 from it, and then prove funmpt 5415 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 16902 The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID ⊥
 
5-Aug-2024bj-dctru 16900 The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
DECID ⊤
 
5-Aug-2024bj-stfal 16889 The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)
STAB ⊥
 
5-Aug-2024bj-sttru 16887 The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)
STAB ⊤
 
5-Aug-2024psrbagcon 15111 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 15109 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 15107 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 12369 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 9622 Version of fcdmnn0fsupp 9620 avoiding ax-coll 4246 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 9621 Version of fcdmnn0supp 9619 avoiding ax-coll 4246 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 6797 The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
2o ⊆ ω
 
5-Aug-2024suppssdc 6500 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 6482 An element of the support of a function with a given domain. This version of elsuppfn 6483 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-coll 4246. (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 6480 The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 6481 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-coll 4246. (Contributed by SN, 5-Aug-2024.)
((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍})
 
4-Aug-2024en3d 7055 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 7054 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 7603 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 7601 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 7600 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 7599 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 7597 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 7596 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 7597), (3) implies (5) (onntri35 7596), (5) implies (1) (onntri51 7599), (2) implies (4) (onntri24 7601), (4) implies (5) (onntri45 7600), and (5) implies (2) (onntri52 7603).

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 7598 and exmidontri2or 7602, respectively. Thus ¬ ¬ EXMID is equivalent to (1) or (2). In addition, ¬ ¬ EXMID is equivalent to (3) by onntri3or 7604 and (4) by onntri2or 7605.

(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 7595 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 7594 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 15103 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 7593 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 7592 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 7591 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 7590 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 7589 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 7588 The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
𝒫 1o ≠ 1o
 
30-Jul-2024pw1ne0 7587 The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
𝒫 1o ≠ ∅
 
30-Jul-2024fsuppeqg 6488 Version of fsuppeq 6487 avoiding ax-coll 4246 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 14371 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 13868 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 7585 The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
𝒫 1o ∈ On
 
29-Jul-2024isfsuppd 7290 Deduction form of isfsupp 7289. (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 7216 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 17231 Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 17230 by means of dceqnconst 17229. (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 17227 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 17233 Lemma for nconstwlpo 17235. 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 17232 Lemma for nconstwlpo 17235. 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)
 
26-Jul-2024fvmptd4 5800 Deduction version of fvmpt 5782 (where the substitution hypothesis does not have the antecedent 𝜑). (Contributed by SN, 26-Jul-2024.)
(𝑥 = 𝐴 → 𝐵 = 𝐶)    &   (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))    &   (𝜑 → 𝐴 ∈ 𝐷)    &   (𝜑 → 𝐶 ∈ 𝑉)    ⇒   (𝜑 → (𝐹‘𝐴) = 𝐶)
 
24-Jul-2024tridceq 17225 Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 17211 and redcwlpo 17224). Thus, this is an analytic analogue to lpowlpo 7508. (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 17220 Weak omniscience stated in terms of equality with 0. Like iswomninn 17219 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 15093 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 7508 LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7507. There is an analogue in terms of analytic omniscience principles at tridceq 17225. (Contributed by Jim Kingdon, 24-Jul-2024.)
(ω ∈ Omni → ω ∈ WOmni)
 
23-Jul-2024nconstwlpolem 17234 Lemma for nconstwlpo 17235. (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 17229 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 17224 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 17226 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 6036 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 17235 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 4892 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 4277 Existence of a difference. (Contributed by SN, 16-Jul-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (𝐴 ∖ 𝐵) ∈ V)
 
15-Jul-2024fprodseq 12366 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 16133 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 12976, 2logb9irrap 16132 and sqrt2cxp2logb9e3 16130. 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 16132 Example for logbgcd1irrap 16125. 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 13702 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 13701 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 13700 Lemma for ercpbl 13701. (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 13699 Value of the function in qusval 13693. (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 13698 Value of the function in qusval 13693. (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 16123 Lemma for logbgcd1irrap 16125. 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 15928 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 16094 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 16124 Lemma for logbgcd1irrap 16125. 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 11170 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 16029 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 16100 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 16030 Deduction form of logrpap0 16029. (Contributed by Jim Kingdon, 3-Jul-2024.)
(𝜑 → 𝐴 ∈ ℝ+)    &   (𝜑 → 𝐴 # 1)    ⇒   (𝜑 → (log‘𝐴) # 0)
 
3-Jul-2024logrpap0b 16028 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 17143 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 17142 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 17203 An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.)
(0(,)1) ≈ ℝ
 
27-Jun-2024iooref1o 17202 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 17236 Lemma for neapmkv 17237. 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 17222 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 17221 Lemma for ismkvnn 17222. 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 7501 Lemma for enmkv 7502. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ Markov → 𝐵 ∈ Markov))
 
24-Jun-2024neapmkv 17237 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 17230 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 17211 for more discussion of decidability of real number apartness.

This is a weaker form of dceqnconst 17229 and in fact this theorem can be proved using dceqnconst 17229 as shown at dcapnconstALT 17231. (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 7502 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 6702 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 17223 Lemma for redcwlpo 17224. A biconditionalized version of trilpolemeq1 17208. (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 17224 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 17223). 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 10689 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)

(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ω ∈ WOmni)
 
20-Jun-2024iswomninn 17219 Weak omniscience stated in terms of natural numbers. Similar to iswomnimap 7506 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 17218 Lemma for iswomnimap 7506. 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 7510 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 6702 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 7509 Lemma for enwomni 7510. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni → 𝐵 ∈ WOmni))
 
19-Jun-2024rpabscxpbnd 16095 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 16077 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 16060 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 16059 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 16057 Closure of the complex power function. (Contributed by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) ∈ ℂ)
 
12-Jun-2024rpcxp0 16053 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 16051 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 16050 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 16049 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 16010 Define the power function on complex numbers. Because df-relog 16009 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 17213 Version of trirec0 17212 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 17212 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 17211). (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 7507 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 7506 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 7505 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 7504 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 14389 A ring is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → 𝑅 ∈ CMnd)
 
1-Jun-2024ringabld 14388 A ring is an Abelian group. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → 𝑅 ∈ Abel)
 
1-Jun-2024cmnmndd 14160 A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝐺 ∈ CMnd)    ⇒   (𝜑 → 𝐺 ∈ Mnd)
 
1-Jun-2024ablcmnd 14144 An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝐺 ∈ Abel)    ⇒   (𝜑 → 𝐺 ∈ CMnd)
 
1-Jun-2024grpmndd 13867 A group is a monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑 → 𝐺 ∈ Grp)    ⇒   (𝜑 → 𝐺 ∈ Mnd)
 
1-Jun-2024fndmi 5481 The domain of a function. (Contributed by Wolf Lammen, 1-Jun-2024.)
𝐹 Fn 𝐴    ⇒   dom 𝐹 = 𝐴
 
29-May-2024pw1nct 17152 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 17151 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 5772 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 5771 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 4294 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 15145 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 15874 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 15924 Lemma for eflt 15925. The converse of efltim 12481 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 15925 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 17215 Lemma for apdiff 17216. (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 17214 Lemma for apdiff 17216. 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 17216 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 14682 A left module is a group. (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑊 ∈ LMod)    ⇒   (𝜑 → 𝑊 ∈ Grp)
 
16-May-2024drnggrpd 14661 A division ring is a group (deduction form). (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑅 ∈ DivRing)    ⇒   (𝜑 → 𝑅 ∈ Grp)
 
16-May-2024drngringd 14660 A division ring is a ring. (Contributed by SN, 16-May-2024.)
(𝜑 → 𝑅 ∈ DivRing)    ⇒   (𝜑 → 𝑅 ∈ Ring)
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