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Theorem List for Metamath Proof Explorer - 42201-42300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaks5lem7 42201* Lemma for aks5. We clean up the hypotheses compared to aks5lem6 42193. (Contributed by metakunt, 9-Aug-2025.)
(𝜑 → (♯‘(Base‘𝐾)) ∈ ℕ)    &   𝑃 = (chr‘𝐾)    &   (𝜑𝐾 ∈ Field)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑅 ∈ ℕ)    &   (𝜑𝑁 ∈ (ℤ‘3))    &   (𝜑𝑃𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   (𝜑 → ((2 logb 𝑁)↑2) < ((od𝑅)‘𝑁))    &   (𝜑𝑅 ∥ ((♯‘(Base‘𝐾)) − 1))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))(𝑋(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))𝑋)(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))    &   (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))𝑋)(-g𝑆)(1r𝑆))})    &   𝑋 = (var1‘(ℤ/nℤ‘𝑁))       (𝜑𝑁 = (𝑃↑(𝑃 pCnt 𝑁)))
 
Theoremaks5lem8 42202* Lemma for aks5. Clean up the conclusion. (Contributed by metakunt, 9-Aug-2025.)
(𝜑 → (♯‘(Base‘𝐾)) ∈ ℕ)    &   𝑃 = (chr‘𝐾)    &   (𝜑𝐾 ∈ Field)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑅 ∈ ℕ)    &   (𝜑𝑁 ∈ (ℤ‘3))    &   (𝜑𝑃𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   (𝜑 → ((2 logb 𝑁)↑2) < ((od𝑅)‘𝑁))    &   (𝜑𝑅 ∥ ((♯‘(Base‘𝐾)) − 1))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))(𝑋(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))𝑋)(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))    &   (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))𝑋)(-g𝑆)(1r𝑆))})    &   𝑋 = (var1‘(ℤ/nℤ‘𝑁))       (𝜑 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ 𝑁 = (𝑝𝑛))
 
Axiomax-exfinfld 42203* Existence axiom for finite fields, eventually we want to construct them. (Contributed by metakunt, 13-Jul-2025.)
𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑝𝑛) ∧ (chr‘𝑘) = 𝑝)
 
Theoremexfinfldd 42204* For any prime 𝑃 and any positive integer 𝑁 there exists a field 𝑘 such that 𝑘 contains 𝑃𝑁 elements. (Contributed by metakunt, 13-Jul-2025.)
(𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃𝑁) ∧ (chr‘𝑘) = 𝑃))
 
Theoremaks5 42205* The AKS Primality test, given an integer 𝑁 greater than or equal to 3, find a coprime 𝑅 such that 𝑅 is big enough. Then, if a bunch of polynomial equalities in the residue ring hold then 𝑁 is a prime power. Currently depends on the axiom ax-exfinfld 42203, since we currently do not have the existence of finite fields in the database. (Contributed by metakunt, 16-Aug-2025.)
𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   𝑋 = (var1‘(ℤ/nℤ‘𝑁))    &   𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))𝑋)(-g𝑆)(1r𝑆))})    &   (𝜑𝑁 ∈ (ℤ‘3))    &   (𝜑𝑅 ∈ ℕ)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → ((2 logb 𝑁)↑2) < ((od𝑅)‘𝑁))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))(𝑋(+g𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))𝑋)(+g𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)(𝑎 gcd 𝑁) = 1)       (𝜑 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ 𝑁 = (𝑝𝑛))
 
21.29.9  Permutation results
 
Theoremmetakunt1 42206* A is an endomapping. (Contributed by metakunt, 23-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))       (𝜑𝐴:(1...𝑀)⟶(1...𝑀))
 
Theoremmetakunt2 42207* A is an endomapping. (Contributed by metakunt, 23-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝐼, if(𝑥 < 𝐼, 𝑥, (𝑥 + 1))))       (𝜑𝐴:(1...𝑀)⟶(1...𝑀))
 
Theoremmetakunt3 42208* Value of A. (Contributed by metakunt, 23-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       (𝜑 → (𝐴𝑋) = if(𝑋 = 𝐼, 𝑀, if(𝑋 < 𝐼, 𝑋, (𝑋 − 1))))
 
Theoremmetakunt4 42209* Value of A. (Contributed by metakunt, 23-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝐼, if(𝑥 < 𝐼, 𝑥, (𝑥 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       (𝜑 → (𝐴𝑋) = if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))))
 
Theoremmetakunt5 42210* C is the left inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝑋 = 𝐼) → (𝐶‘(𝐴𝑋)) = 𝑋)
 
Theoremmetakunt6 42211* C is the left inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝑋 < 𝐼) → (𝐶‘(𝐴𝑋)) = 𝑋)
 
Theoremmetakunt7 42212* C is the left inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝐼 < 𝑋) → ((𝐴𝑋) = (𝑋 − 1) ∧ ¬ (𝐴𝑋) = 𝑀 ∧ ¬ (𝐴𝑋) < 𝐼))
 
Theoremmetakunt8 42213* C is the left inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝐼 < 𝑋) → (𝐶‘(𝐴𝑋)) = 𝑋)
 
Theoremmetakunt9 42214* C is the left inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       (𝜑 → (𝐶‘(𝐴𝑋)) = 𝑋)
 
Theoremmetakunt10 42215* C is the right inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝑋 = 𝑀) → (𝐴‘(𝐶𝑋)) = 𝑋)
 
Theoremmetakunt11 42216* C is the right inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝑋 < 𝐼) → (𝐴‘(𝐶𝑋)) = 𝑋)
 
Theoremmetakunt12 42217* C is the right inverse for A. (Contributed by metakunt, 25-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑 ∧ ¬ (𝑋 = 𝑀𝑋 < 𝐼)) → (𝐴‘(𝐶𝑋)) = 𝑋)
 
Theoremmetakunt13 42218* C is the right inverse for A. (Contributed by metakunt, 25-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       (𝜑 → (𝐴‘(𝐶𝑋)) = 𝑋)
 
Theoremmetakunt14 42219* A is a primitive permutation that moves the I-th element to the end and C is its inverse that moves the last element back to the I-th position. (Contributed by metakunt, 25-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))       (𝜑 → (𝐴:(1...𝑀)–1-1-onto→(1...𝑀) ∧ 𝐴 = 𝐶))
 
Theoremmetakunt15 42220* Construction of another permutation. (Contributed by metakunt, 25-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐹 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))       (𝜑𝐹:(1...(𝐼 − 1))–1-1-onto→(((𝑀𝐼) + 1)...(𝑀 − 1)))
 
Theoremmetakunt16 42221* Construction of another permutation. (Contributed by metakunt, 25-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐹 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))       (𝜑𝐹:(𝐼...(𝑀 − 1))–1-1-onto→(1...(𝑀𝐼)))
 
Theoremmetakunt17 42222 The union of three disjoint bijections is a bijection. (Contributed by metakunt, 28-May-2024.)
(𝜑𝐺:𝐴1-1-onto𝑋)    &   (𝜑𝐻:𝐵1-1-onto𝑌)    &   (𝜑𝐼:𝐶1-1-onto𝑍)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑 → (𝐴𝐶) = ∅)    &   (𝜑 → (𝐵𝐶) = ∅)    &   (𝜑 → (𝑋𝑌) = ∅)    &   (𝜑 → (𝑋𝑍) = ∅)    &   (𝜑 → (𝑌𝑍) = ∅)    &   (𝜑𝐹 = ((𝐺𝐻) ∪ 𝐼))    &   (𝜑𝐷 = ((𝐴𝐵) ∪ 𝐶))    &   (𝜑𝑊 = ((𝑋𝑌) ∪ 𝑍))       (𝜑𝐹:𝐷1-1-onto𝑊)
 
Theoremmetakunt18 42223 Disjoint domains and codomains. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)       (𝜑 → ((((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅ ∧ ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅ ∧ ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅) ∧ (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∩ (1...(𝑀𝐼))) = ∅ ∧ ((((𝑀𝐼) + 1)...(𝑀 − 1)) ∩ {𝑀}) = ∅ ∧ ((1...(𝑀𝐼)) ∩ {𝑀}) = ∅)))
 
Theoremmetakunt19 42224* Domains on restrictions of functions. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))    &   𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))    &   𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))       (𝜑 → ((𝐶 Fn (1...(𝐼 − 1)) ∧ 𝐷 Fn (𝐼...(𝑀 − 1)) ∧ (𝐶𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1)))) ∧ {⟨𝑀, 𝑀⟩} Fn {𝑀}))
 
Theoremmetakunt20 42225* Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))    &   𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))    &   𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))    &   (𝜑𝑋 ∈ (1...𝑀))    &   (𝜑𝑋 = 𝑀)       (𝜑 → (𝐵𝑋) = (((𝐶𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))
 
Theoremmetakunt21 42226* Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))    &   𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))    &   𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))    &   (𝜑𝑋 ∈ (1...𝑀))    &   (𝜑 → ¬ 𝑋 = 𝑀)    &   (𝜑𝑋 < 𝐼)       (𝜑 → (𝐵𝑋) = (((𝐶𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))
 
Theoremmetakunt22 42227* Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))    &   𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))    &   𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))    &   (𝜑𝑋 ∈ (1...𝑀))    &   (𝜑 → ¬ 𝑋 = 𝑀)    &   (𝜑 → ¬ 𝑋 < 𝐼)       (𝜑 → (𝐵𝑋) = (((𝐶𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))
 
Theoremmetakunt23 42228* B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))    &   𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))    &   𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))    &   (𝜑𝑋 ∈ (1...𝑀))       (𝜑 → (𝐵𝑋) = (((𝐶𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))
 
Theoremmetakunt24 42229 Technical condition such that metakunt17 42222 holds. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)       (𝜑 → ((((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅ ∧ (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}) ∧ (1...𝑀) = (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀})))
 
Theoremmetakunt25 42230* B is a permutation. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))       (𝜑𝐵:(1...𝑀)–1-1-onto→(1...𝑀))
 
Theoremmetakunt26 42231* Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀𝐼)), (𝑧 + (1 − 𝐼)))))    &   (𝜑𝑋 = 𝐼)       (𝜑 → (𝐶‘(𝐵‘(𝐴𝑋))) = 𝑋)
 
Theoremmetakunt27 42232* Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   (𝜑𝑋 ∈ (1...𝑀))    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀𝐼)), (𝑧 + (1 − 𝐼)))))    &   (𝜑 → ¬ 𝑋 = 𝐼)    &   (𝜑𝑋 < 𝐼)       (𝜑 → (𝐵‘(𝐴𝑋)) = (𝑋 + (𝑀𝐼)))
 
Theoremmetakunt28 42233* Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   (𝜑𝑋 ∈ (1...𝑀))    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀𝐼)), (𝑧 + (1 − 𝐼)))))    &   (𝜑 → ¬ 𝑋 = 𝐼)    &   (𝜑 → ¬ 𝑋 < 𝐼)       (𝜑 → (𝐵‘(𝐴𝑋)) = (𝑋𝐼))
 
Theoremmetakunt29 42234* Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   (𝜑𝑋 ∈ (1...𝑀))    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀𝐼)), (𝑧 + (1 − 𝐼)))))    &   (𝜑 → ¬ 𝑋 = 𝐼)    &   (𝜑𝑋 < 𝐼)    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   𝐻 = if(𝐼 ≤ (𝑋 + (𝑀𝐼)), 1, 0)       (𝜑 → (𝐶‘(𝐵‘(𝐴𝑋))) = ((𝑋 + (𝑀𝐼)) + 𝐻))
 
Theoremmetakunt30 42235* Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   (𝜑𝑋 ∈ (1...𝑀))    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀𝐼)), (𝑧 + (1 − 𝐼)))))    &   (𝜑 → ¬ 𝑋 = 𝐼)    &   (𝜑 → ¬ 𝑋 < 𝐼)    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   𝐻 = if(𝐼 ≤ (𝑋𝐼), 1, 0)       (𝜑 → (𝐶‘(𝐵‘(𝐴𝑋))) = ((𝑋𝐼) + 𝐻))
 
Theoremmetakunt31 42236* Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   (𝜑𝑋 ∈ (1...𝑀))    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀𝐼)), (𝑧 + (1 − 𝐼)))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   𝐺 = if(𝐼 ≤ (𝑋 + (𝑀𝐼)), 1, 0)    &   𝐻 = if(𝐼 ≤ (𝑋𝐼), 1, 0)    &   𝑅 = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀𝐼)) + 𝐺), ((𝑋𝐼) + 𝐻)))       (𝜑 → (𝐶‘(𝐵‘(𝐴𝑋))) = 𝑅)
 
Theoremmetakunt32 42237* Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   (𝜑𝑋 ∈ (1...𝑀))    &   𝐷 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑥, if(𝑥 < 𝐼, ((𝑥 + (𝑀𝐼)) + if(𝐼 ≤ (𝑥 + (𝑀𝐼)), 1, 0)), ((𝑥𝐼) + if(𝐼 ≤ (𝑥𝐼), 1, 0)))))    &   𝐺 = if(𝐼 ≤ (𝑋 + (𝑀𝐼)), 1, 0)    &   𝐻 = if(𝐼 ≤ (𝑋𝐼), 1, 0)    &   𝑅 = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀𝐼)) + 𝐺), ((𝑋𝐼) + 𝐻)))       (𝜑 → (𝐷𝑋) = 𝑅)
 
Theoremmetakunt33 42238* Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀𝐼)), (𝑧 + (1 − 𝐼)))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   𝐷 = (𝑤 ∈ (1...𝑀) ↦ if(𝑤 = 𝐼, 𝑤, if(𝑤 < 𝐼, ((𝑤 + (𝑀𝐼)) + if(𝐼 ≤ (𝑤 + (𝑀𝐼)), 1, 0)), ((𝑤𝐼) + if(𝐼 ≤ (𝑤𝐼), 1, 0)))))       (𝜑 → (𝐶 ∘ (𝐵𝐴)) = 𝐷)
 
Theoremmetakunt34 42239* 𝐷 is a permutation. (Contributed by metakunt, 18-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐷 = (𝑤 ∈ (1...𝑀) ↦ if(𝑤 = 𝐼, 𝑤, if(𝑤 < 𝐼, ((𝑤 + (𝑀𝐼)) + if(𝐼 ≤ (𝑤 + (𝑀𝐼)), 1, 0)), ((𝑤𝐼) + if(𝐼 ≤ (𝑤𝐼), 1, 0)))))       (𝜑𝐷:(1...𝑀)–1-1-onto→(1...𝑀))
 
21.29.10  Unused lemmas scheduled for deletion
 
Theoremfac2xp3 42240 Factorial of 2x+3, sublemma for sublemma for AKS. (Contributed by metakunt, 19-Apr-2024.)
(𝑥 ∈ ℕ0 → (!‘((2 · 𝑥) + 3)) = ((!‘((2 · 𝑥) + 1)) · (((2 · 𝑥) + 2) · ((2 · 𝑥) + 3))))
 
Theoremprodsplit 42241* Product split into two factors, original by Steven Nguyen. (Contributed by metakunt, 21-Apr-2024.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑀𝑁)    &   (𝜑𝐾 ∈ ℕ0)    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 + 𝐾))) → 𝐴 ∈ ℂ)       (𝜑 → ∏𝑘 ∈ (𝑀...(𝑁 + 𝐾))𝐴 = (∏𝑘 ∈ (𝑀...𝑁)𝐴 · ∏𝑘 ∈ ((𝑁 + 1)...(𝑁 + 𝐾))𝐴))
 
Theorem2xp3dxp2ge1d 42242 2x+3 is greater than or equal to x+2 for x >= -1, a deduction version (Contributed by metakunt, 21-Apr-2024.)
(𝜑𝑋 ∈ (-1[,)+∞))       (𝜑 → 1 ≤ (((2 · 𝑋) + 3) / (𝑋 + 2)))
 
Theoremfactwoffsmonot 42243 A factorial with offset is monotonely increasing. (Contributed by metakunt, 20-Apr-2024.)
(((𝑋 ∈ ℕ0𝑌 ∈ ℕ0𝑋𝑌) ∧ 𝑁 ∈ ℕ0) → (!‘(𝑋 + 𝑁)) ≤ (!‘(𝑌 + 𝑁)))
 
21.30  Mathbox for Steven Nguyen
 
21.30.1  Utility theorems
 
Theoremjarrii 42244 Inference associated with jarri 107. A consequence of ax-mp 5 and ax-1 6. (Contributed by SN, 14-Oct-2025.)
𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒
 
Theoremintnanrt 42245 Introduction of conjunct inside of a contradiction. Would be used in elfvov1 7473. (Contributed by SN, 18-May-2025.)
𝜑 → ¬ (𝜑𝜓))
 
Theoremioin9i8 42246 Miscellaneous inference creating a biconditional from an implied converse implication. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝜑 → (𝜓𝜒))    &   (𝜒 → ¬ 𝜃)    &   (𝜓𝜃)       (𝜑 → (𝜓𝜃))
 
Theoremjaodd 42247 Double deduction form of jaoi 858. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜏𝜃)))       (𝜑 → (𝜓 → ((𝜒𝜏) → 𝜃)))
 
Theoremsyl3an12 42248 A double syllogism inference. (Contributed by SN, 15-Sep-2024.)
(𝜑𝜓)    &   (𝜒𝜃)    &   ((𝜓𝜃𝜏) → 𝜂)       ((𝜑𝜒𝜏) → 𝜂)
 
Theoremexbiii 42249 Inference associated with exbii 1848. Weaker version of eximii 1837. (Contributed by SN, 14-Oct-2025.)
𝑥𝜑    &   (𝜑𝜓)       𝑥𝜓
 
Theoremsbtd 42250* A true statement is true upon substitution (deduction). A similar proof is possible for icht 47439. (Contributed by SN, 4-May-2024.)
(𝜑𝜓)       (𝜑 → [𝑡 / 𝑥]𝜓)
 
Theoremsbor2 42251 One direction of sbor 2307, using fewer axioms. Compare 19.33 1884. (Contributed by Steven Nguyen, 18-Aug-2023.)
(([𝑡 / 𝑥]𝜑 ∨ [𝑡 / 𝑥]𝜓) → [𝑡 / 𝑥](𝜑𝜓))
 
Theoremsbalexi 42252* Inference form of sbalex 2242, avoiding ax-10 2141 by using ax-gen 1795. (Contributed by SN, 12-Aug-2025.)
𝑥(𝑥 = 𝑦𝜑)       𝑥(𝑥 = 𝑦𝜑)
 
Theorem19.9dev 42253* 19.9d 2203 in the case of an existential quantifier, avoiding the ax-10 2141 from nfex 2324 that would be used for the hypothesis of 19.9d 2203, at the cost of an additional DV condition on 𝑦, 𝜑. (Contributed by SN, 26-May-2024.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → (∃𝑥𝑦𝜓 ↔ ∃𝑦𝜓))
 
Theorem3rspcedvd 42254* Triple application of rspcedvd 3624. (Contributed by Steven Nguyen, 27-Feb-2023.)
(𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝐷)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   ((𝜑𝑦 = 𝐵) → (𝜒𝜃))    &   ((𝜑𝑧 = 𝐶) → (𝜃𝜏))    &   (𝜑𝜏)       (𝜑 → ∃𝑥𝐷𝑦𝐷𝑧𝐷 𝜓)
 
Theoremsn-axrep5v 42255* A condensed form of axrep5 5287. (Contributed by SN, 21-Sep-2023.)
(∀𝑤𝑥 ∃*𝑧𝜑 → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 𝜑))
 
Theoremsn-axprlem3 42256* axprlem3 5425 using only Tarski's FOL axiom schemes and ax-rep 5279. (Contributed by SN, 22-Sep-2023.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏))
 
Theoremsn-exelALT 42257* Alternate proof of exel 5438, avoiding ax-pr 5432 but requiring ax-5 1910, ax-9 2118, and ax-pow 5365. This is similar to how elALT2 5369 uses ax-pow 5365 instead of ax-pr 5432 compared to el 5442. (Contributed by SN, 18-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝑥 𝑥𝑦
 
Theoremss2ab1 42258 Class abstractions in a subclass relationship, closed form. One direction of ss2ab 4062 using fewer axioms. (Contributed by SN, 22-Dec-2024.)
(∀𝑥(𝜑𝜓) → {𝑥𝜑} ⊆ {𝑥𝜓})
 
Theoremssabdv 42259* Deduction of abstraction subclass from implication. (Contributed by SN, 22-Dec-2024.)
(𝜑 → (𝑥𝐴𝜓))       (𝜑𝐴 ⊆ {𝑥𝜓})
 
Theoremsn-iotalem 42260* An unused lemma showing that many equivalences involving df-iota 6514 are potentially provable without ax-10 2141, ax-11 2157, ax-12 2177. (Contributed by SN, 6-Nov-2024.)
{𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}}
 
Theoremsn-iotalemcor 42261* Corollary of sn-iotalem 42260. Compare sb8iota 6525. (Contributed by SN, 6-Nov-2024.)
(℩𝑥𝜑) = (℩𝑦{𝑥𝜑} = {𝑦})
 
Theoremabbi1sn 42262* Originally part of uniabio 6528. Convert a theorem about df-iota 6514 to one about dfiota2 6515, without ax-10 2141, ax-11 2157, ax-12 2177. Although, eu6 2574 uses ax-10 2141 and ax-12 2177. (Contributed by SN, 23-Nov-2024.)
(∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
 
Theorembrif2 42263 Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.)
(𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝐶𝑅𝐴, 𝐶𝑅𝐵))
 
Theorembrif12 42264 Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.)
(if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐷))
 
Theorempssexg 42265 The proper subset of a set is also a set. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)
 
Theorempssn0 42266 A proper superset is nonempty. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝐴𝐵𝐵 ≠ ∅)
 
Theorempsspwb 42267 Classes are proper subclasses if and only if their power classes are proper subclasses. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝐴𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)
 
Theoremxppss12 42268 Proper subset theorem for Cartesian product. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐶𝐷) → (𝐴 × 𝐶) ⊊ (𝐵 × 𝐷))
 
Theoremelpwbi 42269 Membership in a power set, biconditional. (Contributed by Steven Nguyen, 17-Jul-2022.) (Proof shortened by Steven Nguyen, 16-Sep-2022.)
𝐵 ∈ V       (𝐴𝐵𝐴 ∈ 𝒫 𝐵)
 
Theoremimaopab 42270* The image of a class of ordered pairs. (Contributed by Steven Nguyen, 6-Jun-2023.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝜑}
 
Theoremfnsnbt 42271 A function's domain is a singleton iff the function is a singleton. (Contributed by Steven Nguyen, 18-Aug-2023.)
(𝐴 ∈ V → (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
 
Theoremfnimasnd 42272 The image of a function by a singleton whose element is in the domain of the function. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝑆𝐴)       (𝜑 → (𝐹 “ {𝑆}) = {(𝐹𝑆)})
 
Theoremeqresfnbd 42273 Property of being the restriction of a function. Note that this is closer to funssres 6610 than fnssres 6691. (Contributed by SN, 11-Mar-2025.)
(𝜑𝐹 Fn 𝐵)    &   (𝜑𝐴𝐵)       (𝜑 → (𝑅 = (𝐹𝐴) ↔ (𝑅 Fn 𝐴𝑅𝐹)))
 
Theoremf1o2d2 42274* Sufficient condition for a binary function expressed in maps-to notation to be bijective. (Contributed by SN, 11-Mar-2025.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝐷)    &   ((𝜑𝑧𝐷) → 𝐼𝐴)    &   ((𝜑𝑧𝐷) → 𝐽𝐵)    &   ((𝜑 ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐷)) → ((𝑥 = 𝐼𝑦 = 𝐽) ↔ 𝑧 = 𝐶))       (𝜑𝐹:(𝐴 × 𝐵)–1-1-onto𝐷)
 
Theoremfmpocos 42275* Composition of two functions. Variation of fmpoco 8120 with more context in the substitution hypothesis for 𝑇. (Contributed by SN, 14-Mar-2025.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑅𝐶)    &   (𝜑𝐹 = (𝑥𝐴, 𝑦𝐵𝑅))    &   (𝜑𝐺 = (𝑧𝐶𝑆))    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑅 / 𝑧𝑆 = 𝑇)       (𝜑 → (𝐺𝐹) = (𝑥𝐴, 𝑦𝐵𝑇))
 
Theoremovmpogad 42276* Value of an operation given by a maps-to rule. Deduction form of ovmpoga 7587. (Contributed by SN, 14-Mar-2025.)
𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)    &   ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)    &   (𝜑𝑆𝑉)       (𝜑 → (𝐴𝐹𝐵) = 𝑆)
 
Theoremofun 42277 A function operation of unions of disjoint functions is a union of function operations. (Contributed by SN, 16-Jun-2024.)
(𝜑𝐴 Fn 𝑀)    &   (𝜑𝐵 Fn 𝑀)    &   (𝜑𝐶 Fn 𝑁)    &   (𝜑𝐷 Fn 𝑁)    &   (𝜑𝑀𝑉)    &   (𝜑𝑁𝑊)    &   (𝜑 → (𝑀𝑁) = ∅)       (𝜑 → ((𝐴𝐶) ∘f 𝑅(𝐵𝐷)) = ((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷)))
 
Theoremdfqs2 42278* Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝐴 / 𝑅) = ran (𝑥𝐴 ↦ [𝑥]𝑅)
 
Theoremdfqs3 42279* Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝐴 / 𝑅) = 𝑥𝐴 {[𝑥]𝑅}
 
Theoremqseq12d 42280 Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))
 
Theoremqsalrel 42281* The quotient set is equal to the singleton of 𝐴 when all elements are related and 𝐴 is nonempty. (Contributed by SN, 8-Jun-2023.)
((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑥 𝑦)    &   (𝜑 Er 𝐴)    &   (𝜑𝑁𝐴)       (𝜑 → (𝐴 / ) = {𝐴})
 
Theoremelmapssresd 42282 A restricted mapping is a mapping. EDITORIAL: Could be used to shorten elpm2r 8885 with some reordering involving mapsspm 8916. (Contributed by SN, 11-Mar-2025.)
(𝜑𝐴 ∈ (𝐵m 𝐶))    &   (𝜑𝐷𝐶)       (𝜑 → (𝐴𝐷) ∈ (𝐵m 𝐷))
 
Theoremsupinf 42283* The supremum is the infimum of the upper bounds. (Contributed by SN, 29-Jun-2025.)
(𝜑< Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦𝐴 (𝑦 < 𝑥 → ∃𝑧𝐵 𝑦 < 𝑧)))       (𝜑 → sup(𝐵, 𝐴, < ) = inf({𝑥𝐴 ∣ ∀𝑤𝐵 ¬ 𝑥 < 𝑤}, 𝐴, < ))
 
Theoremmapcod 42284 Compose two mappings. (Contributed by SN, 11-Mar-2025.)
(𝜑𝐹 ∈ (𝐴m 𝐵))    &   (𝜑𝐺 ∈ (𝐵m 𝐶))       (𝜑 → (𝐹𝐺) ∈ (𝐴m 𝐶))
 
Theoremfisdomnn 42285 A finite set is dominated by the set of natural numbers. (Contributed by SN, 6-Jul-2025.)
(𝐴 ∈ Fin → 𝐴 ≺ ℕ)
 
Theoremltex 42286 The less-than relation is a set. (Contributed by SN, 5-Jun-2025.)
< ∈ V
 
Theoremleex 42287 The less-than-or-equal-to relation is a set. (Contributed by SN, 5-Jun-2025.)
≤ ∈ V
 
Theoremsubex 42288 The subtraction operation is a set. (Contributed by SN, 5-Jun-2025.)
− ∈ V
 
Theoremabsex 42289 The absolute value function is a set. (Contributed by SN, 5-Jun-2025.)
abs ∈ V
 
Theoremcjex 42290 The conjugate function is a set. (Contributed by SN, 5-Jun-2025.)
∗ ∈ V
 
Theoremfzosumm1 42291* Separate out the last term in a finite sum. (Contributed by Steven Nguyen, 22-Aug-2023.)
(𝜑 → (𝑁 − 1) ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀..^𝑁)) → 𝐴 ∈ ℂ)    &   (𝑘 = (𝑁 − 1) → 𝐴 = 𝐵)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝐴 = (Σ𝑘 ∈ (𝑀..^(𝑁 − 1))𝐴 + 𝐵))
 
Theoremccatcan2d 42292 Cancellation law for concatenation. (Contributed by SN, 6-Sep-2023.)
(𝜑𝐴 ∈ Word 𝑉)    &   (𝜑𝐵 ∈ Word 𝑉)    &   (𝜑𝐶 ∈ Word 𝑉)       (𝜑 → ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) ↔ 𝐴 = 𝐵))
 
21.30.2  Arithmetic theorems

Towards the start of this section are several proofs regarding the different complex number axioms that could be used to prove some results.

For example, ax-1rid 11225 is used in mulrid 11259 related theorems, so one could trade off the extra axioms in mulrid 11259 for the axioms needed to prove that something is a real number. Another example is avoiding complex number closure laws by using real number closure laws and then using ax-resscn 11212; in the other direction, real number closure laws can be avoided by using ax-resscn 11212 and then the complex number closure laws. (This only works if the result of (𝐴 + 𝐵) only needs to be a complex number).

The natural numbers are especially amenable to axiom reductions, as the set is the recursive set {1, (1 + 1), ((1 + 1) + 1)}, etc., i.e. the set of numbers formed by only additions of 1. The digits 2 through 9 are defined so that they expand into additions of 1. This conveniently allows for adding natural numbers by rearranging parentheses, as shown below:

(4 + 3) = 7

((3 + 1) + (2 + 1)) = (6 + 1)

((((1 + 1) + 1) + 1) + ((1 + 1) + 1)) =

((((((1 + 1) + 1) + 1) + 1) + 1) + 1)

This only requires ax-addass 11220, ax-1cn 11213, and ax-addcl 11215. (And in practice, the expression isn't fully expanded into ones.)

Multiplication by 1 requires either mullidi 11266 or (ax-1rid 11225 and 1re 11261) as seen in 1t1e1 12428 and 1t1e1ALT 42296. Multiplying with greater natural numbers uses ax-distr 11222. Still, this takes fewer axioms than adding zero, which is often implicit in theorems such as (9 + 1) = 10. Adding zero uses almost every complex number axiom, though notably not ax-mulcom 11219 (see readdrid 42439 and readdlid 42433).

 
Theoremc0exALT 42293 Alternate proof of c0ex 11255 using more set theory axioms but fewer complex number axioms (add ax-10 2141, ax-11 2157, ax-13 2377, ax-nul 5306, and remove ax-1cn 11213, ax-icn 11214, ax-addcl 11215, and ax-mulcl 11217). (Contributed by Steven Nguyen, 4-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ V
 
Theorem0cnALT3 42294 Alternate proof of 0cn 11253 using ax-resscn 11212, ax-addrcl 11216, ax-rnegex 11226, ax-cnre 11228 instead of ax-icn 11214, ax-addcl 11215, ax-mulcl 11217, ax-i2m1 11223. Version of 0cnALT 11496 using ax-1cn 11213 instead of ax-icn 11214. (Contributed by Steven Nguyen, 7-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ ℂ
 
Theoremelre0re 42295 Specialized version of 0red 11264 without using ax-1cn 11213 and ax-cnre 11228. (Contributed by Steven Nguyen, 28-Jan-2023.)
(𝐴 ∈ ℝ → 0 ∈ ℝ)
 
Theorem1t1e1ALT 42296 Alternate proof of 1t1e1 12428 using a different set of axioms (add ax-mulrcl 11218, ax-i2m1 11223, ax-1ne0 11224, ax-rrecex 11227 and remove ax-resscn 11212, ax-mulcom 11219, ax-mulass 11221, ax-distr 11222). (Contributed by Steven Nguyen, 20-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(1 · 1) = 1
 
Theoremlttrii 42297 'Less than' is transitive. (Contributed by SN, 26-Aug-2025.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   𝐴 < 𝐵    &   𝐵 < 𝐶       𝐴 < 𝐶
 
Theoremremulcan2d 42298 mulcan2d 11897 for real numbers using fewer axioms. (Contributed by Steven Nguyen, 15-Apr-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremreaddridaddlidd 42299 Given some real number 𝐵 where 𝐴 acts like a right additive identity, derive that 𝐴 is a left additive identity. Note that the hypothesis is weaker than proving that 𝐴 is a right additive identity (for all numbers). Although, if there is a right additive identity, then by readdcan 11435, 𝐴 is the right additive identity. (Contributed by Steven Nguyen, 14-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐵 + 𝐴) = 𝐵)       ((𝜑𝐶 ∈ ℝ) → (𝐴 + 𝐶) = 𝐶)
 
Theoremsn-1ne2 42300 A proof of 1ne2 12474 without using ax-mulcom 11219, ax-mulass 11221, ax-pre-mulgt0 11232. Based on mul02lem2 11438. (Contributed by SN, 13-Dec-2023.)
1 ≠ 2
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48600 487 48601-48700 488 48701-48800 489 48801-48900 490 48901-49000 491 49001-49100 492 49101-49200 493 49201-49300 494 49301-49324
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