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Theorem List for Metamath Proof Explorer - 41301-41400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmetakunt10 41301* 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 41302* 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 41303* 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 41304* 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 41305* 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 41306* 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 41307* 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 41308 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 41309 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 41310* 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 41311* 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 41312* 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 41313* 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 41314* 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 41315 Technical condition such that metakunt17 41308 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 41316* 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 41317* 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 41318* 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 41319* 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 41320* 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 41321* 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 41322* 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 41323* 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 41324* 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 41325* 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))
 
21.27.9  Unused lemmas scheduled for deletion
 
Theoremandiff 41326 Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.)
(𝜑 → (𝜒𝜃))    &   (𝜓 → (𝜃𝜒))       ((𝜑𝜓) → (𝜒𝜃))
 
Theoremfac2xp3 41327 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 41328* 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 41329 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 41330 A factorial with offset is monotonely increasing. (Contributed by metakunt, 20-Apr-2024.)
(((𝑋 ∈ ℕ0𝑌 ∈ ℕ0𝑋𝑌) ∧ 𝑁 ∈ ℕ0) → (!‘(𝑋 + 𝑁)) ≤ (!‘(𝑌 + 𝑁)))
 
21.28  Mathbox for Steven Nguyen
 
21.28.1  Utility theorems
 
Theoremioin9i8 41331 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 41332 Double deduction form of jaoi 854. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜏𝜃)))       (𝜑 → (𝜓 → ((𝜒𝜏) → 𝜃)))
 
Theoremsyl3an12 41333 A double syllogism inference. (Contributed by SN, 15-Sep-2024.)
(𝜑𝜓)    &   (𝜒𝜃)    &   ((𝜓𝜃𝜏) → 𝜂)       ((𝜑𝜒𝜏) → 𝜂)
 
Theoremsbtd 41334* A true statement is true upon substitution (deduction). A similar proof is possible for icht 46419. (Contributed by SN, 4-May-2024.)
(𝜑𝜓)       (𝜑 → [𝑡 / 𝑥]𝜓)
 
Theoremsbor2 41335 One direction of sbor 2302, using fewer axioms. Compare 19.33 1886. (Contributed by Steven Nguyen, 18-Aug-2023.)
(([𝑡 / 𝑥]𝜑 ∨ [𝑡 / 𝑥]𝜓) → [𝑡 / 𝑥](𝜑𝜓))
 
Theorem19.9dev 41336* 19.9d 2195 in the case of an existential quantifier, avoiding the ax-10 2136 from nfex 2316 that would be used for the hypothesis of 19.9d 2195, at the cost of an additional DV condition on 𝑦, 𝜑. (Contributed by SN, 26-May-2024.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → (∃𝑥𝑦𝜓 ↔ ∃𝑦𝜓))
 
Theorem3rspcedvdw 41337* Triple application of rspcedvdw 3615. (Contributed by SN, 20-Aug-2024.)
(𝑥 = 𝐴 → (𝜓𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜃))    &   (𝑧 = 𝐶 → (𝜃𝜏))    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑𝐶𝑍)    &   (𝜑𝜏)       (𝜑 → ∃𝑥𝑋𝑦𝑌𝑧𝑍 𝜓)
 
Theorem3rspcedvd 41338* Triple application of rspcedvd 3614. (Contributed by Steven Nguyen, 27-Feb-2023.)
(𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝐷)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   ((𝜑𝑦 = 𝐵) → (𝜒𝜃))    &   ((𝜑𝑧 = 𝐶) → (𝜃𝜏))    &   (𝜑𝜏)       (𝜑 → ∃𝑥𝐷𝑦𝐷𝑧𝐷 𝜓)
 
Theoremrabdif 41339* Move difference in and out of a restricted class abstraction. (Contributed by Steven Nguyen, 6-Jun-2023.)
({𝑥𝐴𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
 
Theoremsn-axrep5v 41340* A condensed form of axrep5 5291. (Contributed by SN, 21-Sep-2023.)
(∀𝑤𝑥 ∃*𝑧𝜑 → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 𝜑))
 
Theoremsn-axprlem3 41341* axprlem3 5423 using only Tarski's FOL axiom schemes and ax-rep 5285. (Contributed by SN, 22-Sep-2023.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏))
 
Theoremsn-exelALT 41342* Alternate proof of exel 5433, avoiding ax-pr 5427 but requiring ax-5 1912, ax-9 2115, and ax-pow 5363. This is similar to how elALT2 5367 uses ax-pow 5363 instead of ax-pr 5427 compared to el 5437. (Contributed by SN, 18-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝑥 𝑥𝑦
 
Theoremss2ab1 41343 Class abstractions in a subclass relationship, closed form. One direction of ss2ab 4056 using fewer axioms. (Contributed by SN, 22-Dec-2024.)
(∀𝑥(𝜑𝜓) → {𝑥𝜑} ⊆ {𝑥𝜓})
 
Theoremssabdv 41344* Deduction of abstraction subclass from implication. (Contributed by SN, 22-Dec-2024.)
(𝜑 → (𝑥𝐴𝜓))       (𝜑𝐴 ⊆ {𝑥𝜓})
 
Theoremsn-iotalem 41345* An unused lemma showing that many equivalences involving df-iota 6495 are potentially provable without ax-10 2136, ax-11 2153, ax-12 2170. (Contributed by SN, 6-Nov-2024.)
{𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧}}
 
Theoremsn-iotalemcor 41346* Corollary of sn-iotalem 41345. Compare sb8iota 6507. (Contributed by SN, 6-Nov-2024.)
(℩𝑥𝜑) = (℩𝑦{𝑥𝜑} = {𝑦})
 
Theoremabbi1sn 41347* Originally part of uniabio 6510. Convert a theorem about df-iota 6495 to one about dfiota2 6496, without ax-10 2136, ax-11 2153, ax-12 2170. Although, eu6 2567 uses ax-10 2136 and ax-12 2170. (Contributed by SN, 23-Nov-2024.)
(∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
 
Theorembrif1 41348 Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.)
(if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐶))
 
Theorembrif2 41349 Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.)
(𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝐶𝑅𝐴, 𝐶𝑅𝐵))
 
Theorembrif12 41350 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 41351 The proper subset of a set is also a set. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)
 
Theorempssn0 41352 A proper superset is nonempty. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝐴𝐵𝐵 ≠ ∅)
 
Theorempsspwb 41353 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 41354 Proper subset theorem for Cartesian product. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐶𝐷) → (𝐴 × 𝐶) ⊊ (𝐵 × 𝐷))
 
Theoremcoexd 41355 The composition of two sets is a set. (Contributed by SN, 7-Feb-2025.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐴𝐵) ∈ V)
 
Theoremelpwbi 41356 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 41357* The image of a class of ordered pairs. (Contributed by Steven Nguyen, 6-Jun-2023.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝜑}
 
Theoremfnsnbt 41358 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 41359 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)})
 
Theoremfvmptd4 41360* Deduction version of fvmpt 6998 (where the substitution hypothesis does not have the antecedent 𝜑). (Contributed by SN, 26-Jul-2024.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   (𝜑𝐹 = (𝑥𝐷𝐵))    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)       (𝜑 → (𝐹𝐴) = 𝐶)
 
Theoremeqresfnbd 41361 Property of being the restriction of a function. Note that this is closer to funssres 6592 than fnssres 6673. (Contributed by SN, 11-Mar-2025.)
(𝜑𝐹 Fn 𝐵)    &   (𝜑𝐴𝐵)       (𝜑 → (𝑅 = (𝐹𝐴) ↔ (𝑅 Fn 𝐴𝑅𝐹)))
 
Theoremf1o2d2 41362* 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 41363* Composition of two functions. Variation of fmpoco 8085 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 41364* Value of an operation given by a maps-to rule. Deduction form of ovmpoga 7565. (Contributed by SN, 14-Mar-2025.)
𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)    &   ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)    &   (𝜑𝑆𝑉)       (𝜑 → (𝐴𝐹𝐵) = 𝑆)
 
Theoremofun 41365 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 41366* Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝐴 / 𝑅) = ran (𝑥𝐴 ↦ [𝑥]𝑅)
 
Theoremdfqs3 41367* Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝐴 / 𝑅) = 𝑥𝐴 {[𝑥]𝑅}
 
Theoremqseq12d 41368 Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))
 
Theoremqsalrel 41369* 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})
 
Theoremfsuppfund 41370 A finitely supported function is a function. (Contributed by SN, 8-Mar-2025.)
(𝜑𝐹 finSupp 𝑍)       (𝜑 → Fun 𝐹)
 
Theoremfsuppsssuppgd 41371 If the support of a function is a subset of a finite support, it is finite. Deduction associated with fsuppsssupp 9383. (Contributed by SN, 6-Mar-2025.)
(𝜑𝐺𝑉)    &   (𝜑𝑍𝑊)    &   (𝜑 → Fun 𝐺)    &   (𝜑𝐹 finSupp 𝑂)    &   (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂))       (𝜑𝐺 finSupp 𝑍)
 
Theoremfsuppss 41372 A subset of a finitely supported function is a finitely supported function. (Contributed by SN, 8-Mar-2025.)
(𝜑𝐹𝐺)    &   (𝜑𝐺 finSupp 𝑍)       (𝜑𝐹 finSupp 𝑍)
 
Theoremelmapssresd 41373 A restricted mapping is a mapping. EDITORIAL: Could be used to shorten elpm2r 8843 with some reordering involving mapsspm 8874. (Contributed by SN, 11-Mar-2025.)
(𝜑𝐴 ∈ (𝐵m 𝐶))    &   (𝜑𝐷𝐶)       (𝜑 → (𝐴𝐷) ∈ (𝐵m 𝐷))
 
Theoremmapcod 41374 Compose two mappings. (Contributed by SN, 11-Mar-2025.)
(𝜑𝐹 ∈ (𝐴m 𝐵))    &   (𝜑𝐺 ∈ (𝐵m 𝐶))       (𝜑 → (𝐹𝐺) ∈ (𝐴m 𝐶))
 
Theoremfzosumm1 41375* 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 41376 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))
 
21.28.2  Structures
 
Theoremnelsubginvcld 41377 The inverse of a non-subgroup-member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
(𝜑𝐺 ∈ Grp)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋 ∈ (𝐵𝑆))    &   𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)       (𝜑 → (𝑁𝑋) ∈ (𝐵𝑆))
 
Theoremnelsubgcld 41378 A non-subgroup-member plus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
(𝜑𝐺 ∈ Grp)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋 ∈ (𝐵𝑆))    &   𝐵 = (Base‘𝐺)    &   (𝜑𝑌𝑆)    &    + = (+g𝐺)       (𝜑 → (𝑋 + 𝑌) ∈ (𝐵𝑆))
 
Theoremnelsubgsubcld 41379 A non-subgroup-member minus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
(𝜑𝐺 ∈ Grp)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋 ∈ (𝐵𝑆))    &   𝐵 = (Base‘𝐺)    &   (𝜑𝑌𝑆)    &    = (-g𝐺)       (𝜑 → (𝑋 𝑌) ∈ (𝐵𝑆))
 
Theoremrnasclg 41380 The set of injected scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
𝐴 = (algSc‘𝑊)    &    1 = (1r𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) → ran 𝐴 = (𝑁‘{ 1 }))
 
Theoremfrlmfielbas 41381 The vectors of a finite free module are the functions from 𝐼 to 𝑁. (Contributed by SN, 31-Aug-2023.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)    &   𝐵 = (Base‘𝐹)       ((𝑅𝑉𝐼 ∈ Fin) → (𝑋𝐵𝑋:𝐼𝑁))
 
Theoremfrlmfzwrd 41382 A vector of a module with indices from 0 to 𝑁 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.)
𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       (𝑋𝐵𝑋 ∈ Word 𝑆)
 
Theoremfrlmfzowrd 41383 A vector of a module with indices from 0 to 𝑁 − 1 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.)
𝑊 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       (𝑋𝐵𝑋 ∈ Word 𝑆)
 
Theoremfrlmfzolen 41384 The dimension of a vector of a module with indices from 0 to 𝑁 − 1. (Contributed by SN, 1-Sep-2023.)
𝑊 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       ((𝑁 ∈ ℕ0𝑋𝐵) → (♯‘𝑋) = 𝑁)
 
Theoremfrlmfzowrdb 41385 The vectors of a module with indices 0 to 𝑁 − 1 are the length- 𝑁 words over the scalars of the module. (Contributed by SN, 1-Sep-2023.)
𝑊 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       ((𝐾𝑉𝑁 ∈ ℕ0) → (𝑋𝐵 ↔ (𝑋 ∈ Word 𝑆 ∧ (♯‘𝑋) = 𝑁)))
 
Theoremfrlmfzoccat 41386 The concatenation of two vectors of dimension 𝑁 and 𝑀 forms a vector of dimension 𝑁 + 𝑀. (Contributed by SN, 31-Aug-2023.)
𝑊 = (𝐾 freeLMod (0..^𝐿))    &   𝑋 = (𝐾 freeLMod (0..^𝑀))    &   𝑌 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝐶 = (Base‘𝑋)    &   𝐷 = (Base‘𝑌)    &   (𝜑𝐾𝑍)    &   (𝜑 → (𝑀 + 𝑁) = 𝐿)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑈𝐶)    &   (𝜑𝑉𝐷)       (𝜑 → (𝑈 ++ 𝑉) ∈ 𝐵)
 
Theoremfrlmvscadiccat 41387 Scalar multiplication distributes over concatenation. (Contributed by SN, 6-Sep-2023.)
𝑊 = (𝐾 freeLMod (0..^𝐿))    &   𝑋 = (𝐾 freeLMod (0..^𝑀))    &   𝑌 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝐶 = (Base‘𝑋)    &   𝐷 = (Base‘𝑌)    &   (𝜑𝐾𝑍)    &   (𝜑 → (𝑀 + 𝑁) = 𝐿)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑈𝐶)    &   (𝜑𝑉𝐷)    &   𝑂 = ( ·𝑠𝑊)    &    = ( ·𝑠𝑋)    &    · = ( ·𝑠𝑌)    &   𝑆 = (Base‘𝐾)    &   (𝜑𝐴𝑆)       (𝜑 → (𝐴𝑂(𝑈 ++ 𝑉)) = ((𝐴 𝑈) ++ (𝐴 · 𝑉)))
 
Theoremgrpasscan2d 41388 An associative cancellation law for groups. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑋 + (𝑁𝑌)) + 𝑌) = 𝑋)
 
Theoremgrpcominv1 41389 If two elements commute, then they commute with each other's inverses (case of the first element commuting with the inverse of the second element). (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))       (𝜑 → (𝑋 + (𝑁𝑌)) = ((𝑁𝑌) + 𝑋))
 
Theoremgrpcominv2 41390 If two elements commute, then they commute with each other's inverses (case of the second element commuting with the inverse of the first element). (Contributed by SN, 1-Feb-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))       (𝜑 → (𝑌 + (𝑁𝑋)) = ((𝑁𝑋) + 𝑌))
 
Theoremfinsubmsubg 41391 A submonoid of a finite group is a subgroup. This does not extend to infinite groups, as the submonoid 0 of the group (ℤ, + ) shows. Note also that the union of a submonoid and its inverses need not be a submonoid, as the submonoid (ℕ0 ∖ {1}) of the group (ℤ, + ) shows: 3 is in that submonoid, -2 is the inverse of 2, but 1 is not in their union. Or simply, the subgroup generated by (ℕ0 ∖ {1}) is , not (ℤ ∖ {1, -1}). (Contributed by SN, 31-Jan-2025.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑆 ∈ (SubMnd‘𝐺))    &   (𝜑𝐵 ∈ Fin)       (𝜑𝑆 ∈ (SubGrp‘𝐺))
 
Theoremcrngcomd 41392 Multiplication is commutative in a commutative ring. (Contributed by SN, 8-Mar-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 · 𝑌) = (𝑌 · 𝑋))
 
Theoremcrng12d 41393 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by SN, 8-Mar-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑋 · (𝑌 · 𝑍)) = (𝑌 · (𝑋 · 𝑍)))
 
Theoremimacrhmcl 41394 The image of a commutative ring homomorphism is a commutative ring. (Contributed by SN, 10-Jan-2025.)
𝐶 = (𝑁s (𝐹𝑆))    &   (𝜑𝐹 ∈ (𝑀 RingHom 𝑁))    &   (𝜑𝑀 ∈ CRing)    &   (𝜑𝑆 ∈ (SubRing‘𝑀))       (𝜑𝐶 ∈ CRing)
 
Theoremrimrcl1 41395 Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑅 ∈ Ring)
 
Theoremrimrcl2 41396 Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑆 ∈ Ring)
 
Theoremrimcnv 41397 The converse of a ring isomorphism is a ring isomorphism. (Contributed by SN, 10-Jan-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑆 RingIso 𝑅))
 
Theoremrimco 41398 The composition of ring isomorphisms is a ring isomorphism. (Contributed by SN, 17-Jan-2025.)
((𝐹 ∈ (𝑆 RingIso 𝑇) ∧ 𝐺 ∈ (𝑅 RingIso 𝑆)) → (𝐹𝐺) ∈ (𝑅 RingIso 𝑇))
 
Theoremricsym 41399 Ring isomorphism is symmetric. (Contributed by SN, 10-Jan-2025.)
(𝑅𝑟 𝑆𝑆𝑟 𝑅)
 
Theoremrictr 41400 Ring isomorphism is transitive. (Contributed by SN, 17-Jan-2025.)
((𝑅𝑟 𝑆𝑆𝑟 𝑇) → 𝑅𝑟 𝑇)
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