![]() |
Metamath
Proof Explorer Theorem List (p. 414 of 480) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30439) |
![]() (30440-31962) |
![]() (31963-47940) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | metakunt10 41301* | C is the right inverse for A. (Contributed by metakunt, 24-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 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 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ ((𝜑 ∧ 𝑋 = 𝑀) → (𝐴‘(𝐶‘𝑋)) = 𝑋) | ||
Theorem | metakunt11 41302* | C is the right inverse for A. (Contributed by metakunt, 24-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 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 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐴‘(𝐶‘𝑋)) = 𝑋) | ||
Theorem | metakunt12 41303* | C is the right inverse for A. (Contributed by metakunt, 25-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 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 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ ((𝜑 ∧ ¬ (𝑋 = 𝑀 ∨ 𝑋 < 𝐼)) → (𝐴‘(𝐶‘𝑋)) = 𝑋) | ||
Theorem | metakunt13 41304* | C is the right inverse for A. (Contributed by metakunt, 25-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 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 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ (𝜑 → (𝐴‘(𝐶‘𝑋)) = 𝑋) | ||
Theorem | metakunt14 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 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)) | ||
Theorem | metakunt15 41306* | Construction of another permutation. (Contributed by metakunt, 25-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐹 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) ⇒ ⊢ (𝜑 → 𝐹:(1...(𝐼 − 1))–1-1-onto→(((𝑀 − 𝐼) + 1)...(𝑀 − 1))) | ||
Theorem | metakunt16 41307* | Construction of another permutation. (Contributed by metakunt, 25-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐹 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) ⇒ ⊢ (𝜑 → 𝐹:(𝐼...(𝑀 − 1))–1-1-onto→(1...(𝑀 − 𝐼))) | ||
Theorem | metakunt17 41308 | The union of three disjoint bijections is a bijection. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝐺:𝐴–1-1-onto→𝑋) & ⊢ (𝜑 → 𝐻:𝐵–1-1-onto→𝑌) & ⊢ (𝜑 → 𝐼:𝐶–1-1-onto→𝑍) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) & ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅) & ⊢ (𝜑 → (𝐵 ∩ 𝐶) = ∅) & ⊢ (𝜑 → (𝑋 ∩ 𝑌) = ∅) & ⊢ (𝜑 → (𝑋 ∩ 𝑍) = ∅) & ⊢ (𝜑 → (𝑌 ∩ 𝑍) = ∅) & ⊢ (𝜑 → 𝐹 = ((𝐺 ∪ 𝐻) ∪ 𝐼)) & ⊢ (𝜑 → 𝐷 = ((𝐴 ∪ 𝐵) ∪ 𝐶)) & ⊢ (𝜑 → 𝑊 = ((𝑋 ∪ 𝑌) ∪ 𝑍)) ⇒ ⊢ (𝜑 → 𝐹:𝐷–1-1-onto→𝑊) | ||
Theorem | metakunt18 41309 | Disjoint domains and codomains. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) ⇒ ⊢ (𝜑 → ((((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅ ∧ ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅ ∧ ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅) ∧ (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∩ (1...(𝑀 − 𝐼))) = ∅ ∧ ((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∩ {𝑀}) = ∅ ∧ ((1...(𝑀 − 𝐼)) ∩ {𝑀}) = ∅))) | ||
Theorem | metakunt19 41310* | Domains on restrictions of functions. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) & ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) & ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) ⇒ ⊢ (𝜑 → ((𝐶 Fn (1...(𝐼 − 1)) ∧ 𝐷 Fn (𝐼...(𝑀 − 1)) ∧ (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1)))) ∧ {⟨𝑀, 𝑀⟩} Fn {𝑀})) | ||
Theorem | metakunt20 41311* | Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) & ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) & ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) & ⊢ (𝜑 → 𝑋 = 𝑀) ⇒ ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋)) | ||
Theorem | metakunt21 41312* | Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) & ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) & ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) & ⊢ (𝜑 → ¬ 𝑋 = 𝑀) & ⊢ (𝜑 → 𝑋 < 𝐼) ⇒ ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋)) | ||
Theorem | metakunt22 41313* | Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) & ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) & ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) & ⊢ (𝜑 → ¬ 𝑋 = 𝑀) & ⊢ (𝜑 → ¬ 𝑋 < 𝐼) ⇒ ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋)) | ||
Theorem | metakunt23 41314* | B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) & ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) & ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋)) | ||
Theorem | metakunt24 41315 | Technical condition such that metakunt17 41308 holds. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) ⇒ ⊢ (𝜑 → ((((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅ ∧ (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}) ∧ (1...𝑀) = (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀 − 𝐼))) ∪ {𝑀}))) | ||
Theorem | metakunt25 41316* | B is a permutation. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) ⇒ ⊢ (𝜑 → 𝐵:(1...𝑀)–1-1-onto→(1...𝑀)) | ||
Theorem | metakunt26 41317* | Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 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 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))))) & ⊢ (𝜑 → 𝑋 = 𝐼) ⇒ ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑋) | ||
Theorem | metakunt27 41318* | Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ (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 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))))) & ⊢ (𝜑 → ¬ 𝑋 = 𝐼) & ⊢ (𝜑 → 𝑋 < 𝐼) ⇒ ⊢ (𝜑 → (𝐵‘(𝐴‘𝑋)) = (𝑋 + (𝑀 − 𝐼))) | ||
Theorem | metakunt28 41319* | Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ (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 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))))) & ⊢ (𝜑 → ¬ 𝑋 = 𝐼) & ⊢ (𝜑 → ¬ 𝑋 < 𝐼) ⇒ ⊢ (𝜑 → (𝐵‘(𝐴‘𝑋)) = (𝑋 − 𝐼)) | ||
Theorem | metakunt29 41320* | Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ (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 + (𝑀 − 𝐼)), (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 ≤ (𝑋 + (𝑀 − 𝐼)), 1, 0) ⇒ ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = ((𝑋 + (𝑀 − 𝐼)) + 𝐻)) | ||
Theorem | metakunt30 41321* | Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ (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 + (𝑀 − 𝐼)), (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 ≤ (𝑋 − 𝐼), 1, 0) ⇒ ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = ((𝑋 − 𝐼) + 𝐻)) | ||
Theorem | metakunt31 41322* | Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ (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 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))))) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) & ⊢ 𝐺 = if(𝐼 ≤ (𝑋 + (𝑀 − 𝐼)), 1, 0) & ⊢ 𝐻 = if(𝐼 ≤ (𝑋 − 𝐼), 1, 0) & ⊢ 𝑅 = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))) ⇒ ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅) | ||
Theorem | metakunt32 41323* | Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) & ⊢ 𝐷 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑥, if(𝑥 < 𝐼, ((𝑥 + (𝑀 − 𝐼)) + if(𝐼 ≤ (𝑥 + (𝑀 − 𝐼)), 1, 0)), ((𝑥 − 𝐼) + if(𝐼 ≤ (𝑥 − 𝐼), 1, 0))))) & ⊢ 𝐺 = if(𝐼 ≤ (𝑋 + (𝑀 − 𝐼)), 1, 0) & ⊢ 𝐻 = if(𝐼 ≤ (𝑋 − 𝐼), 1, 0) & ⊢ 𝑅 = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))) ⇒ ⊢ (𝜑 → (𝐷‘𝑋) = 𝑅) | ||
Theorem | metakunt33 41324* | Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 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 + (𝑀 − 𝐼)), (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 + (𝑀 − 𝐼)) + if(𝐼 ≤ (𝑤 + (𝑀 − 𝐼)), 1, 0)), ((𝑤 − 𝐼) + if(𝐼 ≤ (𝑤 − 𝐼), 1, 0))))) ⇒ ⊢ (𝜑 → (𝐶 ∘ (𝐵 ∘ 𝐴)) = 𝐷) | ||
Theorem | metakunt34 41325* | 𝐷 is a permutation. (Contributed by metakunt, 18-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐷 = (𝑤 ∈ (1...𝑀) ↦ if(𝑤 = 𝐼, 𝑤, if(𝑤 < 𝐼, ((𝑤 + (𝑀 − 𝐼)) + if(𝐼 ≤ (𝑤 + (𝑀 − 𝐼)), 1, 0)), ((𝑤 − 𝐼) + if(𝐼 ≤ (𝑤 − 𝐼), 1, 0))))) ⇒ ⊢ (𝜑 → 𝐷:(1...𝑀)–1-1-onto→(1...𝑀)) | ||
Theorem | andiff 41326 | Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.) |
⊢ (𝜑 → (𝜒 → 𝜃)) & ⊢ (𝜓 → (𝜃 → 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) | ||
Theorem | fac2xp3 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)))) | ||
Theorem | prodsplit 41328* | Product split into two factors, original by Steven Nguyen. (Contributed by metakunt, 21-Apr-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 𝐾))) → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...(𝑁 + 𝐾))𝐴 = (∏𝑘 ∈ (𝑀...𝑁)𝐴 · ∏𝑘 ∈ ((𝑁 + 1)...(𝑁 + 𝐾))𝐴)) | ||
Theorem | 2xp3dxp2ge1d 41329 | 2x+3 is greater than or equal to x+2 for x >= -1, a deduction version (Contributed by metakunt, 21-Apr-2024.) |
⊢ (𝜑 → 𝑋 ∈ (-1[,)+∞)) ⇒ ⊢ (𝜑 → 1 ≤ (((2 · 𝑋) + 3) / (𝑋 + 2))) | ||
Theorem | factwoffsmonot 41330 | A factorial with offset is monotonely increasing. (Contributed by metakunt, 20-Apr-2024.) |
⊢ (((𝑋 ∈ ℕ0 ∧ 𝑌 ∈ ℕ0 ∧ 𝑋 ≤ 𝑌) ∧ 𝑁 ∈ ℕ0) → (!‘(𝑋 + 𝑁)) ≤ (!‘(𝑌 + 𝑁))) | ||
Theorem | ioin9i8 41331 | Miscellaneous inference creating a biconditional from an implied converse implication. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ (𝜑 → (𝜓 ∨ 𝜒)) & ⊢ (𝜒 → ¬ 𝜃) & ⊢ (𝜓 → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 ↔ 𝜃)) | ||
Theorem | jaodd 41332 | Double deduction form of jaoi 854. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜏) → 𝜃))) | ||
Theorem | syl3an12 41333 | A double syllogism inference. (Contributed by SN, 15-Sep-2024.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → 𝜃) & ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜏) → 𝜂) ⇒ ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂) | ||
Theorem | sbtd 41334* | A true statement is true upon substitution (deduction). A similar proof is possible for icht 46419. (Contributed by SN, 4-May-2024.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → [𝑡 / 𝑥]𝜓) | ||
Theorem | sbor2 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](𝜑 ∨ 𝜓)) | ||
Theorem | 19.9dev 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 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑦𝜓)) | ||
Theorem | 3rspcedvdw 41337* | Triple application of rspcedvdw 3615. (Contributed by SN, 20-Aug-2024.) |
⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) & ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) & ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜏)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑌) & ⊢ (𝜑 → 𝐶 ∈ 𝑍) & ⊢ (𝜑 → 𝜏) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∃𝑧 ∈ 𝑍 𝜓) | ||
Theorem | 3rspcedvd 41338* | Triple application of rspcedvd 3614. (Contributed by Steven Nguyen, 27-Feb-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝜒 ↔ 𝜃)) & ⊢ ((𝜑 ∧ 𝑧 = 𝐶) → (𝜃 ↔ 𝜏)) & ⊢ (𝜑 → 𝜏) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜓) | ||
Theorem | rabdif 41339* | Move difference in and out of a restricted class abstraction. (Contributed by Steven Nguyen, 6-Jun-2023.) |
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑} | ||
Theorem | sn-axrep5v 41340* | A condensed form of axrep5 5291. (Contributed by SN, 21-Sep-2023.) |
⊢ (∀𝑤 ∈ 𝑥 ∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑)) | ||
Theorem | sn-axprlem3 41341* | axprlem3 5423 using only Tarski's FOL axiom schemes and ax-rep 5285. (Contributed by SN, 22-Sep-2023.) |
⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏)) | ||
Theorem | sn-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.) |
⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦 | ||
Theorem | ss2ab1 41343 | Class abstractions in a subclass relationship, closed form. One direction of ss2ab 4056 using fewer axioms. (Contributed by SN, 22-Dec-2024.) |
⊢ (∀𝑥(𝜑 → 𝜓) → {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓}) | ||
Theorem | ssabdv 41344* | Deduction of abstraction subclass from implication. (Contributed by SN, 22-Dec-2024.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∣ 𝜓}) | ||
Theorem | sn-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 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}} | ||
Theorem | sn-iotalemcor 41346* | Corollary of sn-iotalem 41345. Compare sb8iota 6507. (Contributed by SN, 6-Nov-2024.) |
⊢ (℩𝑥𝜑) = (℩𝑦{𝑥 ∣ 𝜑} = {𝑦}) | ||
Theorem | abbi1sn 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(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦}) | ||
Theorem | brif1 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)) | ||
Theorem | brif2 41349 | Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.) |
⊢ (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝐶𝑅𝐴, 𝐶𝑅𝐵)) | ||
Theorem | brif12 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)) | ||
Theorem | pssexg 41351 | The proper subset of a set is also a set. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) | ||
Theorem | pssn0 41352 | A proper superset is nonempty. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ (𝐴 ⊊ 𝐵 → 𝐵 ≠ ∅) | ||
Theorem | psspwb 41353 | Classes are proper subclasses if and only if their power classes are proper subclasses. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ (𝐴 ⊊ 𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵) | ||
Theorem | xppss12 41354 | Proper subset theorem for Cartesian product. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐶 ⊊ 𝐷) → (𝐴 × 𝐶) ⊊ (𝐵 × 𝐷)) | ||
Theorem | coexd 41355 | The composition of two sets is a set. (Contributed by SN, 7-Feb-2025.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ V) | ||
Theorem | elpwbi 41356 | Membership in a power set, biconditional. (Contributed by Steven Nguyen, 17-Jul-2022.) (Proof shortened by Steven Nguyen, 16-Sep-2022.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵) | ||
Theorem | imaopab 41357* | The image of a class of ordered pairs. (Contributed by Steven Nguyen, 6-Jun-2023.) |
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} | ||
Theorem | fnsnbt 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, (𝐹‘𝐴)⟩})) | ||
Theorem | fnimasnd 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.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝑆 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐹 “ {𝑆}) = {(𝐹‘𝑆)}) | ||
Theorem | fvmptd4 41360* | Deduction version of fvmpt 6998 (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) | ||
Theorem | eqresfnbd 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.) |
⊢ (𝜑 → 𝐹 Fn 𝐵) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑅 = (𝐹 ↾ 𝐴) ↔ (𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹))) | ||
Theorem | f1o2d2 41362* | Sufficient condition for a binary function expressed in maps-to notation to be bijective. (Contributed by SN, 11-Mar-2025.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝐼 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝐽 ∈ 𝐵) & ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐷)) → ((𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ 𝑧 = 𝐶)) ⇒ ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷) | ||
Theorem | fmpocos 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 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ 𝐶) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅)) & ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐶 ↦ 𝑆)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ⦋𝑅 / 𝑧⦌𝑆 = 𝑇) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇)) | ||
Theorem | ovmpogad 41364* | Value of an operation given by a maps-to rule. Deduction form of ovmpoga 7565. (Contributed by SN, 14-Mar-2025.) |
⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) & ⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) | ||
Theorem | ofun 41365 | A function operation of unions of disjoint functions is a union of function operations. (Contributed by SN, 16-Jun-2024.) |
⊢ (𝜑 → 𝐴 Fn 𝑀) & ⊢ (𝜑 → 𝐵 Fn 𝑀) & ⊢ (𝜑 → 𝐶 Fn 𝑁) & ⊢ (𝜑 → 𝐷 Fn 𝑁) & ⊢ (𝜑 → 𝑀 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ 𝑊) & ⊢ (𝜑 → (𝑀 ∩ 𝑁) = ∅) ⇒ ⊢ (𝜑 → ((𝐴 ∪ 𝐶) ∘f 𝑅(𝐵 ∪ 𝐷)) = ((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷))) | ||
Theorem | dfqs2 41366* | Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.) |
⊢ (𝐴 / 𝑅) = ran (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅) | ||
Theorem | dfqs3 41367* | Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.) |
⊢ (𝐴 / 𝑅) = ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅} | ||
Theorem | qseq12d 41368 | Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷)) | ||
Theorem | qsalrel 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 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∼ 𝑦) & ⊢ (𝜑 → ∼ Er 𝐴) & ⊢ (𝜑 → 𝑁 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐴 / ∼ ) = {𝐴}) | ||
Theorem | fsuppfund 41370 | A finitely supported function is a function. (Contributed by SN, 8-Mar-2025.) |
⊢ (𝜑 → 𝐹 finSupp 𝑍) ⇒ ⊢ (𝜑 → Fun 𝐹) | ||
Theorem | fsuppsssuppgd 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.) |
⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑊) & ⊢ (𝜑 → Fun 𝐺) & ⊢ (𝜑 → 𝐹 finSupp 𝑂) & ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂)) ⇒ ⊢ (𝜑 → 𝐺 finSupp 𝑍) | ||
Theorem | fsuppss 41372 | A subset of a finitely supported function is a finitely supported function. (Contributed by SN, 8-Mar-2025.) |
⊢ (𝜑 → 𝐹 ⊆ 𝐺) & ⊢ (𝜑 → 𝐺 finSupp 𝑍) ⇒ ⊢ (𝜑 → 𝐹 finSupp 𝑍) | ||
Theorem | elmapssresd 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.) |
⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m 𝐶)) & ⊢ (𝜑 → 𝐷 ⊆ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷)) | ||
Theorem | mapcod 41374 | Compose two mappings. (Contributed by SN, 11-Mar-2025.) |
⊢ (𝜑 → 𝐹 ∈ (𝐴 ↑m 𝐵)) & ⊢ (𝜑 → 𝐺 ∈ (𝐵 ↑m 𝐶)) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (𝐴 ↑m 𝐶)) | ||
Theorem | fzosumm1 41375* | Separate out the last term in a finite sum. (Contributed by Steven Nguyen, 22-Aug-2023.) |
⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝑘 = (𝑁 − 1) → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝐴 = (Σ𝑘 ∈ (𝑀..^(𝑁 − 1))𝐴 + 𝐵)) | ||
Theorem | ccatcan2d 41376 | Cancellation law for concatenation. (Contributed by SN, 6-Sep-2023.) |
⊢ (𝜑 → 𝐴 ∈ Word 𝑉) & ⊢ (𝜑 → 𝐵 ∈ Word 𝑉) & ⊢ (𝜑 → 𝐶 ∈ Word 𝑉) ⇒ ⊢ (𝜑 → ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | nelsubginvcld 41377 | The inverse of a non-subgroup-member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.) |
⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆)) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ (𝜑 → (𝑁‘𝑋) ∈ (𝐵 ∖ 𝑆)) | ||
Theorem | nelsubgcld 41378 | A non-subgroup-member plus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.) |
⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆)) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝐵 ∖ 𝑆)) | ||
Theorem | nelsubgsubcld 41379 | A non-subgroup-member minus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.) |
⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆)) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝐵 ∖ 𝑆)) | ||
Theorem | rnasclg 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 = (1r‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) → ran 𝐴 = (𝑁‘{ 1 })) | ||
Theorem | frlmfielbas 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) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ Fin) → (𝑋 ∈ 𝐵 ↔ 𝑋:𝐼⟶𝑁)) | ||
Theorem | frlmfzwrd 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) ⇒ ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ Word 𝑆) | ||
Theorem | frlmfzowrd 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) ⇒ ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ Word 𝑆) | ||
Theorem | frlmfzolen 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 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (♯‘𝑋) = 𝑁) | ||
Theorem | frlmfzowrdb 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) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ Word 𝑆 ∧ (♯‘𝑋) = 𝑁))) | ||
Theorem | frlmfzoccat 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) & ⊢ (𝜑 → 𝐾 ∈ 𝑍) & ⊢ (𝜑 → (𝑀 + 𝑁) = 𝐿) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑈 ∈ 𝐶) & ⊢ (𝜑 → 𝑉 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑈 ++ 𝑉) ∈ 𝐵) | ||
Theorem | frlmvscadiccat 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) & ⊢ (𝜑 → 𝐾 ∈ 𝑍) & ⊢ (𝜑 → (𝑀 + 𝑁) = 𝐿) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑈 ∈ 𝐶) & ⊢ (𝜑 → 𝑉 ∈ 𝐷) & ⊢ 𝑂 = ( ·𝑠 ‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝑋) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ 𝑆 = (Base‘𝐾) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐴𝑂(𝑈 ++ 𝑉)) = ((𝐴 ∙ 𝑈) ++ (𝐴 · 𝑉))) | ||
Theorem | grpasscan2d 41388 | An associative cancellation law for groups. (Contributed by SN, 29-Jan-2025.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋) | ||
Theorem | grpcominv1 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) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) ⇒ ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋)) | ||
Theorem | grpcominv2 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) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) ⇒ ⊢ (𝜑 → (𝑌 + (𝑁‘𝑋)) = ((𝑁‘𝑋) + 𝑌)) | ||
Theorem | finsubmsubg 41391 | A submonoid of a finite group is a subgroup. This does not extend to infinite groups, as the submonoid ℕ0 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 ℤ, not (ℤ ∖ {1, -1}). (Contributed by SN, 31-Jan-2025.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) & ⊢ (𝜑 → 𝐵 ∈ Fin) ⇒ ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | ||
Theorem | crngcomd 41392 | Multiplication is commutative in a commutative ring. (Contributed by SN, 8-Mar-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑌 · 𝑋)) | ||
Theorem | crng12d 41393 | Commutative/associative law that swaps the first two factors in a triple product. (Contributed by SN, 8-Mar-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 · (𝑌 · 𝑍)) = (𝑌 · (𝑋 · 𝑍))) | ||
Theorem | imacrhmcl 41394 | The image of a commutative ring homomorphism is a commutative ring. (Contributed by SN, 10-Jan-2025.) |
⊢ 𝐶 = (𝑁 ↾s (𝐹 “ 𝑆)) & ⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁)) & ⊢ (𝜑 → 𝑀 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑀)) ⇒ ⊢ (𝜑 → 𝐶 ∈ CRing) | ||
Theorem | rimrcl1 41395 | Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.) |
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑅 ∈ Ring) | ||
Theorem | rimrcl2 41396 | Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.) |
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑆 ∈ Ring) | ||
Theorem | rimcnv 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)) | ||
Theorem | rimco 41398 | The composition of ring isomorphisms is a ring isomorphism. (Contributed by SN, 17-Jan-2025.) |
⊢ ((𝐹 ∈ (𝑆 RingIso 𝑇) ∧ 𝐺 ∈ (𝑅 RingIso 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 RingIso 𝑇)) | ||
Theorem | ricsym 41399 | Ring isomorphism is symmetric. (Contributed by SN, 10-Jan-2025.) |
⊢ (𝑅 ≃𝑟 𝑆 → 𝑆 ≃𝑟 𝑅) | ||
Theorem | rictr 41400 | Ring isomorphism is transitive. (Contributed by SN, 17-Jan-2025.) |
⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑆 ≃𝑟 𝑇) → 𝑅 ≃𝑟 𝑇) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |