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Type | Label | Description |
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Statement | ||
Theorem | zncrng2 19701 | The value of the ℤ/nℤ structure. It is defined as the quotient ring ℤ / 𝑛ℤ, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 12-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) ⇒ ⊢ (𝑁 ∈ ℤ → 𝑈 ∈ CRing) | ||
Theorem | znval 19702 | The value of the ℤ/nℤ structure. It is defined as the quotient ring ℤ / 𝑛ℤ, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑈) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet 〈(le‘ndx), ≤ 〉)) | ||
Theorem | znle 19703 | The value of the ℤ/nℤ structure. It is defined as the quotient ring ℤ / 𝑛ℤ, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑈) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = (le‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)) | ||
Theorem | znval2 19704 | Self-referential expression for the ℤ/nℤ structure. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ ≤ = (le‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet 〈(le‘ndx), ≤ 〉)) | ||
Theorem | znbaslem 19705 | Lemma for znbas 19711. (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.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐸 = Slot 𝐾 & ⊢ 𝐾 ∈ ℕ & ⊢ 𝐾 < ;10 ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝐸‘𝑈) = (𝐸‘𝑌)) | ||
Theorem | znbaslemOLD 19706 | Obsolete version of znbaslem 19705 as of 28-Apr-2021. (Contributed by Mario Carneiro, 14-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐸 = Slot 𝐾 & ⊢ 𝐾 ∈ ℕ & ⊢ 𝐾 < 10 ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝐸‘𝑈) = (𝐸‘𝑌)) | ||
Theorem | znbas2 19707 | 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.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (Base‘𝑈) = (Base‘𝑌)) | ||
Theorem | znadd 19708 | 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.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (+g‘𝑈) = (+g‘𝑌)) | ||
Theorem | znmul 19709 | 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.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (.r‘𝑈) = (.r‘𝑌)) | ||
Theorem | znzrh 19710 | The ℤ ring homomorphism of ℤ/nℤ is inherited from the quotient ring it is based on. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑈) = (ℤRHom‘𝑌)) | ||
Theorem | znbas 19711 | The base set of ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝑅 = (ℤring ~QG (𝑆‘{𝑁})) ⇒ ⊢ (𝑁 ∈ ℕ0 → (ℤ / 𝑅) = (Base‘𝑌)) | ||
Theorem | zncrng 19712 | ℤ/nℤ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) | ||
Theorem | znzrh2 19713* | The ℤ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ ∼ = (ℤring ~QG (𝑆‘{𝑁})) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥] ∼ )) | ||
Theorem | znzrhval 19714 | The ℤ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ ∼ = (ℤring ~QG (𝑆‘{𝑁})) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝐿‘𝐴) = [𝐴] ∼ ) | ||
Theorem | znzrhfo 19715 | The ℤ ring homomorphism is a surjection onto ℤ / 𝑛ℤ. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→𝐵) | ||
Theorem | zncyg 19716 | The group ℤ / 𝑛ℤ is cyclic for all 𝑛 (including 𝑛 = 0). (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CycGrp) | ||
Theorem | zndvds 19717 | Express equality of equivalence classes in ℤ / 𝑛ℤ in terms of divisibility. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘𝐵) ↔ 𝑁 ∥ (𝐴 − 𝐵))) | ||
Theorem | zndvds0 19718 | Special case of zndvds 19717 when one argument is zero. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑌) & ⊢ 0 = (0g‘𝑌) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = 0 ↔ 𝑁 ∥ 𝐴)) | ||
Theorem | znf1o 19719 | The function 𝐹 enumerates all equivalence classes in ℤ/nℤ for each 𝑛. When 𝑛 = 0, ℤ / 0ℤ = ℤ / {0} ≈ ℤ so we let 𝑊 = ℤ; otherwise 𝑊 = {0, ..., 𝑛 − 1} enumerates all the equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝑊–1-1-onto→𝐵) | ||
Theorem | zzngim 19720 | The ℤ ring homomorphism is an isomorphism for 𝑁 = 0. (We only show group isomorphism here, but ring isomorphism follows, since it is a bijective ring homomorphism.) (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘0) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ 𝐿 ∈ (ℤring GrpIso 𝑌) | ||
Theorem | znle2 19721 | The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = (le‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)) | ||
Theorem | znleval 19722 | The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = (le‘𝑌) & ⊢ 𝑋 = (Base‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝐴 ≤ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵)))) | ||
Theorem | znleval2 19723 | The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = (le‘𝑌) & ⊢ 𝑋 = (Base‘𝑌) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≤ 𝐵 ↔ (◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵))) | ||
Theorem | zntoslem 19724 | Lemma for zntos 19725. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = (le‘𝑌) & ⊢ 𝑋 = (Base‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Toset) | ||
Theorem | zntos 19725 | The ℤ/nℤ structure is a totally ordered set. (The order is not respected by the operations, except in the case 𝑁 = 0 when it coincides with the ordering on ℤ.) (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Toset) | ||
Theorem | znhash 19726 | The ℤ/nℤ structure has 𝑛 elements. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ → (#‘𝐵) = 𝑁) | ||
Theorem | znfi 19727 | The ℤ/nℤ structure is a finite ring. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ → 𝐵 ∈ Fin) | ||
Theorem | znfld 19728 | The ℤ/nℤ structure is a finite field when 𝑛 is prime. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℙ → 𝑌 ∈ Field) | ||
Theorem | znidomb 19729 | The ℤ/nℤ structure is a domain (and hence a field) precisely when 𝑛 is prime. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ → (𝑌 ∈ IDomn ↔ 𝑁 ∈ ℙ)) | ||
Theorem | znchr 19730 | Cyclic rings are defined by their characteristic. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (chr‘𝑌) = 𝑁) | ||
Theorem | znunit 19731 | The units of ℤ/nℤ are the integers coprime to the base. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝑈 = (Unit‘𝑌) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) ∈ 𝑈 ↔ (𝐴 gcd 𝑁) = 1)) | ||
Theorem | znunithash 19732 | The size of the unit group of ℤ/nℤ. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝑈 = (Unit‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ → (#‘𝑈) = (ϕ‘𝑁)) | ||
Theorem | znrrg 19733 | The regular elements of ℤ/nℤ are exactly the units. (This theorem fails for 𝑁 = 0, where all nonzero integers are regular, but only ±1 are units.) (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝑈 = (Unit‘𝑌) & ⊢ 𝐸 = (RLReg‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ → 𝐸 = 𝑈) | ||
Theorem | cygznlem1 19734* | Lemma for cygzn 19738. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = if(𝐵 ∈ Fin, (#‘𝐵), 0) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ · = (.g‘𝐺) & ⊢ 𝐿 = (ℤRHom‘𝑌) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} & ⊢ (𝜑 → 𝐺 ∈ CycGrp) & ⊢ (𝜑 → 𝑋 ∈ 𝐸) ⇒ ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋))) | ||
Theorem | cygznlem2a 19735* | Lemma for cygzn 19738. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = if(𝐵 ∈ Fin, (#‘𝐵), 0) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ · = (.g‘𝐺) & ⊢ 𝐿 = (ℤRHom‘𝑌) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} & ⊢ (𝜑 → 𝐺 ∈ CycGrp) & ⊢ (𝜑 → 𝑋 ∈ 𝐸) & ⊢ 𝐹 = ran (𝑚 ∈ ℤ ↦ 〈(𝐿‘𝑚), (𝑚 · 𝑋)〉) ⇒ ⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵) | ||
Theorem | cygznlem2 19736* | Lemma for cygzn 19738. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = if(𝐵 ∈ Fin, (#‘𝐵), 0) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ · = (.g‘𝐺) & ⊢ 𝐿 = (ℤRHom‘𝑌) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} & ⊢ (𝜑 → 𝐺 ∈ CycGrp) & ⊢ (𝜑 → 𝑋 ∈ 𝐸) & ⊢ 𝐹 = ran (𝑚 ∈ ℤ ↦ 〈(𝐿‘𝑚), (𝑚 · 𝑋)〉) ⇒ ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (𝐹‘(𝐿‘𝑀)) = (𝑀 · 𝑋)) | ||
Theorem | cygznlem3 19737* | A cyclic group with 𝑛 elements is isomorphic to ℤ / 𝑛ℤ. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = if(𝐵 ∈ Fin, (#‘𝐵), 0) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ · = (.g‘𝐺) & ⊢ 𝐿 = (ℤRHom‘𝑌) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} & ⊢ (𝜑 → 𝐺 ∈ CycGrp) & ⊢ (𝜑 → 𝑋 ∈ 𝐸) & ⊢ 𝐹 = ran (𝑚 ∈ ℤ ↦ 〈(𝐿‘𝑚), (𝑚 · 𝑋)〉) ⇒ ⊢ (𝜑 → 𝐺 ≃𝑔 𝑌) | ||
Theorem | cygzn 19738 | A cyclic group with 𝑛 elements is isomorphic to ℤ / 𝑛ℤ, and an infinite cyclic group is isomorphic to ℤ / 0ℤ ≈ ℤ. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = if(𝐵 ∈ Fin, (#‘𝐵), 0) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝐺 ∈ CycGrp → 𝐺 ≃𝑔 𝑌) | ||
Theorem | cygth 19739* | The "fundamental theorem of cyclic groups". Cyclic groups are exactly the additive groups ℤ / 𝑛ℤ, for 0 ≤ 𝑛 (where 𝑛 = 0 is the infinite cyclic group ℤ), up to isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ (𝐺 ∈ CycGrp ↔ ∃𝑛 ∈ ℕ0 𝐺 ≃𝑔 (ℤ/nℤ‘𝑛)) | ||
Theorem | cyggic 19740 | Cyclic groups are isomorphic precisely when they have the same order. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐶 = (Base‘𝐻) ⇒ ⊢ ((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) → (𝐺 ≃𝑔 𝐻 ↔ 𝐵 ≈ 𝐶)) | ||
Theorem | frgpcyg 19741 | A free group is cyclic iff it has zero or one generator. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 18-Apr-2021.) |
⊢ 𝐺 = (freeGrp‘𝐼) ⇒ ⊢ (𝐼 ≼ 1𝑜 ↔ 𝐺 ∈ CycGrp) | ||
Theorem | cnmsgnsubg 19742 | The signs form a multiplicative subgroup of the complex numbers. (Contributed by Stefan O'Rear, 28-Aug-2015.) |
⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ⇒ ⊢ {1, -1} ∈ (SubGrp‘𝑀) | ||
Theorem | cnmsgnbas 19743 | The base set of the sign subgroup of the complex numbers. (Contributed by Stefan O'Rear, 28-Aug-2015.) |
⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) ⇒ ⊢ {1, -1} = (Base‘𝑈) | ||
Theorem | cnmsgngrp 19744 | The group of signs under multiplication. (Contributed by Stefan O'Rear, 28-Aug-2015.) |
⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) ⇒ ⊢ 𝑈 ∈ Grp | ||
Theorem | psgnghm 19745 | The sign is a homomorphism from the finitary permutation group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.) |
⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (pmSgn‘𝐷) & ⊢ 𝐹 = (𝑆 ↾s dom 𝑁) & ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) ⇒ ⊢ (𝐷 ∈ 𝑉 → 𝑁 ∈ (𝐹 GrpHom 𝑈)) | ||
Theorem | psgnghm2 19746 | The sign is a homomorphism from the finite symmetric group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.) |
⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (pmSgn‘𝐷) & ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) ⇒ ⊢ (𝐷 ∈ Fin → 𝑁 ∈ (𝑆 GrpHom 𝑈)) | ||
Theorem | psgninv 19747 | The sign of a permutation equals the sign of the inverse of the permutation. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (pmSgn‘𝐷) & ⊢ 𝑃 = (Base‘𝑆) ⇒ ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃) → (𝑁‘◡𝐹) = (𝑁‘𝐹)) | ||
Theorem | psgnco 19748 | Multiplicativity of the permutation sign function. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (pmSgn‘𝐷) & ⊢ 𝑃 = (Base‘𝑆) ⇒ ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → (𝑁‘(𝐹 ∘ 𝐺)) = ((𝑁‘𝐹) · (𝑁‘𝐺))) | ||
Theorem | zrhpsgnmhm 19749 | Embedding of permutation signs into an arbitrary ring is a homomorphism. (Contributed by SO, 9-Jul-2018.) |
⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝐴)) ∈ ((SymGrp‘𝐴) MndHom (mulGrp‘𝑅))) | ||
Theorem | zrhpsgninv 19750 | The embedded sign of a permutation equals the embedded sign of the inverse of the permutation. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑌 = (ℤRHom‘𝑅) & ⊢ 𝑆 = (pmSgn‘𝑁) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘◡𝐹) = ((𝑌 ∘ 𝑆)‘𝐹)) | ||
Theorem | evpmss 19751 | Even permutations are permutations. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑃 = (Base‘𝑆) ⇒ ⊢ (pmEven‘𝐷) ⊆ 𝑃 | ||
Theorem | psgnevpmb 19752 | A class is an even permutation if it is a permutation with sign 1. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑁 = (pmSgn‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (pmEven‘𝐷) ↔ (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) | ||
Theorem | psgnodpm 19753 | A permutation which is odd (i.e. not even) has sign -1. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑁 = (pmSgn‘𝐷) ⇒ ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑁‘𝐹) = -1) | ||
Theorem | psgnevpm 19754 | A permutation which is even has sign 1. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑁 = (pmSgn‘𝐷) ⇒ ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝐷)) → (𝑁‘𝐹) = 1) | ||
Theorem | psgnodpmr 19755 | If a permutation has sign -1 it is odd (not even). (Contributed by SO, 9-Jul-2018.) |
⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑁 = (pmSgn‘𝐷) ⇒ ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = -1) → 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) | ||
Theorem | zrhpsgnevpm 19756 | The sign of an even permutation embedded into a ring is the multiplicative neutral element of the ring. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝑌 = (ℤRHom‘𝑅) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → ((𝑌 ∘ 𝑆)‘𝐹) = 1 ) | ||
Theorem | zrhpsgnodpm 19757 | The sign of an odd permutation embedded into a ring is the additive inverse of the multiplicative neutral element of the ring. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝑌 = (ℤRHom‘𝑅) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝐼 = (invg‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → ((𝑌 ∘ 𝑆)‘𝐹) = (𝐼‘ 1 )) | ||
Theorem | zrhcofipsgn 19758 | Composition of a ℤRHom homomorphism and the sign function for a finite permutation. (Contributed by AV, 27-Dec-2018.) |
⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑌 = (ℤRHom‘𝑅) & ⊢ 𝑆 = (pmSgn‘𝑁) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) = (𝑌‘(𝑆‘𝑄))) | ||
Theorem | zrhpsgnelbas 19759 | Embedding of permutation signs into a ring results in an element of the ring. (Contributed by AV, 1-Jan-2019.) |
⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ 𝑌 = (ℤRHom‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) ∈ (Base‘𝑅)) | ||
Theorem | zrhcopsgnelbas 19760 | Embedding of permutation signs into a ring results in an element of the ring. (Contributed by AV, 1-Jan-2019.) |
⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ 𝑌 = (ℤRHom‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅)) | ||
Theorem | evpmodpmf1o 19761* | The function for performing an even permutation after a fixed odd permutation is one to one onto all odd permutations. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑃 = (Base‘𝑆) ⇒ ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑓 ∈ (pmEven‘𝐷) ↦ (𝐹(+g‘𝑆)𝑓)):(pmEven‘𝐷)–1-1-onto→(𝑃 ∖ (pmEven‘𝐷))) | ||
Theorem | pmtrodpm 19762 | A transposition is an odd permutation. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑇 = ran (pmTrsp‘𝐷) ⇒ ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) | ||
Theorem | psgnfix1 19763* | A permutation of a finite set fixing one element is generated by transpositions not involving the fixed element. (Contributed by AV, 13-Jan-2019.) |
⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾})) & ⊢ 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾})) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → ∃𝑤 ∈ Word 𝑇(𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑤))) | ||
Theorem | psgnfix2 19764* | A permutation of a finite set fixing one element is generated by transpositions not involving the fixed element. (Contributed by AV, 17-Jan-2019.) |
⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾})) & ⊢ 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾})) & ⊢ 𝑍 = (SymGrp‘𝑁) & ⊢ 𝑅 = ran (pmTrsp‘𝑁) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → ∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σg 𝑤))) | ||
Theorem | psgndiflemB 19765* | Lemma 1 for psgndif 19767. (Contributed by AV, 27-Jan-2019.) |
⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾})) & ⊢ 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾})) & ⊢ 𝑍 = (SymGrp‘𝑁) & ⊢ 𝑅 = ran (pmTrsp‘𝑁) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)) → ((𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛))) → 𝑄 = (𝑍 Σg 𝑈)))) | ||
Theorem | psgndiflemA 19766* | Lemma 2 for psgndif 19767. (Contributed by AV, 31-Jan-2019.) |
⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾})) & ⊢ 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾})) & ⊢ 𝑍 = (SymGrp‘𝑁) & ⊢ 𝑅 = ran (pmTrsp‘𝑁) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → (-1↑(#‘𝑊)) = (-1↑(#‘𝑈))))) | ||
Theorem | psgndif 19767* | Embedding of permutation signs restricted to a set without a single element into a ring. (Contributed by AV, 31-Jan-2019.) |
⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾})) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → (𝑍‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = (𝑆‘𝑄))) | ||
Theorem | zrhcopsgndif 19768* | Embedding of permutation signs restricted to a set without a single element into a ring. (Contributed by AV, 31-Jan-2019.) |
⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾})) & ⊢ 𝑌 = (ℤRHom‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → ((𝑌 ∘ 𝑍)‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = ((𝑌 ∘ 𝑆)‘𝑄))) | ||
Syntax | crefld 19769 | Extend class notation with the field of real numbers. |
class ℝfld | ||
Definition | df-refld 19770 | The field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ ℝfld = (ℂfld ↾s ℝ) | ||
Theorem | rebase 19771 | The base of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ ℝ = (Base‘ℝfld) | ||
Theorem | remulg 19772 | The multiplication (group power) operation of the group of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℝ) → (𝑁(.g‘ℝfld)𝐴) = (𝑁 · 𝐴)) | ||
Theorem | resubdrg 19773 | The real numbers form a division subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.) (Revised by Thierry Arnoux, 30-Jun-2019.) |
⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | ||
Theorem | resubgval 19774 | Subtraction in the field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ − = (-g‘ℝfld) ⇒ ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋 − 𝑌) = (𝑋 − 𝑌)) | ||
Theorem | replusg 19775 | The addition operation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ + = (+g‘ℝfld) | ||
Theorem | remulr 19776 | The multiplication operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ · = (.r‘ℝfld) | ||
Theorem | re0g 19777 | The neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ 0 = (0g‘ℝfld) | ||
Theorem | re1r 19778 | The multiplicative neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ 1 = (1r‘ℝfld) | ||
Theorem | rele2 19779 | The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ ≤ = (le‘ℝfld) | ||
Theorem | relt 19780 | The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ < = (lt‘ℝfld) | ||
Theorem | reds 19781 | The distance of the field of reals. (Contributed by Thierry Arnoux, 20-Jun-2019.) |
⊢ (abs ∘ − ) = (dist‘ℝfld) | ||
Theorem | redvr 19782 | The division operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴(/r‘ℝfld)𝐵) = (𝐴 / 𝐵)) | ||
Theorem | retos 19783 | The real numbers are a totally ordered set. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ ℝfld ∈ Toset | ||
Theorem | refld 19784 | The real numbers form a field. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ ℝfld ∈ Field | ||
Theorem | refldcj 19785 | The conjugation operation of the field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ ∗ = (*𝑟‘ℝfld) | ||
Theorem | recrng 19786 | The real numbers form a star ring. (Contributed by Thierry Arnoux, 19-Apr-2019.) |
⊢ ℝfld ∈ *-Ring | ||
Theorem | regsumsupp 19787* | The group sum over the real numbers, expressed as a finite sum. (Contributed by Thierry Arnoux, 22-Jun-2019.) (Proof shortened by AV, 19-Jul-2019.) |
⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℝfld Σg 𝐹) = Σ𝑥 ∈ (𝐹 supp 0)(𝐹‘𝑥)) | ||
Syntax | cphl 19788 | Extend class notation with class of all pre-Hilbert spaces. |
class PreHil | ||
Syntax | cipf 19789 | Extend class notation with inner product function. |
class ·if | ||
Definition | df-phl 19790* | Define the class of all pre-Hilbert spaces (inner product spaces) over arbitrary fields with involution. (Some textbook definitions are more restrictive and require the field of scalars to be the field of real or complex numbers). (Contributed by NM, 22-Sep-2011.) |
⊢ PreHil = {𝑔 ∈ LVec ∣ [(Base‘𝑔) / 𝑣][(·𝑖‘𝑔) / ℎ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)))} | ||
Definition | df-ipf 19791* | Define the inner product function. Usually we will use ·𝑖 directly instead of ·if, and they have the same behavior in most cases. The main advantage of ·if is that it is a guaranteed function (ipffn 19815), while ·𝑖 only has closure (ipcl 19797). (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) | ||
Theorem | isphl 19792* | The predicate "is a generalized pre-Hilbert (inner product) space". (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ∗ = (*𝑟‘𝐹) & ⊢ 𝑍 = (0g‘𝐹) ⇒ ⊢ (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍 → 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))) | ||
Theorem | phllvec 19793 | A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) | ||
Theorem | phllmod 19794 | A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | ||
Theorem | phlsrng 19795 | The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring) | ||
Theorem | phllmhm 19796* | The inner product of a pre-Hilbert space is linear in its left argument. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹))) | ||
Theorem | ipcl 19797 | Closure of the inner product operation in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ 𝐾) | ||
Theorem | ipcj 19798 | Conjugate of an inner product in a pre-Hilbert space. Equation I1 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ ∗ = (*𝑟‘𝐹) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ( ∗ ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)) | ||
Theorem | iporthcom 19799 | Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑍 = (0g‘𝐹) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 , 𝐵) = 𝑍 ↔ (𝐵 , 𝐴) = 𝑍)) | ||
Theorem | ip0l 19800 | Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑍 = (0g‘𝐹) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ( 0 , 𝐴) = 𝑍) |
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