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Type | Label | Description |
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Statement | ||
Theorem | zntos 20301 | 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 20302 | 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 20303 | 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 20304 | 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 20305 | 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 20306 | 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 20307 | 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 20308 | 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 20309 | 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 20310* | Lemma for cygzn 20314. (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 20311* | Lemma for cygzn 20314. (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 20312* | Lemma for cygzn 20314. (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 20313* | 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 20314 | 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 20315* | 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 20316 | 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 20317 | 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 20318 | 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 20319 | 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 20320 | The group of signs under multiplication. (Contributed by Stefan O'Rear, 28-Aug-2015.) |
⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) ⇒ ⊢ 𝑈 ∈ Grp | ||
Theorem | psgnghm 20321 | 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 20322 | 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 20323 | 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 20324 | 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 20325 | 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 20326 | 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 20327 | Even permutations are permutations. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑃 = (Base‘𝑆) ⇒ ⊢ (pmEven‘𝐷) ⊆ 𝑃 | ||
Theorem | psgnevpmb 20328 | 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 20329 | 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 20330 | 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 20331 | 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 20332 | 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 20333 | 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 | cofipsgn 20334 | Composition of any class 𝑌 and the sign function for a finite permutation. (Contributed by AV, 27-Dec-2018.) (Revised by AV, 3-Jul-2022.) |
⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑆 = (pmSgn‘𝑁) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) = (𝑌‘(𝑆‘𝑄))) | ||
Theorem | zrhcofipsgnOLD 20335 | Obsolete version of cofipsgn 20334 as of 3-Jul-2022 . (Contributed by AV, 27-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑌 = (ℤRHom‘𝑅) & ⊢ 𝑆 = (pmSgn‘𝑁) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) = (𝑌‘(𝑆‘𝑄))) | ||
Theorem | zrhpsgnelbas 20336 | 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 20337 | Embedding of permutation signs into a ring results in an element of the ring. (Contributed by AV, 1-Jan-2019.) (Proof shortened by AV, 3-Jul-2022.) |
⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ 𝑌 = (ℤRHom‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅)) | ||
Theorem | evpmodpmf1o 20338* | 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 20339 | 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 20340* | 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 20341* | 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 20342* | Lemma 1 for psgndif 20344. (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 20343* | Lemma 2 for psgndif 20344. (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 20344* | 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 | copsgndif 20345* | Embedding of permutation signs restricted to a set without a single element into a ring. (Contributed by AV, 31-Jan-2019.) (Revised by AV, 5-Jul-2022.) |
⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾})) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → ((𝑌 ∘ 𝑍)‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = ((𝑌 ∘ 𝑆)‘𝑄))) | ||
Theorem | zrhcopsgndifOLD 20346* | Obsolete version of copsgndif 20345 as of 5-Jul-2022. (Contributed by AV, 31-Jan-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 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 20347 | Extend class notation with the field of real numbers. |
class ℝfld | ||
Definition | df-refld 20348 | The field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ ℝfld = (ℂfld ↾s ℝ) | ||
Theorem | rebase 20349 | The base of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ ℝ = (Base‘ℝfld) | ||
Theorem | remulg 20350 | 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 20351 | 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 20352 | 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 20353 | The addition operation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ + = (+g‘ℝfld) | ||
Theorem | remulr 20354 | The multiplication operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ · = (.r‘ℝfld) | ||
Theorem | re0g 20355 | The neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ 0 = (0g‘ℝfld) | ||
Theorem | re1r 20356 | The multiplicative neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ 1 = (1r‘ℝfld) | ||
Theorem | rele2 20357 | The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ ≤ = (le‘ℝfld) | ||
Theorem | relt 20358 | The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ < = (lt‘ℝfld) | ||
Theorem | reds 20359 | The distance of the field of reals. (Contributed by Thierry Arnoux, 20-Jun-2019.) |
⊢ (abs ∘ − ) = (dist‘ℝfld) | ||
Theorem | redvr 20360 | 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 20361 | The real numbers are a totally ordered set. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ ℝfld ∈ Toset | ||
Theorem | refld 20362 | The real numbers form a field. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ ℝfld ∈ Field | ||
Theorem | refldcj 20363 | The conjugation operation of the field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ ∗ = (*𝑟‘ℝfld) | ||
Theorem | recrng 20364 | The real numbers form a star ring. (Contributed by Thierry Arnoux, 19-Apr-2019.) |
⊢ ℝfld ∈ *-Ring | ||
Theorem | regsumsupp 20365* | 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)) | ||
Theorem | rzgrp 20366 | The quotient group R/Z is a group. (Contributed by Thierry Arnoux, 26-Jan-2020.) |
⊢ 𝑅 = (ℝfld /s (ℝfld ~QG ℤ)) ⇒ ⊢ 𝑅 ∈ Grp | ||
Syntax | cphl 20367 | Extend class notation with class of all pre-Hilbert spaces. |
class PreHil | ||
Syntax | cipf 20368 | Extend class notation with inner product function. |
class ·if | ||
Definition | df-phl 20369* | 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 20370* | 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 20394), while ·𝑖 only has closure (ipcl 20376). (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) | ||
Theorem | isphl 20371* | 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 20372 | A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) | ||
Theorem | phllmod 20373 | A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | ||
Theorem | phlsrng 20374 | 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 20375* | 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 20376 | 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 20377 | 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 20378 | 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 20379 | 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) | ||
Theorem | ip0r 20380 | Inner product with a zero second argument. (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) → (𝐴 , 0 ) = 𝑍) | ||
Theorem | ipeq0 20381 | The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (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) → ((𝐴 , 𝐴) = 𝑍 ↔ 𝐴 = 0 )) | ||
Theorem | ipdir 20382 | Distributive law for inner product (right-distributivity). Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ ⨣ = (+g‘𝐹) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 + 𝐵) , 𝐶) = ((𝐴 , 𝐶) ⨣ (𝐵 , 𝐶))) | ||
Theorem | ipdi 20383 | Distributive law for inner product (left-distributivity). (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ ⨣ = (+g‘𝐹) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , (𝐵 + 𝐶)) = ((𝐴 , 𝐵) ⨣ (𝐴 , 𝐶))) | ||
Theorem | ip2di 20384 | Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ ⨣ = (+g‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ PreHil) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = (((𝐴 , 𝐶) ⨣ (𝐵 , 𝐷)) ⨣ ((𝐴 , 𝐷) ⨣ (𝐵 , 𝐶)))) | ||
Theorem | ipsubdir 20385 | Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝑆 = (-g‘𝐹) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 − 𝐵) , 𝐶) = ((𝐴 , 𝐶)𝑆(𝐵 , 𝐶))) | ||
Theorem | ipsubdi 20386 | Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝑆 = (-g‘𝐹) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , (𝐵 − 𝐶)) = ((𝐴 , 𝐵)𝑆(𝐴 , 𝐶))) | ||
Theorem | ip2subdi 20387 | Distributive law for inner product subtraction. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝑆 = (-g‘𝐹) & ⊢ + = (+g‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ PreHil) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) , (𝐶 − 𝐷)) = (((𝐴 , 𝐶) + (𝐵 , 𝐷))𝑆((𝐴 , 𝐷) + (𝐵 , 𝐶)))) | ||
Theorem | ipass 20388 | Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ × = (.r‘𝐹) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 · 𝐵) , 𝐶) = (𝐴 × (𝐵 , 𝐶))) | ||
Theorem | ipassr 20389 | "Associative" law for second argument of inner product (compare ipass 20388). (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ × = (.r‘𝐹) & ⊢ ∗ = (*𝑟‘𝐹) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐴 , (𝐶 · 𝐵)) = ((𝐴 , 𝐵) × ( ∗ ‘𝐶))) | ||
Theorem | ipassr2 20390 | "Associative" law for inner product. Conjugate version of ipassr 20389. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ × = (.r‘𝐹) & ⊢ ∗ = (*𝑟‘𝐹) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ((𝐴 , 𝐵) × 𝐶) = (𝐴 , (( ∗ ‘𝐶) · 𝐵))) | ||
Theorem | ipffval 20391* | The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ · = (·if‘𝑊) ⇒ ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) | ||
Theorem | ipfval 20392 | The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ · = (·if‘𝑊) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) = (𝑋 , 𝑌)) | ||
Theorem | ipfeq 20393 | If the inner product operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ · = (·if‘𝑊) ⇒ ⊢ ( , Fn (𝑉 × 𝑉) → · = , ) | ||
Theorem | ipffn 20394 | The inner product operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·if‘𝑊) ⇒ ⊢ , Fn (𝑉 × 𝑉) | ||
Theorem | phlipf 20395 | The inner product operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·if‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) ⇒ ⊢ (𝑊 ∈ PreHil → , :(𝑉 × 𝑉)⟶𝐾) | ||
Theorem | ip2eq 20396* | Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ 𝑉 (𝑥 , 𝐴) = (𝑥 , 𝐵))) | ||
Theorem | isphld 20397* | Properties that determine a pre-Hilbert (inner product) space. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ (𝜑 → 𝑉 = (Base‘𝑊)) & ⊢ (𝜑 → + = (+g‘𝑊)) & ⊢ (𝜑 → · = ( ·𝑠 ‘𝑊)) & ⊢ (𝜑 → 𝐼 = (·𝑖‘𝑊)) & ⊢ (𝜑 → 0 = (0g‘𝑊)) & ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊)) & ⊢ (𝜑 → 𝐾 = (Base‘𝐹)) & ⊢ (𝜑 → ⨣ = (+g‘𝐹)) & ⊢ (𝜑 → × = (.r‘𝐹)) & ⊢ (𝜑 → ∗ = (*𝑟‘𝐹)) & ⊢ (𝜑 → 𝑂 = (0g‘𝐹)) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐹 ∈ *-Ring) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥𝐼𝑦) ∈ 𝐾) & ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐾 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) ⨣ (𝑦𝐼𝑧))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑥𝐼𝑥) = 𝑂) → 𝑥 = 0 ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ( ∗ ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥)) ⇒ ⊢ (𝜑 → 𝑊 ∈ PreHil) | ||
Theorem | phlpropd 20398* | If two structures have the same components (properties), one is a pre-Hilbert space iff the other one is. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ (𝜑 → 𝐹 = (Scalar‘𝐾)) & ⊢ (𝜑 → 𝐹 = (Scalar‘𝐿)) & ⊢ 𝑃 = (Base‘𝐹) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(·𝑖‘𝐾)𝑦) = (𝑥(·𝑖‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil)) | ||
Theorem | ssipeq 20399 | The inner product on a subspace equals the inner product on the parent space. (Contributed by AV, 19-Oct-2021.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑃 = (·𝑖‘𝑋) ⇒ ⊢ (𝑈 ∈ 𝑆 → 𝑃 = , ) | ||
Theorem | phssipval 20400 | The inner product on a subspace in terms of the inner product on the parent space. (Contributed by NM, 28-Jan-2008.) (Revised by AV, 19-Oct-2021.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑃 = (·𝑖‘𝑋) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈)) → (𝐴𝑃𝐵) = (𝐴 , 𝐵)) |
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