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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | znfld 21501 | 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 21502 | 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 21503 | 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 21504 | 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 21505 | 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 21506 | 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 21507* | Lemma for cygzn 21511. (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 21508* | Lemma for cygzn 21511. (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 21509* | Lemma for cygzn 21511. (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 21510* | 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 21511 | 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 21512* | 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 21513 | 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 21514 | 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 | freshmansdream 21515 | For a prime number 𝑃, if 𝑋 and 𝑌 are members of a commutative ring 𝑅 of characteristic 𝑃, then ((𝑋 + 𝑌)↑𝑃) = ((𝑋↑𝑃) + (𝑌↑𝑃)). This theorem is sometimes referred to as "the freshman's dream" . (Contributed by Thierry Arnoux, 18-Sep-2023.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ ↑ = (.g‘(mulGrp‘𝑅)) & ⊢ 𝑃 = (chr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝑌)) = ((𝑃 ↑ 𝑋) + (𝑃 ↑ 𝑌))) | ||
Theorem | cnmsgnsubg 21516 | 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 21517 | 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 21518 | The group of signs under multiplication. (Contributed by Stefan O'Rear, 28-Aug-2015.) |
⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) ⇒ ⊢ 𝑈 ∈ Grp | ||
Theorem | psgnghm 21519 | 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 21520 | 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 21521 | 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 21522 | 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 21523 | 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 21524 | 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 21525 | Even permutations are permutations. (Contributed by SO, 9-Jul-2018.) |
⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑃 = (Base‘𝑆) ⇒ ⊢ (pmEven‘𝐷) ⊆ 𝑃 | ||
Theorem | psgnevpmb 21526 | 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 21527 | 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 21528 | 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 21529 | 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 21530 | The sign of an even permutation embedded into a ring is the unity 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 21531 | The sign of an odd permutation embedded into a ring is the additive inverse of the unity 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 21532 | 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 | zrhpsgnelbas 21533 | 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 21534 | 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 21535* | 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 21536 | 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 21537* | 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 21538* | 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 21539* | Lemma 1 for psgndif 21541. (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 21540* | Lemma 2 for psgndif 21541. (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 21541* | 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 21542* | 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))) | ||
Syntax | crefld 21543 | Extend class notation with the field of real numbers. |
class ℝfld | ||
Definition | df-refld 21544 | The field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ ℝfld = (ℂfld ↾s ℝ) | ||
Theorem | rebase 21545 | The base of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ ℝ = (Base‘ℝfld) | ||
Theorem | remulg 21546 | 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 21547 | 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 21548 | 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 21549 | The addition operation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ + = (+g‘ℝfld) | ||
Theorem | remulr 21550 | The multiplication operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ · = (.r‘ℝfld) | ||
Theorem | re0g 21551 | The zero element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ 0 = (0g‘ℝfld) | ||
Theorem | re1r 21552 | The unity element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ 1 = (1r‘ℝfld) | ||
Theorem | rele2 21553 | The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ ≤ = (le‘ℝfld) | ||
Theorem | relt 21554 | The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ < = (lt‘ℝfld) | ||
Theorem | reds 21555 | The distance of the field of reals. (Contributed by Thierry Arnoux, 20-Jun-2019.) |
⊢ (abs ∘ − ) = (dist‘ℝfld) | ||
Theorem | redvr 21556 | 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 21557 | The real numbers are a totally ordered set. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ ℝfld ∈ Toset | ||
Theorem | refld 21558 | The real numbers form a field. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ ℝfld ∈ Field | ||
Theorem | refldcj 21559 | The conjugation operation of the field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ ∗ = (*𝑟‘ℝfld) | ||
Theorem | resrng 21560 | The real numbers form a star ring. (Contributed by Thierry Arnoux, 19-Apr-2019.) (Proof shortened by Thierry Arnoux, 11-Jan-2025.) |
⊢ ℝfld ∈ *-Ring | ||
Theorem | regsumsupp 21561* | 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 21562 | 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 21563 | Extend class notation with class of all pre-Hilbert spaces. |
class PreHil | ||
Syntax | cipf 21564 | Extend class notation with inner product function. |
class ·if | ||
Definition | df-phl 21565* | 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 21566* | 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 21590), while ·𝑖 only has closure (ipcl 21572). (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) | ||
Theorem | isphl 21567* | 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 21568 | A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) | ||
Theorem | phllmod 21569 | A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | ||
Theorem | phlsrng 21570 | 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 21571* | 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 21572 | 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 21573 | 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 21574 | 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 21575 | 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 21576 | 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 21577 | 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 21578 | 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 21579 | 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 21580 | 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 21581 | 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 21582 | 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 21583 | 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 21584 | 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 21585 | "Associative" law for second argument of inner product (compare ipass 21584). (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 21586 | "Associative" law for inner product. Conjugate version of ipassr 21585. (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 21587* | The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ · = (·if‘𝑊) ⇒ ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) | ||
Theorem | ipfval 21588 | 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 21589 | 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 21590 | The inner product operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·if‘𝑊) ⇒ ⊢ , Fn (𝑉 × 𝑉) | ||
Theorem | phlipf 21591 | 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 21592* | 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 21593* | 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 21594* | 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 21595 | 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 21596 | 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)) | ||
Theorem | phssip 21597 | The inner product (as a function) on a subspace is a restriction of the inner product (as a function) on the parent space. (Contributed by NM, 28-Jan-2008.) (Revised by AV, 19-Oct-2021.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ · = (·if‘𝑊) & ⊢ 𝑃 = (·if‘𝑋) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑃 = ( · ↾ (𝑈 × 𝑈))) | ||
Theorem | phlssphl 21598 | A subspace of an inner product space (pre-Hilbert space) is an inner product space. (Contributed by AV, 25-Sep-2022.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ PreHil) | ||
Syntax | cocv 21599 | Extend class notation with orthocomplement of a subset. |
class ocv | ||
Syntax | ccss 21600 | Extend class notation with set of closed subspaces. |
class ClSubSp |
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