P. 216 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21501-21600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	znleval 21501	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 21502	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 21503	Lemma for zntos 21504. (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 21504	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 21505	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 21506	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 21507	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 21508	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 21509	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 21510	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 21511	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 21512	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 21513*	Lemma for cygzn 21517. (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 21514*	Lemma for cygzn 21517. (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 21515*	Lemma for cygzn 21517. (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 21516*	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 21517	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 21518*	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 21519	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 21520	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 21521	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	frobrhm 21522*	In a commutative ring with prime characteristic, the Frobenius function 𝐹 is a ring endomorphism, thus named the Frobenius endomorphism. (Contributed by Thierry Arnoux, 31-May-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (chr‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑃 ↑ 𝑥))    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑅))
Theorem	ofldchr 21523	The characteristic of an ordered field is zero. (Contributed by Thierry Arnoux, 21-Jan-2018.) (Proof shortened by AV, 6-Oct-2020.)
⊢ (𝐹 ∈ oField → (chr‘𝐹) = 0)
10.8.4  Signs as subgroup of the complex numbers
Theorem	cnmsgnsubg 21524	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 21525	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 21526	The group of signs under multiplication. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1})    ⇒   ⊢ 𝑈 ∈ Grp
Theorem	psgnghm 21527	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 21528	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 21529	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 21530	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)))
10.8.5  Embedding of permutation signs into a ring
Theorem	zrhpsgnmhm 21531	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 21532	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 21533	Even permutations are permutations. (Contributed by SO, 9-Jul-2018.)
⊢ 𝑆 = (SymGrp‘𝐷)    &   ⊢ 𝑃 = (Base‘𝑆)    ⇒   ⊢ (pmEven‘𝐷) ⊆ 𝑃
Theorem	psgnevpmb 21534	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 21535	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 21536	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 21537	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 21538	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 21539	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 21540	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 21541	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 21542	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 21543*	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 21544	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 21545*	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 21546*	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 21547*	Lemma 1 for psgndif 21549. (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 21548*	Lemma 2 for psgndif 21549. (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 21549*	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 21550*	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)))
10.8.6  The ordered field of real numbers
Syntax	crefld 21551	Extend class notation with the field of real numbers.
class ℝfld
Definition	df-refld 21552	The field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
⊢ ℝfld = (ℂfld ↾s ℝ)
Theorem	rebase 21553	The base of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
⊢ ℝ = (Base‘ℝfld)
Theorem	remulg 21554	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 21555	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 21556	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 21557	The addition operation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
⊢ + = (+g‘ℝfld)
Theorem	remulr 21558	The multiplication operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
⊢ · = (.r‘ℝfld)
Theorem	re0g 21559	The zero element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
⊢ 0 = (0g‘ℝfld)
Theorem	re1r 21560	The unity element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
⊢ 1 = (1r‘ℝfld)
Theorem	rele2 21561	The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
⊢ ≤ = (le‘ℝfld)
Theorem	relt 21562	The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
⊢ < = (lt‘ℝfld)
Theorem	reds 21563	The distance of the field of reals. (Contributed by Thierry Arnoux, 20-Jun-2019.)
⊢ (abs ∘ − ) = (dist‘ℝfld)
Theorem	redvr 21564	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 21565	The real numbers are a totally ordered set. (Contributed by Thierry Arnoux, 21-Jan-2018.)
⊢ ℝfld ∈ Toset
Theorem	refld 21566	The real numbers form a field. (Contributed by Thierry Arnoux, 1-Nov-2017.)
⊢ ℝfld ∈ Field
Theorem	refldcj 21567	The conjugation operation of the field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
⊢ ∗ = (*𝑟‘ℝfld)
Theorem	resrng 21568	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 21569*	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 21570	The quotient group ℝ / ℤ is a group. (Contributed by Thierry Arnoux, 26-Jan-2020.)
⊢ 𝑅 = (ℝfld /s (ℝfld ~QG ℤ))    ⇒   ⊢ 𝑅 ∈ Grp
10.9  Generalized pre-Hilbert and Hilbert spaces
10.9.1  Definition and basic properties
Syntax	cphl 21571	Extend class notation with class of all pre-Hilbert spaces.
class PreHil
Syntax	cipf 21572	Extend class notation with inner product function.
class ·if
Definition	df-phl 21573*	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 21574*	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 21598), while ·𝑖 only has closure (ipcl 21580). (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)))
Theorem	isphl 21575*	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 21576	A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
Theorem	phllmod 21577	A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
Theorem	phlsrng 21578	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 21579*	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 21580	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 21581	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 21582	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 21583	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 21584	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 21585	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 21586	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 21587	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 21588	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 21589	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 21590	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 21591	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 21592	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 21593	"Associative" law for second argument of inner product (compare ipass 21592). (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 21594	"Associative" law for inner product. Conjugate version of ipassr 21593. (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 21595*	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 21596	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 21597	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 21598	The inner product operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·if‘𝑊)    ⇒   ⊢ , Fn (𝑉 × 𝑉)
Theorem	phlipf 21599	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 21600*	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)))