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Theorem List for Metamath Proof Explorer - 19701-19800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremzncrng2 19701 The value of the ℤ/n structure. It is defined as the quotient ring ℤ / 𝑛, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/n 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)

Theoremznval 19702 The value of the ℤ/n structure. It is defined as the quotient ring ℤ / 𝑛, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/n 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 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘ndx), ⟩))

Theoremznle 19703 The value of the ℤ/n structure. It is defined as the quotient ring ℤ / 𝑛, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/n 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))

Theoremznval2 19704 Self-referential expression for the ℤ/n 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 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘ndx), ⟩))

Theoremznbaslem 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 → (𝐸𝑈) = (𝐸𝑌))

TheoremznbaslemOLD 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 → (𝐸𝑈) = (𝐸𝑌))

Theoremznbas2 19707 The base set of ℤ/n 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))

Theoremznadd 19708 The additive structure of ℤ/n 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 → (+g𝑈) = (+g𝑌))

Theoremznmul 19709 The multiplicative structure of ℤ/n 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 → (.r𝑈) = (.r𝑌))

Theoremznzrh 19710 The ring homomorphism of ℤ/n 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))

Theoremznbas 19711 The base set of ℤ/n 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))

Theoremzncrng 19712 ℤ/n is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (ℤ/nℤ‘𝑁)       (𝑁 ∈ ℕ0𝑌 ∈ CRing)

Theoremznzrh2 19713* The 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 ∈ ℕ0𝐿 = (𝑥 ∈ ℤ ↦ [𝑥] ))

Theoremznzrhval 19714 The 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 ∈ ℕ0𝐴 ∈ ℤ) → (𝐿𝐴) = [𝐴] )

Theoremznzrhfo 19715 The ring homomorphism is a surjection onto ℤ / 𝑛. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑌)    &   𝐿 = (ℤRHom‘𝑌)       (𝑁 ∈ ℕ0𝐿:ℤ–onto𝐵)

Theoremzncyg 19716 The group ℤ / 𝑛 is cyclic for all 𝑛 (including 𝑛 = 0). (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑌 = (ℤ/nℤ‘𝑁)       (𝑁 ∈ ℕ0𝑌 ∈ CycGrp)

Theoremzndvds 19717 Express equality of equivalence classes in ℤ / 𝑛 in terms of divisibility. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑌)       ((𝑁 ∈ ℕ0𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐿𝐴) = (𝐿𝐵) ↔ 𝑁 ∥ (𝐴𝐵)))

Theoremzndvds0 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 = (0g𝑌)       ((𝑁 ∈ ℕ0𝐴 ∈ ℤ) → ((𝐿𝐴) = 0𝑁𝐴))

Theoremznf1o 19719 The function 𝐹 enumerates all equivalence classes in ℤ/n 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 ∈ ℕ0𝐹:𝑊1-1-onto𝐵)

Theoremzzngim 19720 The 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)

Theoremznle2 19721 The ordering of the ℤ/n 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))

Theoremznleval 19722 The ordering of the ℤ/n 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 ↔ (𝐴𝑋𝐵𝑋 ∧ (𝐹𝐴) ≤ (𝐹𝐵))))

Theoremznleval2 19723 The ordering of the ℤ/n 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 ∈ ℕ0𝐴𝑋𝐵𝑋) → (𝐴 𝐵 ↔ (𝐹𝐴) ≤ (𝐹𝐵)))

Theoremzntoslem 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 ∈ ℕ0𝑌 ∈ Toset)

Theoremzntos 19725 The ℤ/n 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 .) (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (ℤ/nℤ‘𝑁)       (𝑁 ∈ ℕ0𝑌 ∈ Toset)

Theoremznhash 19726 The ℤ/n structure has 𝑛 elements. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑌)       (𝑁 ∈ ℕ → (#‘𝐵) = 𝑁)

Theoremznfi 19727 The ℤ/n structure is a finite ring. (Contributed by Mario Carneiro, 2-May-2016.)
𝑌 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑌)       (𝑁 ∈ ℕ → 𝐵 ∈ Fin)

Theoremznfld 19728 The ℤ/n structure is a finite field when 𝑛 is prime. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (ℤ/nℤ‘𝑁)       (𝑁 ∈ ℙ → 𝑌 ∈ Field)

Theoremznidomb 19729 The ℤ/n 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))

Theoremznchr 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)

Theoremznunit 19731 The units of ℤ/n 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 ∈ ℕ0𝐴 ∈ ℤ) → ((𝐿𝐴) ∈ 𝑈 ↔ (𝐴 gcd 𝑁) = 1))

Theoremznunithash 19732 The size of the unit group of ℤ/n. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑌 = (ℤ/nℤ‘𝑁)    &   𝑈 = (Unit‘𝑌)       (𝑁 ∈ ℕ → (#‘𝑈) = (ϕ‘𝑁))

Theoremznrrg 19733 The regular elements of ℤ/n 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)

Theoremcygznlem1 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)    &    · = (.g𝐺)    &   𝐿 = (ℤRHom‘𝑌)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   (𝜑𝐺 ∈ CycGrp)    &   (𝜑𝑋𝐸)       ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝐿𝐾) = (𝐿𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋)))

Theoremcygznlem2a 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)    &    · = (.g𝐺)    &   𝐿 = (ℤRHom‘𝑌)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   (𝜑𝐺 ∈ CycGrp)    &   (𝜑𝑋𝐸)    &   𝐹 = ran (𝑚 ∈ ℤ ↦ ⟨(𝐿𝑚), (𝑚 · 𝑋)⟩)       (𝜑𝐹:(Base‘𝑌)⟶𝐵)

Theoremcygznlem2 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)    &    · = (.g𝐺)    &   𝐿 = (ℤRHom‘𝑌)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   (𝜑𝐺 ∈ CycGrp)    &   (𝜑𝑋𝐸)    &   𝐹 = ran (𝑚 ∈ ℤ ↦ ⟨(𝐿𝑚), (𝑚 · 𝑋)⟩)       ((𝜑𝑀 ∈ ℤ) → (𝐹‘(𝐿𝑀)) = (𝑀 · 𝑋))

Theoremcygznlem3 19737* A cyclic group with 𝑛 elements is isomorphic to ℤ / 𝑛. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑁 = if(𝐵 ∈ Fin, (#‘𝐵), 0)    &   𝑌 = (ℤ/nℤ‘𝑁)    &    · = (.g𝐺)    &   𝐿 = (ℤRHom‘𝑌)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   (𝜑𝐺 ∈ CycGrp)    &   (𝜑𝑋𝐸)    &   𝐹 = ran (𝑚 ∈ ℤ ↦ ⟨(𝐿𝑚), (𝑚 · 𝑋)⟩)       (𝜑𝐺𝑔 𝑌)

Theoremcygzn 19738 A cyclic group with 𝑛 elements is isomorphic to ℤ / 𝑛, 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 → 𝐺𝑔 𝑌)

Theoremcygth 19739* The "fundamental theorem of cyclic groups". Cyclic groups are exactly the additive groups ℤ / 𝑛, for 0 ≤ 𝑛 (where 𝑛 = 0 is the infinite cyclic group ), up to isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝐺 ∈ CycGrp ↔ ∃𝑛 ∈ ℕ0 𝐺𝑔 (ℤ/nℤ‘𝑛))

Theoremcyggic 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) → (𝐺𝑔 𝐻𝐵𝐶))

Theoremfrgpcyg 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 ≼ 1𝑜𝐺 ∈ CycGrp)

10.11.4  Signs as subgroup of the complex numbers

Theoremcnmsgnsubg 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)

Theoremcnmsgnbas 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)

Theoremcnmsgngrp 19744 The group of signs under multiplication. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1})       𝑈 ∈ Grp

Theorempsgnghm 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 = (𝑆s dom 𝑁)    &   𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1})       (𝐷𝑉𝑁 ∈ (𝐹 GrpHom 𝑈))

Theorempsgnghm2 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))

Theorempsgninv 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 ∧ 𝐹𝑃) → (𝑁𝐹) = (𝑁𝐹))

Theorempsgnco 19748 Multiplicativity of the permutation sign function. (Contributed by SO, 9-Jul-2018.)
𝑆 = (SymGrp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)    &   𝑃 = (Base‘𝑆)       ((𝐷 ∈ Fin ∧ 𝐹𝑃𝐺𝑃) → (𝑁‘(𝐹𝐺)) = ((𝑁𝐹) · (𝑁𝐺)))

10.11.5  Embedding of permutation signs into a ring

Theoremzrhpsgnmhm 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)))

Theoremzrhpsgninv 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 ∧ 𝐹𝑃) → ((𝑌𝑆)‘𝐹) = ((𝑌𝑆)‘𝐹))

Theoremevpmss 19751 Even permutations are permutations. (Contributed by SO, 9-Jul-2018.)
𝑆 = (SymGrp‘𝐷)    &   𝑃 = (Base‘𝑆)       (pmEven‘𝐷) ⊆ 𝑃

Theorempsgnevpmb 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) ↔ (𝐹𝑃 ∧ (𝑁𝐹) = 1)))

Theorempsgnodpm 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))) → (𝑁𝐹) = -1)

Theorempsgnevpm 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)) → (𝑁𝐹) = 1)

Theorempsgnodpmr 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 ∧ 𝐹𝑃 ∧ (𝑁𝐹) = -1) → 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)))

Theoremzrhpsgnevpm 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 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → ((𝑌𝑆)‘𝐹) = 1 )

Theoremzrhpsgnodpm 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 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐼 = (invg𝑅)       ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → ((𝑌𝑆)‘𝐹) = (𝐼1 ))

Theoremzrhcofipsgn 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 ∧ 𝑄𝑃) → ((𝑌𝑆)‘𝑄) = (𝑌‘(𝑆𝑄)))

Theoremzrhpsgnelbas 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 ∧ 𝑄𝑃) → (𝑌‘(𝑆𝑄)) ∈ (Base‘𝑅))

Theoremzrhcopsgnelbas 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 ∧ 𝑄𝑃) → ((𝑌𝑆)‘𝑄) ∈ (Base‘𝑅))

Theoremevpmodpmf1o 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(+g𝑆)𝑓)):(pmEven‘𝐷)–1-1-onto→(𝑃 ∖ (pmEven‘𝐷)))

Theorempmtrodpm 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 ∧ 𝐹𝑇) → 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)))

Theorempsgnfix1 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 ∧ 𝐾𝑁) → (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} → ∃𝑤 ∈ Word 𝑇(𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑤)))

Theorempsgnfix2 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 ∧ 𝐾𝑁) → (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} → ∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σg 𝑤)))

TheorempsgndiflemB 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 ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)) → ((𝑈 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑈) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑈𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))) → 𝑄 = (𝑍 Σg 𝑈))))

TheorempsgndiflemA 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 ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → (-1↑(#‘𝑊)) = (-1↑(#‘𝑈)))))

Theorempsgndif 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 ∧ 𝐾𝑁) → (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} → (𝑍‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = (𝑆𝑄)))

Theoremzrhcopsgndif 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 ∧ 𝐾𝑁) → (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} → ((𝑌𝑍)‘(𝑄 ↾ (𝑁 ∖ {𝐾}))) = ((𝑌𝑆)‘𝑄)))

10.11.6  The ordered field of real numbers

Syntaxcrefld 19769 Extend class notation with the field of real numbers.
class fld

Definitiondf-refld 19770 The field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
fld = (ℂflds ℝ)

Theoremrebase 19771 The base of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
ℝ = (Base‘ℝfld)

Theoremremulg 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))

Theoremresubdrg 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)

Theoremresubgval 19774 Subtraction in the field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
= (-g‘ℝfld)       ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋𝑌) = (𝑋 𝑌))

Theoremreplusg 19775 The addition operation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
+ = (+g‘ℝfld)

Theoremremulr 19776 The multiplication operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
· = (.r‘ℝfld)

Theoremre0g 19777 The neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
0 = (0g‘ℝfld)

Theoremre1r 19778 The multiplicative neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
1 = (1r‘ℝfld)

Theoremrele2 19779 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
≤ = (le‘ℝfld)

Theoremrelt 19780 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
< = (lt‘ℝfld)

Theoremreds 19781 The distance of the field of reals. (Contributed by Thierry Arnoux, 20-Jun-2019.)
(abs ∘ − ) = (dist‘ℝfld)

Theoremredvr 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))

Theoremretos 19783 The real numbers are a totally ordered set. (Contributed by Thierry Arnoux, 21-Jan-2018.)
fld ∈ Toset

Theoremrefld 19784 The real numbers form a field. (Contributed by Thierry Arnoux, 1-Nov-2017.)
fld ∈ Field

Theoremrefldcj 19785 The conjugation operation of the field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
∗ = (*𝑟‘ℝfld)

Theoremrecrng 19786 The real numbers form a star ring. (Contributed by Thierry Arnoux, 19-Apr-2019.)
fld ∈ *-Ring

Theoremregsumsupp 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 ∧ 𝐼𝑉) → (ℝfld Σg 𝐹) = Σ𝑥 ∈ (𝐹 supp 0)(𝐹𝑥))

10.12  Generalized pre-Hilbert and Hilbert spaces

10.12.1  Definition and basic properties

Syntaxcphl 19788 Extend class notation with class of all pre-Hilbert spaces.
class PreHil

Syntaxcipf 19789 Extend class notation with inner product function.
class ·if

Definitiondf-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][(·𝑖𝑔) / ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥)))}

Definitiondf-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(·𝑖𝑔)𝑦)))

Theoremisphl 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)    &    , = (·𝑖𝑊)    &    0 = (0g𝑊)    &    = (*𝑟𝐹)    &   𝑍 = (0g𝐹)       (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 ) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))))

Theoremphllvec 19793 A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ PreHil → 𝑊 ∈ LVec)

Theoremphllmod 19794 A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ PreHil → 𝑊 ∈ LMod)

Theoremphlsrng 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)

Theoremphllmhm 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)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐺 = (𝑥𝑉 ↦ (𝑥 , 𝐴))       ((𝑊 ∈ PreHil ∧ 𝐴𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))

Theoremipcl 19797 Closure of the inner product operation in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → (𝐴 , 𝐵) ∈ 𝐾)

Theoremipcj 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)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (*𝑟𝐹)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))

Theoremiporthcom 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)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑍 = (0g𝐹)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → ((𝐴 , 𝐵) = 𝑍 ↔ (𝐵 , 𝐴) = 𝑍))

Theoremip0l 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)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑍 = (0g𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉) → ( 0 , 𝐴) = 𝑍)

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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