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Theorem List for Metamath Proof Explorer - 19601-19700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremchrdvds 19601 The ring homomorphism is zero only at multiples of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝐶 = (chr‘𝑅)    &   𝐿 = (ℤRHom‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (𝐶𝑁 ↔ (𝐿𝑁) = 0 ))
 
Theoremchrcong 19602 If two integers are congruent relative to the ring characteristic, their images in the ring are the same. (Contributed by Mario Carneiro, 24-Sep-2015.)
𝐶 = (chr‘𝑅)    &   𝐿 = (ℤRHom‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐶 ∥ (𝑀𝑁) ↔ (𝐿𝑀) = (𝐿𝑁)))
 
Theoremchrnzr 19603 Nonzero rings are precisely those with characteristic not 1. (Contributed by Stefan O'Rear, 6-Sep-2015.)
(𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ (chr‘𝑅) ≠ 1))
 
Theoremchrrhm 19604 The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)
(𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑆) ∥ (chr‘𝑅))
 
Theoremdomnchr 19605 The characteristic of a domain can only be zero or a prime. (Contributed by Stefan O'Rear, 6-Sep-2015.)
(𝑅 ∈ Domn → ((chr‘𝑅) = 0 ∨ (chr‘𝑅) ∈ ℙ))
 
Theoremznlidl 19606 The set 𝑛 is an ideal in . (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
𝑆 = (RSpan‘ℤring)       (𝑁 ∈ ℤ → (𝑆‘{𝑁}) ∈ (LIdeal‘ℤring))
 
Theoremzncrng2 19607 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 19608 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 19609 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 19610 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 19611 Lemma for znbas 19617. (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 19612 Obsolete version of znbaslem 19611 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 19613 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 19614 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 19615 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 19616 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 19617 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 19618 ℤ/n is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (ℤ/nℤ‘𝑁)       (𝑁 ∈ ℕ0𝑌 ∈ CRing)
 
Theoremznzrh2 19619* 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 19620 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 19621 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 19622 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 19623 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 19624 Special case of zndvds 19623 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 19625 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 19626 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 19627 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 19628 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 19629 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 19630 Lemma for zntos 19631. (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 19631 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 19632 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 19633 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 19634 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 19635 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 19636 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 19637 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 19638 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 19639 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 19640* Lemma for cygzn 19644. (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 19641* Lemma for cygzn 19644. (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 19642* Lemma for cygzn 19644. (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 19643* 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 19644 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 19645* 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 19646 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 19647 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 19648 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 19649 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 19650 The group of signs under multiplication. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1})       𝑈 ∈ Grp
 
Theorempsgnghm 19651 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 19652 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 19653 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 19654 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 19655 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 19656 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 19657 Even permutations are permutations. (Contributed by SO, 9-Jul-2018.)
𝑆 = (SymGrp‘𝐷)    &   𝑃 = (Base‘𝑆)       (pmEven‘𝐷) ⊆ 𝑃
 
Theorempsgnevpmb 19658 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 19659 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 19660 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 19661 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 19662 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 19663 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 19664 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 19665 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 19666 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 19667* 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 19668 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 19669* 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 19670* 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 19671* Lemma 1 for psgndif 19673. (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 19672* Lemma 2 for psgndif 19673. (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 19673* 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 19674* 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 19675 Extend class notation with the field of real numbers.
class fld
 
Definitiondf-refld 19676 The field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
fld = (ℂflds ℝ)
 
Theoremrebase 19677 The base of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
ℝ = (Base‘ℝfld)
 
Theoremremulg 19678 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 19679 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 19680 Subtraction in the field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
= (-g‘ℝfld)       ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋𝑌) = (𝑋 𝑌))
 
Theoremreplusg 19681 The addition operation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
+ = (+g‘ℝfld)
 
Theoremremulr 19682 The multiplication operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
· = (.r‘ℝfld)
 
Theoremre0g 19683 The neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
0 = (0g‘ℝfld)
 
Theoremre1r 19684 The multiplicative neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
1 = (1r‘ℝfld)
 
Theoremrele2 19685 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
≤ = (le‘ℝfld)
 
Theoremrelt 19686 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
< = (lt‘ℝfld)
 
Theoremreds 19687 The distance of the field of reals. (Contributed by Thierry Arnoux, 20-Jun-2019.)
(abs ∘ − ) = (dist‘ℝfld)
 
Theoremredvr 19688 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 19689 The real numbers are a totally ordered set. (Contributed by Thierry Arnoux, 21-Jan-2018.)
fld ∈ Toset
 
Theoremrefld 19690 The real numbers form a field. (Contributed by Thierry Arnoux, 1-Nov-2017.)
fld ∈ Field
 
Theoremrefldcj 19691 The conjugation operation of the field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
∗ = (*𝑟‘ℝfld)
 
Theoremrecrng 19692 The real numbers form a star ring. (Contributed by Thierry Arnoux, 19-Apr-2019.)
fld ∈ *-Ring
 
Theoremregsumsupp 19693* 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 19694 Extend class notation with class all pre-Hilbert spaces.
class PreHil
 
Syntaxcipf 19695 Extend class notation with inner product function.
class ·if
 
Definitiondf-phl 19696* Define class all generalized pre-Hilbert (inner product) spaces. (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 19697* Define group addition function. Usually we will use +g directly instead of +𝑓, and they have the same behavior in most cases. The main advantage of +𝑓 is that it is a guaranteed function (mndplusf 17024), while +g only has closure (mndcl 17016). (Contributed by Mario Carneiro, 12-Aug-2015.)
·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖𝑔)𝑦)))
 
Theoremisphl 19698* 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 19699 A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ PreHil → 𝑊 ∈ LVec)
 
Theoremphllmod 19700 A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ PreHil → 𝑊 ∈ LMod)
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
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