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Theorem List for Metamath Proof Explorer - 19901-20000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremznhash 19901 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 19902 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 19903 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 19904 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 19905 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 19906 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 19907 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 19908 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 19909* Lemma for cygzn 19913. (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 19910* Lemma for cygzn 19913. (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 19911* Lemma for cygzn 19913. (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 19912* 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 19913 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 19914* 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 19915 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 19916 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 19917 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 19918 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 19919 The group of signs under multiplication. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1})       𝑈 ∈ Grp
 
Theorempsgnghm 19920 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 19921 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 19922 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 19923 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 19924 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 19925 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 19926 Even permutations are permutations. (Contributed by SO, 9-Jul-2018.)
𝑆 = (SymGrp‘𝐷)    &   𝑃 = (Base‘𝑆)       (pmEven‘𝐷) ⊆ 𝑃
 
Theorempsgnevpmb 19927 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 19928 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 19929 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 19930 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 19931 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 19932 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 19933 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 19934 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 19935 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 19936* 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 19937 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 19938* 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 19939* 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 19940* Lemma 1 for psgndif 19942. (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 19941* Lemma 2 for psgndif 19942. (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 19942* 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 19943* 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 19944 Extend class notation with the field of real numbers.
class fld
 
Definitiondf-refld 19945 The field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
fld = (ℂflds ℝ)
 
Theoremrebase 19946 The base of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
ℝ = (Base‘ℝfld)
 
Theoremremulg 19947 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 19948 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 19949 Subtraction in the field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
= (-g‘ℝfld)       ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋𝑌) = (𝑋 𝑌))
 
Theoremreplusg 19950 The addition operation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
+ = (+g‘ℝfld)
 
Theoremremulr 19951 The multiplication operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
· = (.r‘ℝfld)
 
Theoremre0g 19952 The neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
0 = (0g‘ℝfld)
 
Theoremre1r 19953 The multiplicative neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
1 = (1r‘ℝfld)
 
Theoremrele2 19954 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
≤ = (le‘ℝfld)
 
Theoremrelt 19955 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
< = (lt‘ℝfld)
 
Theoremreds 19956 The distance of the field of reals. (Contributed by Thierry Arnoux, 20-Jun-2019.)
(abs ∘ − ) = (dist‘ℝfld)
 
Theoremredvr 19957 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 19958 The real numbers are a totally ordered set. (Contributed by Thierry Arnoux, 21-Jan-2018.)
fld ∈ Toset
 
Theoremrefld 19959 The real numbers form a field. (Contributed by Thierry Arnoux, 1-Nov-2017.)
fld ∈ Field
 
Theoremrefldcj 19960 The conjugation operation of the field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
∗ = (*𝑟‘ℝfld)
 
Theoremrecrng 19961 The real numbers form a star ring. (Contributed by Thierry Arnoux, 19-Apr-2019.)
fld ∈ *-Ring
 
Theoremregsumsupp 19962* 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 19963 Extend class notation with class of all pre-Hilbert spaces.
class PreHil
 
Syntaxcipf 19964 Extend class notation with inner product function.
class ·if
 
Definitiondf-phl 19965* 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 19966* 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 19990), while ·𝑖 only has closure (ipcl 19972). (Contributed by Mario Carneiro, 12-Aug-2015.)
·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖𝑔)𝑦)))
 
Theoremisphl 19967* 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 19968 A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ PreHil → 𝑊 ∈ LVec)
 
Theoremphllmod 19969 A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ PreHil → 𝑊 ∈ LMod)
 
Theoremphlsrng 19970 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 19971* 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 19972 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 19973 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 19974 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 19975 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)
 
Theoremip0r 19976 Inner product with a zero second argument. (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) → (𝐴 , 0 ) = 𝑍)
 
Theoremipeq0 19977 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)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑍 = (0g𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉) → ((𝐴 , 𝐴) = 𝑍𝐴 = 0 ))
 
Theoremipdir 19978 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)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (+g𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 + 𝐵) , 𝐶) = ((𝐴 , 𝐶) (𝐵 , 𝐶)))
 
Theoremipdi 19979 Distributive law for inner product (left-distributivity). (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (+g𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴 , (𝐵 + 𝐶)) = ((𝐴 , 𝐵) (𝐴 , 𝐶)))
 
Theoremip2di 19980 Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (+g𝐹)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = (((𝐴 , 𝐶) (𝐵 , 𝐷)) ((𝐴 , 𝐷) (𝐵 , 𝐶))))
 
Theoremipsubdir 19981 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &   𝑆 = (-g𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) , 𝐶) = ((𝐴 , 𝐶)𝑆(𝐵 , 𝐶)))
 
Theoremipsubdi 19982 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &   𝑆 = (-g𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴 , (𝐵 𝐶)) = ((𝐴 , 𝐵)𝑆(𝐴 , 𝐶)))
 
Theoremip2subdi 19983 Distributive law for inner product subtraction. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &   𝑆 = (-g𝐹)    &    + = (+g𝐹)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → ((𝐴 𝐵) , (𝐶 𝐷)) = (((𝐴 , 𝐶) + (𝐵 , 𝐷))𝑆((𝐴 , 𝐷) + (𝐵 , 𝐶))))
 
Theoremipass 19984 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)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → ((𝐴 · 𝐵) , 𝐶) = (𝐴 × (𝐵 , 𝐶)))
 
Theoremipassr 19985 "Associative" law for second argument of inner product (compare ipass 19984). (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝐹)    &    = (*𝑟𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝐾)) → (𝐴 , (𝐶 · 𝐵)) = ((𝐴 , 𝐵) × ( 𝐶)))
 
Theoremipassr2 19986 "Associative" law for inner product. Conjugate version of ipassr 19985. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝐹)    &    = (*𝑟𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝐾)) → ((𝐴 , 𝐵) × 𝐶) = (𝐴 , (( 𝐶) · 𝐵)))
 
Theoremipffval 19987* The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &    · = (·if𝑊)        · = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
 
Theoremipfval 19988 The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &    · = (·if𝑊)       ((𝑋𝑉𝑌𝑉) → (𝑋 · 𝑌) = (𝑋 , 𝑌))
 
Theoremipfeq 19989 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)    &    , = (·𝑖𝑊)    &    · = (·if𝑊)       ( , Fn (𝑉 × 𝑉) → · = , )
 
Theoremipffn 19990 The inner product operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
𝑉 = (Base‘𝑊)    &    , = (·if𝑊)        , Fn (𝑉 × 𝑉)
 
Theoremphlipf 19991 The inner product operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝑉 = (Base‘𝑊)    &    , = (·if𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)       (𝑊 ∈ PreHil → , :(𝑉 × 𝑉)⟶𝐾)
 
Theoremip2eq 19992* 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.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → (𝐴 = 𝐵 ↔ ∀𝑥𝑉 (𝑥 , 𝐴) = (𝑥 , 𝐵)))
 
Theoremisphld 19993* Properties that determine a pre-Hilbert (inner product) space. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 7-Oct-2015.)
(𝜑𝑉 = (Base‘𝑊))    &   (𝜑+ = (+g𝑊))    &   (𝜑· = ( ·𝑠𝑊))    &   (𝜑𝐼 = (·𝑖𝑊))    &   (𝜑0 = (0g𝑊))    &   (𝜑𝐹 = (Scalar‘𝑊))    &   (𝜑𝐾 = (Base‘𝐹))    &   (𝜑 = (+g𝐹))    &   (𝜑× = (.r𝐹))    &   (𝜑 = (*𝑟𝐹))    &   (𝜑𝑂 = (0g𝐹))    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐹 ∈ *-Ring)    &   ((𝜑𝑥𝑉𝑦𝑉) → (𝑥𝐼𝑦) ∈ 𝐾)    &   ((𝜑𝑞𝐾 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) (𝑦𝐼𝑧)))    &   ((𝜑𝑥𝑉 ∧ (𝑥𝐼𝑥) = 𝑂) → 𝑥 = 0 )    &   ((𝜑𝑥𝑉𝑦𝑉) → ( ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥))       (𝜑𝑊 ∈ PreHil)
 
Theoremphlpropd 19994* 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.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   (𝜑𝐹 = (Scalar‘𝐾))    &   (𝜑𝐹 = (Scalar‘𝐿))    &   𝑃 = (Base‘𝐹)    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(·𝑖𝐾)𝑦) = (𝑥(·𝑖𝐿)𝑦))       (𝜑 → (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))
 
Theoremssipeq 19995 The inner product on a subspace equals the inner product on the parent space. (Contributed by AV, 19-Oct-2021.)
𝑋 = (𝑊s 𝑈)    &    , = (·𝑖𝑊)    &   𝑃 = (·𝑖𝑋)       (𝑈𝑆𝑃 = , )
 
Theoremphssipval 19996 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 = (𝑊s 𝑈)    &    , = (·𝑖𝑊)    &   𝑃 = (·𝑖𝑋)    &   𝑆 = (LSubSp‘𝑊)       (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ (𝐴𝑈𝐵𝑈)) → (𝐴𝑃𝐵) = (𝐴 , 𝐵))
 
Theoremphssip 19997 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 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)    &    · = (·if𝑊)    &   𝑃 = (·if𝑋)       ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑃 = ( · ↾ (𝑈 × 𝑈)))
 
10.12.2  Orthocomplements and closed subspaces
 
Syntaxcocv 19998 Extend class notation with orthocomplement of a subspace.
class ocv
 
Syntaxccss 19999 Extend class notation with set of closed subspaces.
class CSubSp
 
Syntaxcthl 20000 Extend class notation with the Hilbert lattice.
class toHL
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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 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