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Theorem List for Metamath Proof Explorer - 19901-20000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-zlm 19901 Augment an abelian group with vector space operations to turn it into a -module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝑔)⟩))

Definitiondf-chr 19902 The characteristic of a ring is the smallest positive integer which is equal to 0 when interpreted in the ring, or 0 if there is no such positive integer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
chr = (𝑔 ∈ V ↦ ((od‘𝑔)‘(1r𝑔)))

Definitiondf-zn 19903* Define the ring of integers mod 𝑛. This is literally the quotient ring of by the ideal 𝑛, but we augment it with a total order. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.)
ℤ/nℤ = (𝑛 ∈ ℕ0ring / 𝑧(𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) / 𝑠(𝑠 sSet ⟨(le‘ndx), ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓)⟩))

Theoremzrhval 19904 Define the unique homomorphism from the integers to a ring or field. (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
𝐿 = (ℤRHom‘𝑅)       𝐿 = (ℤring RingHom 𝑅)

Theoremzrhval2 19905* Alternate value of the ℤRHom homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝐿 = (ℤRHom‘𝑅)    &    · = (.g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → 𝐿 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 )))

Theoremzrhmulg 19906 Value of the ℤRHom homomorphism. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐿 = (ℤRHom‘𝑅)    &    · = (.g𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (𝐿𝑁) = (𝑁 · 1 ))

Theoremzrhrhmb 19907 The ℤRHom homomorphism is the unique ring homomorphism from 𝑍. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 12-Jun-2019.)
𝐿 = (ℤRHom‘𝑅)       (𝑅 ∈ Ring → (𝐹 ∈ (ℤring RingHom 𝑅) ↔ 𝐹 = 𝐿))

Theoremzrhrhm 19908 The ℤRHom homomorphism is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) (Revised by AV, 12-Jun-2019.)
𝐿 = (ℤRHom‘𝑅)       (𝑅 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑅))

Theoremzrh1 19909 Interpretation of 1 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝐿 = (ℤRHom‘𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (𝐿‘1) = 1 )

Theoremzrh0 19910 Interpretation of 0 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝐿 = (ℤRHom‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → (𝐿‘0) = 0 )

Theoremzrhpropd 19911* The ring homomorphism depends only on the ring attributes of a structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (ℤRHom‘𝐾) = (ℤRHom‘𝐿))

Theoremzlmval 19912 Augment an abelian group with vector space operations to turn it into a -module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
𝑊 = (ℤMod‘𝐺)    &    · = (.g𝐺)       (𝐺𝑉𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))

Theoremzlmlem 19913 Lemma for zlmbas 19914 and zlmplusg 19915. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = (ℤMod‘𝐺)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 < 5       (𝐸𝐺) = (𝐸𝑊)

Theoremzlmbas 19914 Base set of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = (ℤMod‘𝐺)    &   𝐵 = (Base‘𝐺)       𝐵 = (Base‘𝑊)

Theoremzlmplusg 19915 Group operation of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = (ℤMod‘𝐺)    &    + = (+g𝐺)        + = (+g𝑊)

Theoremzlmmulr 19916 Ring operation of a -module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = (ℤMod‘𝐺)    &    · = (.r𝐺)        · = (.r𝑊)

Theoremzlmsca 19917 Scalar ring of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
𝑊 = (ℤMod‘𝐺)       (𝐺𝑉 → ℤring = (Scalar‘𝑊))

Theoremzlmvsca 19918 Scalar multiplication operation of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = (ℤMod‘𝐺)    &    · = (.g𝐺)        · = ( ·𝑠𝑊)

Theoremzlmlmod 19919 The -module operation turns an arbitrary abelian group into a left module over . (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = (ℤMod‘𝐺)       (𝐺 ∈ Abel ↔ 𝑊 ∈ LMod)

Theoremzlmassa 19920 The -module operation turns a ring into an associative algebra over . (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = (ℤMod‘𝐺)       (𝐺 ∈ Ring ↔ 𝑊 ∈ AssAlg)

Theoremchrval 19921 Definition substitution of the ring characteristic. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝑂 = (od‘𝑅)    &    1 = (1r𝑅)    &   𝐶 = (chr‘𝑅)       (𝑂1 ) = 𝐶

Theoremchrcl 19922 Closure of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝐶 = (chr‘𝑅)       (𝑅 ∈ Ring → 𝐶 ∈ ℕ0)

Theoremchrid 19923 The canonical ring homomorphism applied to a ring's characteristic is zero. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝐶 = (chr‘𝑅)    &   𝐿 = (ℤRHom‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → (𝐿𝐶) = 0 )

Theoremchrdvds 19924 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 19925 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 19926 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 19927 The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)
(𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑆) ∥ (chr‘𝑅))

Theoremdomnchr 19928 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 19929 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 19930 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 19931 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 19932 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 19933 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 19934 Lemma for znbas 19940. (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 19935 Obsolete version of znbaslem 19934 as of 9-Sep-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 19936 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 19937 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 19938 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 19939 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 19940 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 19941 ℤ/n is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (ℤ/nℤ‘𝑁)       (𝑁 ∈ ℕ0𝑌 ∈ CRing)

Theoremznzrh2 19942* 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 19943 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 19944 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 19945 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 19946 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 19947 Special case of zndvds 19946 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 19948 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 19949 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 19950 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 19951 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 19952 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 19953 Lemma for zntos 19954. (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 19954 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 19955 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 19956 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 19957 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 19958 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 19959 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 19960 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 19961 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 19962 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 19963* Lemma for cygzn 19967. (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 19964* Lemma for cygzn 19967. (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 19965* Lemma for cygzn 19967. (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 19966* 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 19967 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 19968* 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 19969 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 19970 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 19971 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 19972 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 19973 The group of signs under multiplication. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1})       𝑈 ∈ Grp

Theorempsgnghm 19974 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 19975 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 19976 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 19977 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 19978 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 19979 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 19980 Even permutations are permutations. (Contributed by SO, 9-Jul-2018.)
𝑆 = (SymGrp‘𝐷)    &   𝑃 = (Base‘𝑆)       (pmEven‘𝐷) ⊆ 𝑃

Theorempsgnevpmb 19981 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 19982 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 19983 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 19984 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 19985 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 19986 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 19987 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 19988 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 19989 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 19990* 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 19991 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 19992* 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 19993* 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 19994* Lemma 1 for psgndif 19996. (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 19995* Lemma 2 for psgndif 19996. (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 19996* 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 19997* 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 19998 Extend class notation with the field of real numbers.
class fld

Definitiondf-refld 19999 The field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
fld = (ℂflds ℝ)

Theoremrebase 20000 The base of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
ℝ = (Base‘ℝfld)

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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 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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