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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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