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Mirrors > Home > MPE Home > Th. List > chrdvds | Structured version Visualization version GIF version |
Description: The ℤ ring homomorphism is zero only at multiples of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.) |
Ref | Expression |
---|---|
chrcl.c | ⊢ 𝐶 = (chr‘𝑅) |
chrid.l | ⊢ 𝐿 = (ℤRHom‘𝑅) |
chrid.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
chrdvds | ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (𝐶 ∥ 𝑁 ↔ (𝐿‘𝑁) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . . 5 ⊢ (od‘𝑅) = (od‘𝑅) | |
2 | eqid 2737 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | chrcl.c | . . . . 5 ⊢ 𝐶 = (chr‘𝑅) | |
4 | 1, 2, 3 | chrval 20880 | . . . 4 ⊢ ((od‘𝑅)‘(1r‘𝑅)) = 𝐶 |
5 | 4 | breq1i 5110 | . . 3 ⊢ (((od‘𝑅)‘(1r‘𝑅)) ∥ 𝑁 ↔ 𝐶 ∥ 𝑁) |
6 | ringgrp 19922 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
7 | 6 | adantr 481 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → 𝑅 ∈ Grp) |
8 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 8, 2 | ringidcl 19942 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
10 | 9 | adantr 481 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (1r‘𝑅) ∈ (Base‘𝑅)) |
11 | simpr 485 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
12 | eqid 2737 | . . . . 5 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
13 | chrid.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
14 | 8, 1, 12, 13 | oddvds 19287 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ (1r‘𝑅) ∈ (Base‘𝑅) ∧ 𝑁 ∈ ℤ) → (((od‘𝑅)‘(1r‘𝑅)) ∥ 𝑁 ↔ (𝑁(.g‘𝑅)(1r‘𝑅)) = 0 )) |
15 | 7, 10, 11, 14 | syl3anc 1371 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (((od‘𝑅)‘(1r‘𝑅)) ∥ 𝑁 ↔ (𝑁(.g‘𝑅)(1r‘𝑅)) = 0 )) |
16 | 5, 15 | bitr3id 284 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (𝐶 ∥ 𝑁 ↔ (𝑁(.g‘𝑅)(1r‘𝑅)) = 0 )) |
17 | chrid.l | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
18 | 17, 12, 2 | zrhmulg 20862 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (𝐿‘𝑁) = (𝑁(.g‘𝑅)(1r‘𝑅))) |
19 | 18 | eqeq1d 2739 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → ((𝐿‘𝑁) = 0 ↔ (𝑁(.g‘𝑅)(1r‘𝑅)) = 0 )) |
20 | 16, 19 | bitr4d 281 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (𝐶 ∥ 𝑁 ↔ (𝐿‘𝑁) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 ‘cfv 6493 (class class class)co 7351 ℤcz 12457 ∥ cdvds 16095 Basecbs 17042 0gc0g 17280 Grpcgrp 18707 .gcmg 18830 odcod 19264 1rcur 19871 Ringcrg 19917 ℤRHomczrh 20852 chrcchr 20854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-rp 12870 df-fz 13379 df-fl 13651 df-mod 13729 df-seq 13861 df-exp 13922 df-cj 14943 df-re 14944 df-im 14945 df-sqrt 15079 df-abs 15080 df-dvds 16096 df-struct 16978 df-sets 16995 df-slot 17013 df-ndx 17025 df-base 17043 df-ress 17072 df-plusg 17105 df-mulr 17106 df-starv 17107 df-tset 17111 df-ple 17112 df-ds 17114 df-unif 17115 df-0g 17282 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-mhm 18560 df-grp 18710 df-minusg 18711 df-sbg 18712 df-mulg 18831 df-subg 18883 df-ghm 18964 df-od 19268 df-cmn 19522 df-mgp 19855 df-ur 19872 df-ring 19919 df-cring 19920 df-rnghom 20098 df-subrg 20172 df-cnfld 20749 df-zring 20822 df-zrh 20856 df-chr 20858 |
This theorem is referenced by: chrnzr 20885 chrrhm 20886 domnchr 20887 znchr 20921 fermltlchr 31977 |
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