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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 2770 | . . . . 5 ⊢ (od‘𝑅) = (od‘𝑅) | |
2 | eqid 2770 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | chrcl.c | . . . . 5 ⊢ 𝐶 = (chr‘𝑅) | |
4 | 1, 2, 3 | chrval 20087 | . . . 4 ⊢ ((od‘𝑅)‘(1r‘𝑅)) = 𝐶 |
5 | 4 | breq1i 4791 | . . 3 ⊢ (((od‘𝑅)‘(1r‘𝑅)) ∥ 𝑁 ↔ 𝐶 ∥ 𝑁) |
6 | ringgrp 18759 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
7 | 6 | adantr 466 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → 𝑅 ∈ Grp) |
8 | eqid 2770 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 8, 2 | ringidcl 18775 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
10 | 9 | adantr 466 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (1r‘𝑅) ∈ (Base‘𝑅)) |
11 | simpr 471 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
12 | eqid 2770 | . . . . 5 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
13 | chrid.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
14 | 8, 1, 12, 13 | oddvds 18172 | . . . 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 1475 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (((od‘𝑅)‘(1r‘𝑅)) ∥ 𝑁 ↔ (𝑁(.g‘𝑅)(1r‘𝑅)) = 0 )) |
16 | 5, 15 | syl5bbr 274 | . 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 20072 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (𝐿‘𝑁) = (𝑁(.g‘𝑅)(1r‘𝑅))) |
19 | 18 | eqeq1d 2772 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → ((𝐿‘𝑁) = 0 ↔ (𝑁(.g‘𝑅)(1r‘𝑅)) = 0 )) |
20 | 16, 19 | bitr4d 271 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (𝐶 ∥ 𝑁 ↔ (𝐿‘𝑁) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1630 ∈ wcel 2144 class class class wbr 4784 ‘cfv 6031 (class class class)co 6792 ℤcz 11578 ∥ cdvds 15188 Basecbs 16063 0gc0g 16307 Grpcgrp 17629 .gcmg 17747 odcod 18150 1rcur 18708 Ringcrg 18754 ℤRHomczrh 20062 chrcchr 20064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-inf2 8701 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-pre-sup 10215 ax-addf 10216 ax-mulf 10217 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-int 4610 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-oadd 7716 df-er 7895 df-map 8010 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-sup 8503 df-inf 8504 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-div 10886 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-7 11285 df-8 11286 df-9 11287 df-n0 11494 df-z 11579 df-dec 11695 df-uz 11888 df-rp 12035 df-fz 12533 df-fl 12800 df-mod 12876 df-seq 13008 df-exp 13067 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-dvds 15189 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-starv 16163 df-tset 16167 df-ple 16168 df-ds 16171 df-unif 16172 df-0g 16309 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-mhm 17542 df-grp 17632 df-minusg 17633 df-sbg 17634 df-mulg 17748 df-subg 17798 df-ghm 17865 df-od 18154 df-cmn 18401 df-mgp 18697 df-ur 18709 df-ring 18756 df-cring 18757 df-rnghom 18924 df-subrg 18987 df-cnfld 19961 df-zring 20033 df-zrh 20066 df-chr 20068 |
This theorem is referenced by: chrnzr 20092 chrrhm 20093 domnchr 20094 znchr 20125 |
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