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Mirrors > Home > MPE Home > Th. List > chrid | Structured version Visualization version GIF version |
Description: The canonical ℤ ring homomorphism applied to a ring's characteristic is zero. (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 |
---|---|
chrid | ⊢ (𝑅 ∈ Ring → (𝐿‘𝐶) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chrcl.c | . . . . 5 ⊢ 𝐶 = (chr‘𝑅) | |
2 | 1 | chrcl 20279 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝐶 ∈ ℕ0) |
3 | 2 | nn0zd 12109 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐶 ∈ ℤ) |
4 | chrid.l | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
5 | eqid 2759 | . . . 4 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
6 | eqid 2759 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | 4, 5, 6 | zrhmulg 20264 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ ℤ) → (𝐿‘𝐶) = (𝐶(.g‘𝑅)(1r‘𝑅))) |
8 | 3, 7 | mpdan 687 | . 2 ⊢ (𝑅 ∈ Ring → (𝐿‘𝐶) = (𝐶(.g‘𝑅)(1r‘𝑅))) |
9 | eqid 2759 | . . . . 5 ⊢ (od‘𝑅) = (od‘𝑅) | |
10 | 9, 6, 1 | chrval 20278 | . . . 4 ⊢ ((od‘𝑅)‘(1r‘𝑅)) = 𝐶 |
11 | 10 | oveq1i 7153 | . . 3 ⊢ (((od‘𝑅)‘(1r‘𝑅))(.g‘𝑅)(1r‘𝑅)) = (𝐶(.g‘𝑅)(1r‘𝑅)) |
12 | eqid 2759 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
13 | 12, 6 | ringidcl 19374 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
14 | chrid.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
15 | 12, 9, 5, 14 | odid 18718 | . . . 4 ⊢ ((1r‘𝑅) ∈ (Base‘𝑅) → (((od‘𝑅)‘(1r‘𝑅))(.g‘𝑅)(1r‘𝑅)) = 0 ) |
16 | 13, 15 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → (((od‘𝑅)‘(1r‘𝑅))(.g‘𝑅)(1r‘𝑅)) = 0 ) |
17 | 11, 16 | syl5eqr 2808 | . 2 ⊢ (𝑅 ∈ Ring → (𝐶(.g‘𝑅)(1r‘𝑅)) = 0 ) |
18 | 8, 17 | eqtrd 2794 | 1 ⊢ (𝑅 ∈ Ring → (𝐿‘𝐶) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2112 ‘cfv 6328 (class class class)co 7143 ℤcz 12005 Basecbs 16526 0gc0g 16756 .gcmg 18276 odcod 18704 1rcur 19304 Ringcrg 19350 ℤRHomczrh 20254 chrcchr 20256 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-addf 10639 ax-mulf 10640 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-sup 8924 df-inf 8925 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-5 11725 df-6 11726 df-7 11727 df-8 11728 df-9 11729 df-n0 11920 df-z 12006 df-dec 12123 df-uz 12268 df-fz 12925 df-seq 13404 df-struct 16528 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-ress 16534 df-plusg 16621 df-mulr 16622 df-starv 16623 df-tset 16627 df-ple 16628 df-ds 16630 df-unif 16631 df-0g 16758 df-mgm 17903 df-sgrp 17952 df-mnd 17963 df-mhm 18007 df-grp 18157 df-minusg 18158 df-mulg 18277 df-subg 18328 df-ghm 18408 df-od 18708 df-cmn 18960 df-mgp 19293 df-ur 19305 df-ring 19352 df-cring 19353 df-rnghom 19523 df-subrg 19586 df-cnfld 20152 df-zring 20224 df-zrh 20258 df-chr 20260 |
This theorem is referenced by: chrrhm 20284 |
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