Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zrhchr | Structured version Visualization version GIF version |
Description: The kernel of the homomorphism from the integers to a ring is injective if and only if the ring has characteristic 0 . (Contributed by Thierry Arnoux, 8-Nov-2017.) |
Ref | Expression |
---|---|
zrhker.0 | ⊢ 𝐵 = (Base‘𝑅) |
zrhker.1 | ⊢ 𝐿 = (ℤRHom‘𝑅) |
zrhker.2 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
zrhchr | ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) = 0 ↔ 𝐿:ℤ–1-1→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zrhker.1 | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
2 | eqid 2738 | . . . 4 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
3 | eqid 2738 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
4 | 1, 2, 3 | zrhval2 20329 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐿 = (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅)))) |
5 | f1eq1 6569 | . . 3 ⊢ (𝐿 = (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅))) → (𝐿:ℤ–1-1→𝐵 ↔ (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅))):ℤ–1-1→𝐵)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝑅 ∈ Ring → (𝐿:ℤ–1-1→𝐵 ↔ (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅))):ℤ–1-1→𝐵)) |
7 | ringgrp 19421 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
8 | zrhker.0 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
9 | 8, 3 | ringidcl 19440 | . . 3 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
10 | eqid 2738 | . . . 4 ⊢ (od‘𝑅) = (od‘𝑅) | |
11 | eqid 2738 | . . . 4 ⊢ (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅))) = (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅))) | |
12 | 8, 10, 2, 11 | odf1 18807 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (1r‘𝑅) ∈ 𝐵) → (((od‘𝑅)‘(1r‘𝑅)) = 0 ↔ (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅))):ℤ–1-1→𝐵)) |
13 | 7, 9, 12 | syl2anc 587 | . 2 ⊢ (𝑅 ∈ Ring → (((od‘𝑅)‘(1r‘𝑅)) = 0 ↔ (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝑅)(1r‘𝑅))):ℤ–1-1→𝐵)) |
14 | eqid 2738 | . . . . 5 ⊢ (chr‘𝑅) = (chr‘𝑅) | |
15 | 10, 3, 14 | chrval 20344 | . . . 4 ⊢ ((od‘𝑅)‘(1r‘𝑅)) = (chr‘𝑅) |
16 | 15 | eqeq1i 2743 | . . 3 ⊢ (((od‘𝑅)‘(1r‘𝑅)) = 0 ↔ (chr‘𝑅) = 0) |
17 | 16 | a1i 11 | . 2 ⊢ (𝑅 ∈ Ring → (((od‘𝑅)‘(1r‘𝑅)) = 0 ↔ (chr‘𝑅) = 0)) |
18 | 6, 13, 17 | 3bitr2rd 311 | 1 ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) = 0 ↔ 𝐿:ℤ–1-1→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1542 ∈ wcel 2114 ↦ cmpt 5110 –1-1→wf1 6336 ‘cfv 6339 (class class class)co 7170 0cc0 10615 ℤcz 12062 Basecbs 16586 0gc0g 16816 Grpcgrp 18219 .gcmg 18342 odcod 18770 1rcur 19370 Ringcrg 19416 ℤRHomczrh 20320 chrcchr 20322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-inf 8980 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-rp 12473 df-fz 12982 df-fl 13253 df-mod 13329 df-seq 13461 df-exp 13522 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-dvds 15700 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-0g 16818 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mhm 18072 df-grp 18222 df-minusg 18223 df-sbg 18224 df-mulg 18343 df-subg 18394 df-ghm 18474 df-od 18774 df-cmn 19026 df-mgp 19359 df-ur 19371 df-ring 19418 df-cring 19419 df-rnghom 19589 df-subrg 19652 df-cnfld 20218 df-zring 20290 df-zrh 20324 df-chr 20326 |
This theorem is referenced by: zrhker 31497 qqhre 31540 |
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