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Mirrors > Home > MPE Home > Th. List > chrnzr | Structured version Visualization version GIF version |
Description: Nonzero rings are precisely those with characteristic not 1. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
chrnzr | ⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ (chr‘𝑅) ≠ 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | eqid 2738 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
3 | 1, 2 | isnzr 20151 | . . 3 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
4 | 3 | baib 539 | . 2 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ (1r‘𝑅) ≠ (0g‘𝑅))) |
5 | 1z 12093 | . . . . 5 ⊢ 1 ∈ ℤ | |
6 | eqid 2738 | . . . . . 6 ⊢ (chr‘𝑅) = (chr‘𝑅) | |
7 | eqid 2738 | . . . . . 6 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅) | |
8 | 6, 7, 2 | chrdvds 20347 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ ℤ) → ((chr‘𝑅) ∥ 1 ↔ ((ℤRHom‘𝑅)‘1) = (0g‘𝑅))) |
9 | 5, 8 | mpan2 691 | . . . 4 ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) ∥ 1 ↔ ((ℤRHom‘𝑅)‘1) = (0g‘𝑅))) |
10 | 6 | chrcl 20345 | . . . . 5 ⊢ (𝑅 ∈ Ring → (chr‘𝑅) ∈ ℕ0) |
11 | dvds1 15764 | . . . . 5 ⊢ ((chr‘𝑅) ∈ ℕ0 → ((chr‘𝑅) ∥ 1 ↔ (chr‘𝑅) = 1)) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) ∥ 1 ↔ (chr‘𝑅) = 1)) |
13 | 7, 1 | zrh1 20333 | . . . . 5 ⊢ (𝑅 ∈ Ring → ((ℤRHom‘𝑅)‘1) = (1r‘𝑅)) |
14 | 13 | eqeq1d 2740 | . . . 4 ⊢ (𝑅 ∈ Ring → (((ℤRHom‘𝑅)‘1) = (0g‘𝑅) ↔ (1r‘𝑅) = (0g‘𝑅))) |
15 | 9, 12, 14 | 3bitr3d 312 | . . 3 ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) = 1 ↔ (1r‘𝑅) = (0g‘𝑅))) |
16 | 15 | necon3bid 2978 | . 2 ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) ≠ 1 ↔ (1r‘𝑅) ≠ (0g‘𝑅))) |
17 | 4, 16 | bitr4d 285 | 1 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ (chr‘𝑅) ≠ 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 class class class wbr 5030 ‘cfv 6339 1c1 10616 ℕ0cn0 11976 ℤcz 12062 ∥ cdvds 15699 0gc0g 16816 1rcur 19370 Ringcrg 19416 NzRingcnzr 20149 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-nzr 20150 df-cnfld 20218 df-zring 20290 df-zrh 20324 df-chr 20326 |
This theorem is referenced by: domnchr 20351 |
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