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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 2731 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | eqid 2731 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
3 | 1, 2 | isnzr 20409 | . . 3 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
4 | 3 | baib 535 | . 2 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ (1r‘𝑅) ≠ (0g‘𝑅))) |
5 | 1z 12599 | . . . . 5 ⊢ 1 ∈ ℤ | |
6 | eqid 2731 | . . . . . 6 ⊢ (chr‘𝑅) = (chr‘𝑅) | |
7 | eqid 2731 | . . . . . 6 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅) | |
8 | 6, 7, 2 | chrdvds 21303 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ ℤ) → ((chr‘𝑅) ∥ 1 ↔ ((ℤRHom‘𝑅)‘1) = (0g‘𝑅))) |
9 | 5, 8 | mpan2 688 | . . . 4 ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) ∥ 1 ↔ ((ℤRHom‘𝑅)‘1) = (0g‘𝑅))) |
10 | 6 | chrcl 21301 | . . . . 5 ⊢ (𝑅 ∈ Ring → (chr‘𝑅) ∈ ℕ0) |
11 | dvds1 16269 | . . . . 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 21285 | . . . . 5 ⊢ (𝑅 ∈ Ring → ((ℤRHom‘𝑅)‘1) = (1r‘𝑅)) |
14 | 13 | eqeq1d 2733 | . . . 4 ⊢ (𝑅 ∈ Ring → (((ℤRHom‘𝑅)‘1) = (0g‘𝑅) ↔ (1r‘𝑅) = (0g‘𝑅))) |
15 | 9, 12, 14 | 3bitr3d 309 | . . 3 ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) = 1 ↔ (1r‘𝑅) = (0g‘𝑅))) |
16 | 15 | necon3bid 2984 | . 2 ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) ≠ 1 ↔ (1r‘𝑅) ≠ (0g‘𝑅))) |
17 | 4, 16 | bitr4d 282 | 1 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ (chr‘𝑅) ≠ 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 class class class wbr 5148 ‘cfv 6543 1c1 11117 ℕ0cn0 12479 ℤcz 12565 ∥ cdvds 16204 0gc0g 17392 1rcur 20079 Ringcrg 20131 NzRingcnzr 20407 ℤRHomczrh 21272 chrcchr 21274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-rp 12982 df-fz 13492 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-dvds 16205 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-0g 17394 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-mhm 18708 df-grp 18861 df-minusg 18862 df-sbg 18863 df-mulg 18991 df-subg 19043 df-ghm 19132 df-od 19441 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-ring 20133 df-cring 20134 df-rhm 20367 df-nzr 20408 df-subrng 20438 df-subrg 20463 df-cnfld 21149 df-zring 21222 df-zrh 21276 df-chr 21278 |
This theorem is referenced by: domnchr 21307 |
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