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Mirrors > Home > MPE Home > Th. List > znchr | Structured version Visualization version GIF version |
Description: Cyclic rings are defined by their characteristic. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
znchr.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
Ref | Expression |
---|---|
znchr | ⊢ (𝑁 ∈ ℕ0 → (chr‘𝑌) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znchr.y | . . . . . . 7 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
2 | 1 | zncrng 20259 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) |
3 | crngring 18919 | . . . . . 6 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Ring) |
5 | nn0z 11735 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℤ) | |
6 | eqid 2825 | . . . . . 6 ⊢ (chr‘𝑌) = (chr‘𝑌) | |
7 | eqid 2825 | . . . . . 6 ⊢ (ℤRHom‘𝑌) = (ℤRHom‘𝑌) | |
8 | eqid 2825 | . . . . . 6 ⊢ (0g‘𝑌) = (0g‘𝑌) | |
9 | 6, 7, 8 | chrdvds 20243 | . . . . 5 ⊢ ((𝑌 ∈ Ring ∧ 𝑥 ∈ ℤ) → ((chr‘𝑌) ∥ 𝑥 ↔ ((ℤRHom‘𝑌)‘𝑥) = (0g‘𝑌))) |
10 | 4, 5, 9 | syl2an 589 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℕ0) → ((chr‘𝑌) ∥ 𝑥 ↔ ((ℤRHom‘𝑌)‘𝑥) = (0g‘𝑌))) |
11 | 1, 7, 8 | zndvds0 20265 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℤ) → (((ℤRHom‘𝑌)‘𝑥) = (0g‘𝑌) ↔ 𝑁 ∥ 𝑥)) |
12 | 5, 11 | sylan2 586 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℕ0) → (((ℤRHom‘𝑌)‘𝑥) = (0g‘𝑌) ↔ 𝑁 ∥ 𝑥)) |
13 | 10, 12 | bitrd 271 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℕ0) → ((chr‘𝑌) ∥ 𝑥 ↔ 𝑁 ∥ 𝑥)) |
14 | 13 | ralrimiva 3175 | . 2 ⊢ (𝑁 ∈ ℕ0 → ∀𝑥 ∈ ℕ0 ((chr‘𝑌) ∥ 𝑥 ↔ 𝑁 ∥ 𝑥)) |
15 | 6 | chrcl 20241 | . . . 4 ⊢ (𝑌 ∈ Ring → (chr‘𝑌) ∈ ℕ0) |
16 | 4, 15 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (chr‘𝑌) ∈ ℕ0) |
17 | dvdsext 15427 | . . 3 ⊢ (((chr‘𝑌) ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((chr‘𝑌) = 𝑁 ↔ ∀𝑥 ∈ ℕ0 ((chr‘𝑌) ∥ 𝑥 ↔ 𝑁 ∥ 𝑥))) | |
18 | 16, 17 | mpancom 679 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((chr‘𝑌) = 𝑁 ↔ ∀𝑥 ∈ ℕ0 ((chr‘𝑌) ∥ 𝑥 ↔ 𝑁 ∥ 𝑥))) |
19 | 14, 18 | mpbird 249 | 1 ⊢ (𝑁 ∈ ℕ0 → (chr‘𝑌) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∀wral 3117 class class class wbr 4875 ‘cfv 6127 ℕ0cn0 11625 ℤcz 11711 ∥ cdvds 15364 0gc0g 16460 Ringcrg 18908 CRingccrg 18909 ℤRHomczrh 20215 chrcchr 20217 ℤ/nℤczn 20218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-ec 8016 df-qs 8020 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-inf 8624 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-rp 12120 df-fz 12627 df-fl 12895 df-mod 12971 df-seq 13103 df-exp 13162 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-dvds 15365 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-0g 16462 df-imas 16528 df-qus 16529 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-grp 17786 df-minusg 17787 df-sbg 17788 df-mulg 17902 df-subg 17949 df-nsg 17950 df-eqg 17951 df-ghm 18016 df-od 18306 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-oppr 18984 df-dvdsr 19002 df-rnghom 19078 df-subrg 19141 df-lmod 19228 df-lss 19296 df-lsp 19338 df-sra 19540 df-rgmod 19541 df-lidl 19542 df-rsp 19543 df-2idl 19600 df-cnfld 20114 df-zring 20186 df-zrh 20219 df-chr 20221 df-zn 20222 |
This theorem is referenced by: (None) |
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