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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 21534 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) |
3 | crngring 20223 | . . . . . 6 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Ring) |
5 | nn0z 12630 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℤ) | |
6 | eqid 2725 | . . . . . 6 ⊢ (chr‘𝑌) = (chr‘𝑌) | |
7 | eqid 2725 | . . . . . 6 ⊢ (ℤRHom‘𝑌) = (ℤRHom‘𝑌) | |
8 | eqid 2725 | . . . . . 6 ⊢ (0g‘𝑌) = (0g‘𝑌) | |
9 | 6, 7, 8 | chrdvds 21512 | . . . . 5 ⊢ ((𝑌 ∈ Ring ∧ 𝑥 ∈ ℤ) → ((chr‘𝑌) ∥ 𝑥 ↔ ((ℤRHom‘𝑌)‘𝑥) = (0g‘𝑌))) |
10 | 4, 5, 9 | syl2an 594 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℕ0) → ((chr‘𝑌) ∥ 𝑥 ↔ ((ℤRHom‘𝑌)‘𝑥) = (0g‘𝑌))) |
11 | 1, 7, 8 | zndvds0 21540 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℤ) → (((ℤRHom‘𝑌)‘𝑥) = (0g‘𝑌) ↔ 𝑁 ∥ 𝑥)) |
12 | 5, 11 | sylan2 591 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℕ0) → (((ℤRHom‘𝑌)‘𝑥) = (0g‘𝑌) ↔ 𝑁 ∥ 𝑥)) |
13 | 10, 12 | bitrd 278 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℕ0) → ((chr‘𝑌) ∥ 𝑥 ↔ 𝑁 ∥ 𝑥)) |
14 | 13 | ralrimiva 3135 | . 2 ⊢ (𝑁 ∈ ℕ0 → ∀𝑥 ∈ ℕ0 ((chr‘𝑌) ∥ 𝑥 ↔ 𝑁 ∥ 𝑥)) |
15 | 6 | chrcl 21510 | . . . 4 ⊢ (𝑌 ∈ Ring → (chr‘𝑌) ∈ ℕ0) |
16 | 4, 15 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (chr‘𝑌) ∈ ℕ0) |
17 | dvdsext 16318 | . . 3 ⊢ (((chr‘𝑌) ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((chr‘𝑌) = 𝑁 ↔ ∀𝑥 ∈ ℕ0 ((chr‘𝑌) ∥ 𝑥 ↔ 𝑁 ∥ 𝑥))) | |
18 | 16, 17 | mpancom 686 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((chr‘𝑌) = 𝑁 ↔ ∀𝑥 ∈ ℕ0 ((chr‘𝑌) ∥ 𝑥 ↔ 𝑁 ∥ 𝑥))) |
19 | 14, 18 | mpbird 256 | 1 ⊢ (𝑁 ∈ ℕ0 → (chr‘𝑌) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 class class class wbr 5152 ‘cfv 6553 ℕ0cn0 12519 ℤcz 12605 ∥ cdvds 16251 0gc0g 17449 Ringcrg 20211 CRingccrg 20212 ℤRHomczrh 21481 chrcchr 21483 ℤ/nℤczn 21484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232 ax-addf 11233 ax-mulf 11234 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-ec 8735 df-qs 8739 df-map 8856 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9481 df-inf 9482 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12520 df-z 12606 df-dec 12725 df-uz 12870 df-rp 13024 df-fz 13534 df-fl 13807 df-mod 13885 df-seq 14017 df-exp 14077 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-dvds 16252 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-0g 17451 df-imas 17518 df-qus 17519 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-grp 18926 df-minusg 18927 df-sbg 18928 df-mulg 19057 df-subg 19112 df-nsg 19113 df-eqg 19114 df-ghm 19202 df-od 19521 df-cmn 19775 df-abl 19776 df-mgp 20113 df-rng 20131 df-ur 20160 df-ring 20213 df-cring 20214 df-oppr 20311 df-dvdsr 20334 df-rhm 20449 df-subrng 20523 df-subrg 20548 df-lmod 20785 df-lss 20856 df-lsp 20896 df-sra 21098 df-rgmod 21099 df-lidl 21144 df-rsp 21145 df-2idl 21186 df-cnfld 21336 df-zring 21429 df-zrh 21485 df-chr 21487 df-zn 21488 |
This theorem is referenced by: ply1fermltl 33431 |
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