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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 20740 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) |
3 | crngring 19783 | . . . . . 6 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Ring) |
5 | nn0z 12331 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℤ) | |
6 | eqid 2738 | . . . . . 6 ⊢ (chr‘𝑌) = (chr‘𝑌) | |
7 | eqid 2738 | . . . . . 6 ⊢ (ℤRHom‘𝑌) = (ℤRHom‘𝑌) | |
8 | eqid 2738 | . . . . . 6 ⊢ (0g‘𝑌) = (0g‘𝑌) | |
9 | 6, 7, 8 | chrdvds 20720 | . . . . 5 ⊢ ((𝑌 ∈ Ring ∧ 𝑥 ∈ ℤ) → ((chr‘𝑌) ∥ 𝑥 ↔ ((ℤRHom‘𝑌)‘𝑥) = (0g‘𝑌))) |
10 | 4, 5, 9 | syl2an 596 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℕ0) → ((chr‘𝑌) ∥ 𝑥 ↔ ((ℤRHom‘𝑌)‘𝑥) = (0g‘𝑌))) |
11 | 1, 7, 8 | zndvds0 20746 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℤ) → (((ℤRHom‘𝑌)‘𝑥) = (0g‘𝑌) ↔ 𝑁 ∥ 𝑥)) |
12 | 5, 11 | sylan2 593 | . . . 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 3113 | . 2 ⊢ (𝑁 ∈ ℕ0 → ∀𝑥 ∈ ℕ0 ((chr‘𝑌) ∥ 𝑥 ↔ 𝑁 ∥ 𝑥)) |
15 | 6 | chrcl 20718 | . . . 4 ⊢ (𝑌 ∈ Ring → (chr‘𝑌) ∈ ℕ0) |
16 | 4, 15 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (chr‘𝑌) ∈ ℕ0) |
17 | dvdsext 16018 | . . 3 ⊢ (((chr‘𝑌) ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((chr‘𝑌) = 𝑁 ↔ ∀𝑥 ∈ ℕ0 ((chr‘𝑌) ∥ 𝑥 ↔ 𝑁 ∥ 𝑥))) | |
18 | 16, 17 | mpancom 685 | . 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 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 class class class wbr 5074 ‘cfv 6427 ℕ0cn0 12221 ℤcz 12307 ∥ cdvds 15951 0gc0g 17138 Ringcrg 19771 CRingccrg 19772 ℤRHomczrh 20689 chrcchr 20691 ℤ/nℤczn 20692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-pre-sup 10937 ax-addf 10938 ax-mulf 10939 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-tpos 8030 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-er 8486 df-ec 8488 df-qs 8492 df-map 8605 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-sup 9189 df-inf 9190 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-3 12025 df-4 12026 df-5 12027 df-6 12028 df-7 12029 df-8 12030 df-9 12031 df-n0 12222 df-z 12308 df-dec 12426 df-uz 12571 df-rp 12719 df-fz 13228 df-fl 13500 df-mod 13578 df-seq 13710 df-exp 13771 df-cj 14798 df-re 14799 df-im 14800 df-sqrt 14934 df-abs 14935 df-dvds 15952 df-struct 16836 df-sets 16853 df-slot 16871 df-ndx 16883 df-base 16901 df-ress 16930 df-plusg 16963 df-mulr 16964 df-starv 16965 df-sca 16966 df-vsca 16967 df-ip 16968 df-tset 16969 df-ple 16970 df-ds 16972 df-unif 16973 df-0g 17140 df-imas 17207 df-qus 17208 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-mhm 18418 df-grp 18568 df-minusg 18569 df-sbg 18570 df-mulg 18689 df-subg 18740 df-nsg 18741 df-eqg 18742 df-ghm 18820 df-od 19124 df-cmn 19376 df-abl 19377 df-mgp 19709 df-ur 19726 df-ring 19773 df-cring 19774 df-oppr 19850 df-dvdsr 19871 df-rnghom 19947 df-subrg 20010 df-lmod 20113 df-lss 20182 df-lsp 20222 df-sra 20422 df-rgmod 20423 df-lidl 20424 df-rsp 20425 df-2idl 20491 df-cnfld 20586 df-zring 20659 df-zrh 20693 df-chr 20695 df-zn 20696 |
This theorem is referenced by: ply1fermltl 31656 |
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