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Mirrors > Home > MPE Home > Th. List > zncyg | Structured version Visualization version GIF version |
Description: The group ℤ / 𝑛ℤ is cyclic for all 𝑛 (including 𝑛 = 0). (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
zncyg.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
Ref | Expression |
---|---|
zncyg | ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CycGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zncyg.y | . . . . 5 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
2 | 1 | zncrng 20288 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) |
3 | crngring 18945 | . . . 4 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Ring) |
5 | ringgrp 18939 | . . 3 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Grp) |
7 | eqid 2778 | . . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
8 | eqid 2778 | . . . . 5 ⊢ (1r‘𝑌) = (1r‘𝑌) | |
9 | 7, 8 | ringidcl 18955 | . . . 4 ⊢ (𝑌 ∈ Ring → (1r‘𝑌) ∈ (Base‘𝑌)) |
10 | 4, 9 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (1r‘𝑌) ∈ (Base‘𝑌)) |
11 | eqid 2778 | . . . . . . 7 ⊢ (ℤRHom‘𝑌) = (ℤRHom‘𝑌) | |
12 | eqid 2778 | . . . . . . 7 ⊢ (.g‘𝑌) = (.g‘𝑌) | |
13 | 11, 12, 8 | zrhval2 20253 | . . . . . 6 ⊢ (𝑌 ∈ Ring → (ℤRHom‘𝑌) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
14 | 4, 13 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑌) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
15 | 14 | rneqd 5598 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ran (ℤRHom‘𝑌) = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
16 | 1, 7, 11 | znzrhfo 20291 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌)) |
17 | forn 6369 | . . . . 5 ⊢ ((ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌) → ran (ℤRHom‘𝑌) = (Base‘𝑌)) | |
18 | 16, 17 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ran (ℤRHom‘𝑌) = (Base‘𝑌)) |
19 | 15, 18 | eqtr3d 2816 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌))) = (Base‘𝑌)) |
20 | oveq2 6930 | . . . . . . 7 ⊢ (𝑥 = (1r‘𝑌) → (𝑛(.g‘𝑌)𝑥) = (𝑛(.g‘𝑌)(1r‘𝑌))) | |
21 | 20 | mpteq2dv 4980 | . . . . . 6 ⊢ (𝑥 = (1r‘𝑌) → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
22 | 21 | rneqd 5598 | . . . . 5 ⊢ (𝑥 = (1r‘𝑌) → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
23 | 22 | eqeq1d 2780 | . . . 4 ⊢ (𝑥 = (1r‘𝑌) → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = (Base‘𝑌) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌))) = (Base‘𝑌))) |
24 | 23 | rspcev 3511 | . . 3 ⊢ (((1r‘𝑌) ∈ (Base‘𝑌) ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌))) = (Base‘𝑌)) → ∃𝑥 ∈ (Base‘𝑌)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = (Base‘𝑌)) |
25 | 10, 19, 24 | syl2anc 579 | . 2 ⊢ (𝑁 ∈ ℕ0 → ∃𝑥 ∈ (Base‘𝑌)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = (Base‘𝑌)) |
26 | 7, 12 | iscyg 18667 | . 2 ⊢ (𝑌 ∈ CycGrp ↔ (𝑌 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝑌)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = (Base‘𝑌))) |
27 | 6, 25, 26 | sylanbrc 578 | 1 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CycGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 ↦ cmpt 4965 ran crn 5356 –onto→wfo 6133 ‘cfv 6135 (class class class)co 6922 ℕ0cn0 11642 ℤcz 11728 Basecbs 16255 Grpcgrp 17809 .gcmg 17927 CycGrpccyg 18665 1rcur 18888 Ringcrg 18934 CRingccrg 18935 ℤRHomczrh 20244 ℤ/nℤczn 20247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-ec 8028 df-qs 8032 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-seq 13120 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-0g 16488 df-imas 16554 df-qus 16555 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mulg 17928 df-subg 17975 df-nsg 17976 df-eqg 17977 df-ghm 18042 df-cmn 18581 df-abl 18582 df-cyg 18666 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-oppr 19010 df-rnghom 19104 df-subrg 19170 df-lmod 19257 df-lss 19325 df-lsp 19367 df-sra 19569 df-rgmod 19570 df-lidl 19571 df-rsp 19572 df-2idl 19629 df-cnfld 20143 df-zring 20215 df-zrh 20248 df-zn 20251 |
This theorem is referenced by: cygth 20315 |
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