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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 21428 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) |
3 | crngring 20146 | . . . 4 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Ring) |
5 | ringgrp 20139 | . . 3 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Grp) |
7 | eqid 2724 | . . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
8 | eqid 2724 | . . . . 5 ⊢ (1r‘𝑌) = (1r‘𝑌) | |
9 | 7, 8 | ringidcl 20161 | . . . 4 ⊢ (𝑌 ∈ Ring → (1r‘𝑌) ∈ (Base‘𝑌)) |
10 | 4, 9 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (1r‘𝑌) ∈ (Base‘𝑌)) |
11 | eqid 2724 | . . . . . . 7 ⊢ (ℤRHom‘𝑌) = (ℤRHom‘𝑌) | |
12 | eqid 2724 | . . . . . . 7 ⊢ (.g‘𝑌) = (.g‘𝑌) | |
13 | 11, 12, 8 | zrhval2 21384 | . . . . . 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 5928 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ran (ℤRHom‘𝑌) = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
16 | 1, 7, 11 | znzrhfo 21431 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌)) |
17 | forn 6799 | . . . . 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 2766 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌))) = (Base‘𝑌)) |
20 | oveq2 7410 | . . . . . . 7 ⊢ (𝑥 = (1r‘𝑌) → (𝑛(.g‘𝑌)𝑥) = (𝑛(.g‘𝑌)(1r‘𝑌))) | |
21 | 20 | mpteq2dv 5241 | . . . . . 6 ⊢ (𝑥 = (1r‘𝑌) → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
22 | 21 | rneqd 5928 | . . . . 5 ⊢ (𝑥 = (1r‘𝑌) → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
23 | 22 | eqeq1d 2726 | . . . 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 3604 | . . 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 583 | . 2 ⊢ (𝑁 ∈ ℕ0 → ∃𝑥 ∈ (Base‘𝑌)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = (Base‘𝑌)) |
26 | 7, 12 | iscyg 19795 | . 2 ⊢ (𝑌 ∈ CycGrp ↔ (𝑌 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝑌)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = (Base‘𝑌))) |
27 | 6, 25, 26 | sylanbrc 582 | 1 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CycGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∃wrex 3062 ↦ cmpt 5222 ran crn 5668 –onto→wfo 6532 ‘cfv 6534 (class class class)co 7402 ℕ0cn0 12471 ℤcz 12557 Basecbs 17149 Grpcgrp 18859 .gcmg 18991 CycGrpccyg 19793 1rcur 20082 Ringcrg 20134 CRingccrg 20135 ℤRHomczrh 21375 ℤ/nℤczn 21378 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-ec 8702 df-qs 8706 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-seq 13968 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-0g 17392 df-imas 17459 df-qus 17460 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18709 df-grp 18862 df-minusg 18863 df-sbg 18864 df-mulg 18992 df-subg 19046 df-nsg 19047 df-eqg 19048 df-ghm 19135 df-cmn 19698 df-abl 19699 df-cyg 19794 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-rhm 20370 df-subrng 20442 df-subrg 20467 df-lmod 20704 df-lss 20775 df-lsp 20815 df-sra 21017 df-rgmod 21018 df-lidl 21063 df-rsp 21064 df-2idl 21103 df-cnfld 21235 df-zring 21323 df-zrh 21379 df-zn 21382 |
This theorem is referenced by: cygth 21455 |
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