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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 21320 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) |
3 | crngring 20140 | . . . 4 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Ring) |
5 | ringgrp 20133 | . . 3 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Grp) |
7 | eqid 2731 | . . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
8 | eqid 2731 | . . . . 5 ⊢ (1r‘𝑌) = (1r‘𝑌) | |
9 | 7, 8 | ringidcl 20155 | . . . 4 ⊢ (𝑌 ∈ Ring → (1r‘𝑌) ∈ (Base‘𝑌)) |
10 | 4, 9 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (1r‘𝑌) ∈ (Base‘𝑌)) |
11 | eqid 2731 | . . . . . . 7 ⊢ (ℤRHom‘𝑌) = (ℤRHom‘𝑌) | |
12 | eqid 2731 | . . . . . . 7 ⊢ (.g‘𝑌) = (.g‘𝑌) | |
13 | 11, 12, 8 | zrhval2 21278 | . . . . . 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 5937 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ran (ℤRHom‘𝑌) = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
16 | 1, 7, 11 | znzrhfo 21323 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌)) |
17 | forn 6808 | . . . . 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 2773 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌))) = (Base‘𝑌)) |
20 | oveq2 7420 | . . . . . . 7 ⊢ (𝑥 = (1r‘𝑌) → (𝑛(.g‘𝑌)𝑥) = (𝑛(.g‘𝑌)(1r‘𝑌))) | |
21 | 20 | mpteq2dv 5250 | . . . . . 6 ⊢ (𝑥 = (1r‘𝑌) → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
22 | 21 | rneqd 5937 | . . . . 5 ⊢ (𝑥 = (1r‘𝑌) → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑌)(1r‘𝑌)))) |
23 | 22 | eqeq1d 2733 | . . . 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 3612 | . . 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 19789 | . 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 1540 ∈ wcel 2105 ∃wrex 3069 ↦ cmpt 5231 ran crn 5677 –onto→wfo 6541 ‘cfv 6543 (class class class)co 7412 ℕ0cn0 12477 ℤcz 12563 Basecbs 17149 Grpcgrp 18856 .gcmg 18987 CycGrpccyg 19787 1rcur 20076 Ringcrg 20128 CRingccrg 20129 ℤRHomczrh 21269 ℤ/nℤczn 21272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-addf 11193 ax-mulf 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-ec 8709 df-qs 8713 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-seq 13972 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 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-nsg 19041 df-eqg 19042 df-ghm 19129 df-cmn 19692 df-abl 19693 df-cyg 19788 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-oppr 20226 df-rhm 20364 df-subrng 20435 df-subrg 20460 df-lmod 20617 df-lss 20688 df-lsp 20728 df-sra 20931 df-rgmod 20932 df-lidl 20933 df-rsp 20934 df-2idl 21007 df-cnfld 21146 df-zring 21219 df-zrh 21273 df-zn 21276 |
This theorem is referenced by: cygth 21347 |
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