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Mirrors > Home > MPE Home > Th. List > zringcyg | Structured version Visualization version GIF version |
Description: The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 9-Jun-2019.) |
Ref | Expression |
---|---|
zringcyg | ⊢ ℤring ∈ CycGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zringbas 20169 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
2 | eqid 2798 | . . 3 ⊢ (.g‘ℤring) = (.g‘ℤring) | |
3 | zsubrg 20144 | . . . . 5 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
4 | subrgsubg 19534 | . . . . 5 ⊢ (ℤ ∈ (SubRing‘ℂfld) → ℤ ∈ (SubGrp‘ℂfld)) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ℤ ∈ (SubGrp‘ℂfld) |
6 | df-zring 20164 | . . . . 5 ⊢ ℤring = (ℂfld ↾s ℤ) | |
7 | 6 | subggrp 18274 | . . . 4 ⊢ (ℤ ∈ (SubGrp‘ℂfld) → ℤring ∈ Grp) |
8 | 5, 7 | mp1i 13 | . . 3 ⊢ (⊤ → ℤring ∈ Grp) |
9 | 1zzd 12001 | . . 3 ⊢ (⊤ → 1 ∈ ℤ) | |
10 | ax-1cn 10584 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
11 | cnfldmulg 20123 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 1 ∈ ℂ) → (𝑥(.g‘ℂfld)1) = (𝑥 · 1)) | |
12 | 10, 11 | mpan2 690 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥(.g‘ℂfld)1) = (𝑥 · 1)) |
13 | 1z 12000 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
14 | eqid 2798 | . . . . . . . 8 ⊢ (.g‘ℂfld) = (.g‘ℂfld) | |
15 | 14, 6, 2 | subgmulg 18285 | . . . . . . 7 ⊢ ((ℤ ∈ (SubGrp‘ℂfld) ∧ 𝑥 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑥(.g‘ℂfld)1) = (𝑥(.g‘ℤring)1)) |
16 | 5, 13, 15 | mp3an13 1449 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥(.g‘ℂfld)1) = (𝑥(.g‘ℤring)1)) |
17 | zcn 11974 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
18 | 17 | mulid1d 10647 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥 · 1) = 𝑥) |
19 | 12, 16, 18 | 3eqtr3rd 2842 | . . . . 5 ⊢ (𝑥 ∈ ℤ → 𝑥 = (𝑥(.g‘ℤring)1)) |
20 | oveq1 7142 | . . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧(.g‘ℤring)1) = (𝑥(.g‘ℤring)1)) | |
21 | 20 | rspceeqv 3586 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑥 = (𝑥(.g‘ℤring)1)) → ∃𝑧 ∈ ℤ 𝑥 = (𝑧(.g‘ℤring)1)) |
22 | 19, 21 | mpdan 686 | . . . 4 ⊢ (𝑥 ∈ ℤ → ∃𝑧 ∈ ℤ 𝑥 = (𝑧(.g‘ℤring)1)) |
23 | 22 | adantl 485 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℤ) → ∃𝑧 ∈ ℤ 𝑥 = (𝑧(.g‘ℤring)1)) |
24 | 1, 2, 8, 9, 23 | iscygd 18999 | . 2 ⊢ (⊤ → ℤring ∈ CycGrp) |
25 | 24 | mptru 1545 | 1 ⊢ ℤring ∈ CycGrp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 ∃wrex 3107 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 1c1 10527 · cmul 10531 ℤcz 11969 Grpcgrp 18095 .gcmg 18216 SubGrpcsubg 18265 CycGrpccyg 18989 SubRingcsubrg 19524 ℂfldccnfld 20091 ℤringzring 20163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-seq 13365 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-mulg 18217 df-subg 18268 df-cmn 18900 df-cyg 18990 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-subrg 19526 df-cnfld 20092 df-zring 20164 |
This theorem is referenced by: (None) |
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