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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 20623 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
2 | eqid 2821 | . . 3 ⊢ (.g‘ℤring) = (.g‘ℤring) | |
3 | zsubrg 20598 | . . . . 5 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
4 | subrgsubg 19541 | . . . . 5 ⊢ (ℤ ∈ (SubRing‘ℂfld) → ℤ ∈ (SubGrp‘ℂfld)) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ℤ ∈ (SubGrp‘ℂfld) |
6 | df-zring 20618 | . . . . 5 ⊢ ℤring = (ℂfld ↾s ℤ) | |
7 | 6 | subggrp 18282 | . . . 4 ⊢ (ℤ ∈ (SubGrp‘ℂfld) → ℤring ∈ Grp) |
8 | 5, 7 | mp1i 13 | . . 3 ⊢ (⊤ → ℤring ∈ Grp) |
9 | 1zzd 12014 | . . 3 ⊢ (⊤ → 1 ∈ ℤ) | |
10 | ax-1cn 10595 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
11 | cnfldmulg 20577 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 1 ∈ ℂ) → (𝑥(.g‘ℂfld)1) = (𝑥 · 1)) | |
12 | 10, 11 | mpan2 689 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥(.g‘ℂfld)1) = (𝑥 · 1)) |
13 | 1z 12013 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
14 | eqid 2821 | . . . . . . . 8 ⊢ (.g‘ℂfld) = (.g‘ℂfld) | |
15 | 14, 6, 2 | subgmulg 18293 | . . . . . . 7 ⊢ ((ℤ ∈ (SubGrp‘ℂfld) ∧ 𝑥 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑥(.g‘ℂfld)1) = (𝑥(.g‘ℤring)1)) |
16 | 5, 13, 15 | mp3an13 1448 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥(.g‘ℂfld)1) = (𝑥(.g‘ℤring)1)) |
17 | zcn 11987 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
18 | 17 | mulid1d 10658 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥 · 1) = 𝑥) |
19 | 12, 16, 18 | 3eqtr3rd 2865 | . . . . 5 ⊢ (𝑥 ∈ ℤ → 𝑥 = (𝑥(.g‘ℤring)1)) |
20 | oveq1 7163 | . . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧(.g‘ℤring)1) = (𝑥(.g‘ℤring)1)) | |
21 | 20 | rspceeqv 3638 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑥 = (𝑥(.g‘ℤring)1)) → ∃𝑧 ∈ ℤ 𝑥 = (𝑧(.g‘ℤring)1)) |
22 | 19, 21 | mpdan 685 | . . . 4 ⊢ (𝑥 ∈ ℤ → ∃𝑧 ∈ ℤ 𝑥 = (𝑧(.g‘ℤring)1)) |
23 | 22 | adantl 484 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℤ) → ∃𝑧 ∈ ℤ 𝑥 = (𝑧(.g‘ℤring)1)) |
24 | 1, 2, 8, 9, 23 | iscygd 19006 | . 2 ⊢ (⊤ → ℤring ∈ CycGrp) |
25 | 24 | mptru 1544 | 1 ⊢ ℤring ∈ CycGrp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ∃wrex 3139 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 1c1 10538 · cmul 10542 ℤcz 11982 Grpcgrp 18103 .gcmg 18224 SubGrpcsubg 18273 CycGrpccyg 18996 SubRingcsubrg 19531 ℂfldccnfld 20545 ℤringzring 20617 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-seq 13371 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-mulg 18225 df-subg 18276 df-cmn 18908 df-cyg 18997 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-subrg 19533 df-cnfld 20546 df-zring 20618 |
This theorem is referenced by: (None) |
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