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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 20657 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
2 | eqid 2739 | . . 3 ⊢ (.g‘ℤring) = (.g‘ℤring) | |
3 | zsubrg 20632 | . . . . 5 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
4 | subrgsubg 20011 | . . . . 5 ⊢ (ℤ ∈ (SubRing‘ℂfld) → ℤ ∈ (SubGrp‘ℂfld)) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ℤ ∈ (SubGrp‘ℂfld) |
6 | df-zring 20652 | . . . . 5 ⊢ ℤring = (ℂfld ↾s ℤ) | |
7 | 6 | subggrp 18739 | . . . 4 ⊢ (ℤ ∈ (SubGrp‘ℂfld) → ℤring ∈ Grp) |
8 | 5, 7 | mp1i 13 | . . 3 ⊢ (⊤ → ℤring ∈ Grp) |
9 | 1zzd 12334 | . . 3 ⊢ (⊤ → 1 ∈ ℤ) | |
10 | ax-1cn 10913 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
11 | cnfldmulg 20611 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 1 ∈ ℂ) → (𝑥(.g‘ℂfld)1) = (𝑥 · 1)) | |
12 | 10, 11 | mpan2 687 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥(.g‘ℂfld)1) = (𝑥 · 1)) |
13 | 1z 12333 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
14 | eqid 2739 | . . . . . . . 8 ⊢ (.g‘ℂfld) = (.g‘ℂfld) | |
15 | 14, 6, 2 | subgmulg 18750 | . . . . . . 7 ⊢ ((ℤ ∈ (SubGrp‘ℂfld) ∧ 𝑥 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑥(.g‘ℂfld)1) = (𝑥(.g‘ℤring)1)) |
16 | 5, 13, 15 | mp3an13 1450 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥(.g‘ℂfld)1) = (𝑥(.g‘ℤring)1)) |
17 | zcn 12307 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
18 | 17 | mulid1d 10976 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥 · 1) = 𝑥) |
19 | 12, 16, 18 | 3eqtr3rd 2788 | . . . . 5 ⊢ (𝑥 ∈ ℤ → 𝑥 = (𝑥(.g‘ℤring)1)) |
20 | oveq1 7275 | . . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧(.g‘ℤring)1) = (𝑥(.g‘ℤring)1)) | |
21 | 20 | rspceeqv 3575 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑥 = (𝑥(.g‘ℤring)1)) → ∃𝑧 ∈ ℤ 𝑥 = (𝑧(.g‘ℤring)1)) |
22 | 19, 21 | mpdan 683 | . . . 4 ⊢ (𝑥 ∈ ℤ → ∃𝑧 ∈ ℤ 𝑥 = (𝑧(.g‘ℤring)1)) |
23 | 22 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℤ) → ∃𝑧 ∈ ℤ 𝑥 = (𝑧(.g‘ℤring)1)) |
24 | 1, 2, 8, 9, 23 | iscygd 19468 | . 2 ⊢ (⊤ → ℤring ∈ CycGrp) |
25 | 24 | mptru 1548 | 1 ⊢ ℤring ∈ CycGrp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ⊤wtru 1542 ∈ wcel 2109 ∃wrex 3066 ‘cfv 6430 (class class class)co 7268 ℂcc 10853 1c1 10856 · cmul 10860 ℤcz 12302 Grpcgrp 18558 .gcmg 18681 SubGrpcsubg 18730 CycGrpccyg 19458 SubRingcsubrg 20001 ℂfldccnfld 20578 ℤringczring 20651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-addf 10934 ax-mulf 10935 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-fz 13222 df-seq 13703 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-starv 16958 df-tset 16962 df-ple 16963 df-ds 16965 df-unif 16966 df-0g 17133 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-grp 18561 df-minusg 18562 df-mulg 18682 df-subg 18733 df-cmn 19369 df-cyg 19459 df-mgp 19702 df-ur 19719 df-ring 19766 df-cring 19767 df-subrg 20003 df-cnfld 20579 df-zring 20652 |
This theorem is referenced by: (None) |
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