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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 21413 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
2 | eqid 2725 | . . 3 ⊢ (.g‘ℤring) = (.g‘ℤring) | |
3 | zsubrg 21387 | . . . . 5 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
4 | subrgsubg 20545 | . . . . 5 ⊢ (ℤ ∈ (SubRing‘ℂfld) → ℤ ∈ (SubGrp‘ℂfld)) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ℤ ∈ (SubGrp‘ℂfld) |
6 | df-zring 21407 | . . . . 5 ⊢ ℤring = (ℂfld ↾s ℤ) | |
7 | 6 | subggrp 19109 | . . . 4 ⊢ (ℤ ∈ (SubGrp‘ℂfld) → ℤring ∈ Grp) |
8 | 5, 7 | mp1i 13 | . . 3 ⊢ (⊤ → ℤring ∈ Grp) |
9 | 1zzd 12631 | . . 3 ⊢ (⊤ → 1 ∈ ℤ) | |
10 | ax-1cn 11203 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
11 | cnfldmulg 21365 | . . . . . . 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 12630 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
14 | eqid 2725 | . . . . . . . 8 ⊢ (.g‘ℂfld) = (.g‘ℂfld) | |
15 | 14, 6, 2 | subgmulg 19120 | . . . . . . 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 12601 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
18 | 17 | mulridd 11268 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥 · 1) = 𝑥) |
19 | 12, 16, 18 | 3eqtr3rd 2774 | . . . . 5 ⊢ (𝑥 ∈ ℤ → 𝑥 = (𝑥(.g‘ℤring)1)) |
20 | oveq1 7426 | . . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧(.g‘ℤring)1) = (𝑥(.g‘ℤring)1)) | |
21 | 20 | rspceeqv 3628 | . . . . 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 480 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℤ) → ∃𝑧 ∈ ℤ 𝑥 = (𝑧(.g‘ℤring)1)) |
24 | 1, 2, 8, 9, 23 | iscygd 19871 | . 2 ⊢ (⊤ → ℤring ∈ CycGrp) |
25 | 24 | mptru 1540 | 1 ⊢ ℤring ∈ CycGrp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ∃wrex 3059 ‘cfv 6549 (class class class)co 7419 ℂcc 11143 1c1 11146 · cmul 11150 ℤcz 12596 Grpcgrp 18914 .gcmg 19047 SubGrpcsubg 19100 CycGrpccyg 19861 SubRingcsubrg 20535 ℂfldccnfld 21313 ℤringczring 21406 |
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 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-addf 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-seq 14008 df-struct 17135 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-ress 17229 df-plusg 17265 df-mulr 17266 df-starv 17267 df-tset 17271 df-ple 17272 df-ds 17274 df-unif 17275 df-0g 17442 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-grp 18917 df-minusg 18918 df-mulg 19048 df-subg 19103 df-cmn 19766 df-abl 19767 df-cyg 19862 df-mgp 20104 df-rng 20122 df-ur 20151 df-ring 20204 df-cring 20205 df-subrng 20512 df-subrg 20537 df-cnfld 21314 df-zring 21407 |
This theorem is referenced by: (None) |
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