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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 19643 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
2 | eqid 2610 | . . 3 ⊢ (.g‘ℤring) = (.g‘ℤring) | |
3 | zsubrg 19618 | . . . . 5 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
4 | subrgsubg 18609 | . . . . 5 ⊢ (ℤ ∈ (SubRing‘ℂfld) → ℤ ∈ (SubGrp‘ℂfld)) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ℤ ∈ (SubGrp‘ℂfld) |
6 | df-zring 19638 | . . . . 5 ⊢ ℤring = (ℂfld ↾s ℤ) | |
7 | 6 | subggrp 17420 | . . . 4 ⊢ (ℤ ∈ (SubGrp‘ℂfld) → ℤring ∈ Grp) |
8 | 5, 7 | mp1i 13 | . . 3 ⊢ (⊤ → ℤring ∈ Grp) |
9 | 1zzd 11285 | . . 3 ⊢ (⊤ → 1 ∈ ℤ) | |
10 | ax-1cn 9873 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
11 | cnfldmulg 19597 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 1 ∈ ℂ) → (𝑥(.g‘ℂfld)1) = (𝑥 · 1)) | |
12 | 10, 11 | mpan2 703 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥(.g‘ℂfld)1) = (𝑥 · 1)) |
13 | 1z 11284 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
14 | eqid 2610 | . . . . . . . 8 ⊢ (.g‘ℂfld) = (.g‘ℂfld) | |
15 | 14, 6, 2 | subgmulg 17431 | . . . . . . 7 ⊢ ((ℤ ∈ (SubGrp‘ℂfld) ∧ 𝑥 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑥(.g‘ℂfld)1) = (𝑥(.g‘ℤring)1)) |
16 | 5, 13, 15 | mp3an13 1407 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥(.g‘ℂfld)1) = (𝑥(.g‘ℤring)1)) |
17 | zcn 11259 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
18 | 17 | mulid1d 9936 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥 · 1) = 𝑥) |
19 | 12, 16, 18 | 3eqtr3rd 2653 | . . . . 5 ⊢ (𝑥 ∈ ℤ → 𝑥 = (𝑥(.g‘ℤring)1)) |
20 | oveq1 6556 | . . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧(.g‘ℤring)1) = (𝑥(.g‘ℤring)1)) | |
21 | 20 | eqeq2d 2620 | . . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑥 = (𝑧(.g‘ℤring)1) ↔ 𝑥 = (𝑥(.g‘ℤring)1))) |
22 | 21 | rspcev 3282 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑥 = (𝑥(.g‘ℤring)1)) → ∃𝑧 ∈ ℤ 𝑥 = (𝑧(.g‘ℤring)1)) |
23 | 19, 22 | mpdan 699 | . . . 4 ⊢ (𝑥 ∈ ℤ → ∃𝑧 ∈ ℤ 𝑥 = (𝑧(.g‘ℤring)1)) |
24 | 23 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℤ) → ∃𝑧 ∈ ℤ 𝑥 = (𝑧(.g‘ℤring)1)) |
25 | 1, 2, 8, 9, 24 | iscygd 18112 | . 2 ⊢ (⊤ → ℤring ∈ CycGrp) |
26 | 25 | trud 1484 | 1 ⊢ ℤring ∈ CycGrp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ∃wrex 2897 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 1c1 9816 · cmul 9820 ℤcz 11254 Grpcgrp 17245 .gcmg 17363 SubGrpcsubg 17411 CycGrpccyg 18102 SubRingcsubrg 18599 ℂfldccnfld 19567 ℤringzring 19637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-seq 12664 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-mulg 17364 df-subg 17414 df-cmn 18018 df-cyg 18103 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-subrg 18601 df-cnfld 19568 df-zring 19638 |
This theorem is referenced by: (None) |
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