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Mirrors > Home > MPE Home > Th. List > iscyg | Structured version Visualization version GIF version |
Description: Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
iscyg.1 | ⊢ 𝐵 = (Base‘𝐺) |
iscyg.2 | ⊢ · = (.g‘𝐺) |
Ref | Expression |
---|---|
iscyg | ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6664 | . . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
2 | iscyg.1 | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | 1, 2 | syl6eqr 2874 | . . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
4 | fveq2 6664 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺)) | |
5 | iscyg.2 | . . . . . . . 8 ⊢ · = (.g‘𝐺) | |
6 | 4, 5 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · ) |
7 | 6 | oveqd 7167 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑛(.g‘𝑔)𝑥) = (𝑛 · 𝑥)) |
8 | 7 | mpteq2dv 5154 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥))) |
9 | 8 | rneqd 5802 | . . . 4 ⊢ (𝑔 = 𝐺 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥))) |
10 | 9, 3 | eqeq12d 2837 | . . 3 ⊢ (𝑔 = 𝐺 → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)) |
11 | 3, 10 | rexeqbidv 3402 | . 2 ⊢ (𝑔 = 𝐺 → (∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔) ↔ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)) |
12 | df-cyg 18991 | . 2 ⊢ CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔)} | |
13 | 11, 12 | elrab2 3682 | 1 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 ↦ cmpt 5138 ran crn 5550 ‘cfv 6349 (class class class)co 7150 ℤcz 11975 Basecbs 16477 Grpcgrp 18097 .gcmg 18218 CycGrpccyg 18990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-cnv 5557 df-dm 5559 df-rn 5560 df-iota 6308 df-fv 6357 df-ov 7153 df-cyg 18991 |
This theorem is referenced by: iscyg2 18995 iscyg3 18999 cyggrp 19003 cygctb 19006 ghmcyg 19010 ablfac2 19205 fincygsubgodexd 19229 zncyg 20689 |
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