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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 6663 | . . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
2 | iscyg.1 | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | 1, 2 | syl6eqr 2873 | . . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
4 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺)) | |
5 | iscyg.2 | . . . . . . . 8 ⊢ · = (.g‘𝐺) | |
6 | 4, 5 | syl6eqr 2873 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · ) |
7 | 6 | oveqd 7166 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑛(.g‘𝑔)𝑥) = (𝑛 · 𝑥)) |
8 | 7 | mpteq2dv 5155 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥))) |
9 | 8 | rneqd 5801 | . . . 4 ⊢ (𝑔 = 𝐺 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥))) |
10 | 9, 3 | eqeq12d 2836 | . . 3 ⊢ (𝑔 = 𝐺 → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)) |
11 | 3, 10 | rexeqbidv 3401 | . 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 18992 | . 2 ⊢ CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔)} | |
13 | 11, 12 | elrab2 3679 | 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 1536 ∈ wcel 2113 ∃wrex 3138 ↦ cmpt 5139 ran crn 5549 ‘cfv 6348 (class class class)co 7149 ℤcz 11975 Basecbs 16478 Grpcgrp 18098 .gcmg 18219 CycGrpccyg 18991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-cnv 5556 df-dm 5558 df-rn 5559 df-iota 6307 df-fv 6356 df-ov 7152 df-cyg 18992 |
This theorem is referenced by: iscyg2 18996 iscyg3 19000 cyggrp 19004 cygctb 19007 ghmcyg 19011 ablfac2 19206 fincygsubgodexd 19230 zncyg 20690 |
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