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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 6499 | . . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
2 | iscyg.1 | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | 1, 2 | syl6eqr 2833 | . . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
4 | fveq2 6499 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺)) | |
5 | iscyg.2 | . . . . . . . 8 ⊢ · = (.g‘𝐺) | |
6 | 4, 5 | syl6eqr 2833 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · ) |
7 | 6 | oveqd 6993 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑛(.g‘𝑔)𝑥) = (𝑛 · 𝑥)) |
8 | 7 | mpteq2dv 5023 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥))) |
9 | 8 | rneqd 5651 | . . . 4 ⊢ (𝑔 = 𝐺 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥))) |
10 | 9, 3 | eqeq12d 2794 | . . 3 ⊢ (𝑔 = 𝐺 → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)) |
11 | 3, 10 | rexeqbidv 3343 | . 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 18753 | . 2 ⊢ CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔)} | |
13 | 11, 12 | elrab2 3600 | 1 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∃wrex 3090 ↦ cmpt 5008 ran crn 5408 ‘cfv 6188 (class class class)co 6976 ℤcz 11793 Basecbs 16339 Grpcgrp 17891 .gcmg 18011 CycGrpccyg 18752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2751 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-cnv 5415 df-dm 5417 df-rn 5418 df-iota 6152 df-fv 6196 df-ov 6979 df-cyg 18753 |
This theorem is referenced by: iscyg2 18757 iscyg3 18761 cyggrp 18764 cygctb 18766 ghmcyg 18770 ablfac2 18961 zncyg 20397 fincygsubgodexd 40045 |
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