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| Mirrors > Home > MPE Home > Th. List > cyggrp | Structured version Visualization version GIF version | ||
| Description: A cyclic group is a group. (Contributed by Mario Carneiro, 21-Apr-2016.) | 
| Ref | Expression | 
|---|---|
| cyggrp | ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 2 | eqid 2737 | . . 3 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
| 3 | 1, 2 | iscyg 19897 | . 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (Base‘𝐺))) | 
| 4 | 3 | simplbi 497 | 1 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ℤcz 12613 Basecbs 17247 Grpcgrp 18951 .gcmg 19085 CycGrpccyg 19895 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-cnv 5693 df-dm 5695 df-rn 5696 df-iota 6514 df-fv 6569 df-ov 7434 df-cyg 19896 | 
| This theorem is referenced by: fincygsubgodexd 20133 cygznlem1 21585 cygznlem2a 21586 cygznlem3 21588 prmsimpcyc 33234 | 
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