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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 2735 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2735 | . . 3 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
3 | 1, 2 | iscyg 19912 | . 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 1537 ∈ wcel 2106 ∃wrex 3068 ↦ cmpt 5231 ran crn 5690 ‘cfv 6563 (class class class)co 7431 ℤcz 12611 Basecbs 17245 Grpcgrp 18964 .gcmg 19098 CycGrpccyg 19910 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-cnv 5697 df-dm 5699 df-rn 5700 df-iota 6516 df-fv 6571 df-ov 7434 df-cyg 19911 |
This theorem is referenced by: fincygsubgodexd 20148 cygznlem1 21603 cygznlem2a 21604 cygznlem3 21606 prmsimpcyc 33217 |
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