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Mirrors > Home > MPE Home > Th. List > simpggrp | Structured version Visualization version GIF version |
Description: A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
simpggrp | ⊢ (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issimpg 19346 | . 2 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o)) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5040 ‘cfv 6350 2oc2o 8138 ≈ cen 8565 Grpcgrp 18232 NrmSGrpcnsg 18405 SimpGrpcsimpg 19344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-rab 3063 df-v 3402 df-un 3858 df-in 3860 df-ss 3870 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-iota 6308 df-fv 6358 df-simpg 19345 |
This theorem is referenced by: simpggrpd 19349 prmsimpcyc 31071 |
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