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Mirrors > Home > MPE Home > Th. List > simpg2nsg | Structured version Visualization version GIF version |
Description: A simple group has two normal subgroups. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
simpg2nsg | ⊢ (𝐺 ∈ SimpGrp → (NrmSGrp‘𝐺) ≈ 2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issimpg 19207 | . 2 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o)) | |
2 | 1 | simprbi 499 | 1 ⊢ (𝐺 ∈ SimpGrp → (NrmSGrp‘𝐺) ≈ 2o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 class class class wbr 5059 ‘cfv 6348 2oc2o 8089 ≈ cen 8499 Grpcgrp 18096 NrmSGrpcnsg 18267 SimpGrpcsimpg 19205 |
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-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-iota 6307 df-fv 6356 df-simpg 19206 |
This theorem is referenced by: trivnsimpgd 19212 simpgnsgd 19215 |
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