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| Description: The predicate "is a simple group". (Contributed by Rohan Ridenour, 3-Aug-2023.) | 
| Ref | Expression | 
|---|---|
| issimpg | ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq2 6906 | . . 3 ⊢ (𝑔 = 𝐺 → (NrmSGrp‘𝑔) = (NrmSGrp‘𝐺)) | |
| 2 | 1 | breq1d 5153 | . 2 ⊢ (𝑔 = 𝐺 → ((NrmSGrp‘𝑔) ≈ 2o ↔ (NrmSGrp‘𝐺) ≈ 2o)) | 
| 3 | df-simpg 20111 | . 2 ⊢ SimpGrp = {𝑔 ∈ Grp ∣ (NrmSGrp‘𝑔) ≈ 2o} | |
| 4 | 2, 3 | elrab2 3695 | 1 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 2oc2o 8500 ≈ cen 8982 Grpcgrp 18951 NrmSGrpcnsg 19139 SimpGrpcsimpg 20110 | 
| 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-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-iota 6514 df-fv 6569 df-simpg 20111 | 
| This theorem is referenced by: issimpgd 20113 simpggrp 20114 simpg2nsg 20116 | 
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