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Mirrors > Home > MPE Home > Th. List > issimpg | Structured version Visualization version GIF version |
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 6656 | . . 3 ⊢ (𝑔 = 𝐺 → (NrmSGrp‘𝑔) = (NrmSGrp‘𝐺)) | |
2 | 1 | breq1d 5062 | . 2 ⊢ (𝑔 = 𝐺 → ((NrmSGrp‘𝑔) ≈ 2o ↔ (NrmSGrp‘𝐺) ≈ 2o)) |
3 | df-simpg 19196 | . 2 ⊢ SimpGrp = {𝑔 ∈ Grp ∣ (NrmSGrp‘𝑔) ≈ 2o} | |
4 | 2, 3 | elrab2 3674 | 1 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5052 ‘cfv 6341 2oc2o 8082 ≈ cen 8492 Grpcgrp 18086 NrmSGrpcnsg 18257 SimpGrpcsimpg 19195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-iota 6300 df-fv 6349 df-simpg 19196 |
This theorem is referenced by: issimpgd 19198 simpggrp 19199 simpg2nsg 19201 |
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