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Mirrors > Home > MPE Home > Th. List > simpggrpd | Structured version Visualization version GIF version |
Description: A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
simpggrpd.1 | ⊢ (𝜑 → 𝐺 ∈ SimpGrp) |
Ref | Expression |
---|---|
simpggrpd | ⊢ (𝜑 → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpggrpd.1 | . 2 ⊢ (𝜑 → 𝐺 ∈ SimpGrp) | |
2 | simpggrp 19199 | . 2 ⊢ (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Grpcgrp 18086 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: simpgntrivd 19203 simpgnideld 19204 simpgnsgd 19205 ablsimpg1gend 19210 ablsimpgcygd 19211 ablsimpgfindlem1 19212 ablsimpgfindlem2 19213 ablsimpgfind 19215 ablsimpgprmd 19220 simpcntrab 43217 |
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