Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pgpprm | Structured version Visualization version GIF version |
Description: Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.) |
Ref | Expression |
---|---|
pgpprm | ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2821 | . . 3 ⊢ (od‘𝐺) = (od‘𝐺) | |
3 | 1, 2 | ispgp 18717 | . 2 ⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛))) |
4 | 3 | simp1bi 1141 | 1 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℕ0cn0 11898 ↑cexp 13430 ℙcprime 16015 Basecbs 16483 Grpcgrp 18103 odcod 18652 pGrp cpgp 18654 |
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 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
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-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-iota 6314 df-fv 6363 df-ov 7159 df-pgp 18658 |
This theorem is referenced by: subgpgp 18722 pgpssslw 18739 sylow2blem3 18747 pgpfac1lem2 19197 pgpfac1lem3a 19198 pgpfac1lem3 19199 pgpfac1lem4 19200 pgpfaclem1 19203 |
Copyright terms: Public domain | W3C validator |