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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 2610 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2610 | . . 3 ⊢ (od‘𝐺) = (od‘𝐺) | |
3 | 1, 2 | ispgp 17830 | . 2 ⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛))) |
4 | 3 | simp1bi 1069 | 1 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℕ0cn0 11169 ↑cexp 12722 ℙcprime 15223 Basecbs 15695 Grpcgrp 17245 odcod 17767 pGrp cpgp 17769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-iota 5768 df-fv 5812 df-ov 6552 df-pgp 17773 |
This theorem is referenced by: subgpgp 17835 pgpssslw 17852 sylow2blem3 17860 pgpfac1lem2 18297 pgpfac1lem3a 18298 pgpfac1lem3 18299 pgpfac1lem4 18300 pgpfaclem1 18303 |
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