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Mirrors > Home > MPE Home > Th. List > pgpgrp | Structured version Visualization version GIF version |
Description: Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.) |
Ref | Expression |
---|---|
pgpgrp | ⊢ (𝑃 pGrp 𝐺 → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2738 | . . 3 ⊢ (od‘𝐺) = (od‘𝐺) | |
3 | 1, 2 | ispgp 18828 | . 2 ⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛))) |
4 | 3 | simp2bi 1147 | 1 ⊢ (𝑃 pGrp 𝐺 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ∀wral 3053 ∃wrex 3054 class class class wbr 5027 ‘cfv 6333 (class class class)co 7164 ℕ0cn0 11969 ↑cexp 13514 ℙcprime 16105 Basecbs 16579 Grpcgrp 18212 odcod 18763 pGrp cpgp 18765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-xp 5525 df-iota 6291 df-fv 6341 df-ov 7167 df-pgp 18769 |
This theorem is referenced by: pgphash 18843 |
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