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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 2823 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2823 | . . 3 ⊢ (od‘𝐺) = (od‘𝐺) | |
3 | 1, 2 | ispgp 18719 | . 2 ⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛))) |
4 | 3 | simp2bi 1142 | 1 ⊢ (𝑃 pGrp 𝐺 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℕ0cn0 11900 ↑cexp 13432 ℙcprime 16017 Basecbs 16485 Grpcgrp 18105 odcod 18654 pGrp cpgp 18656 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-iota 6316 df-fv 6365 df-ov 7161 df-pgp 18660 |
This theorem is referenced by: pgphash 18734 |
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