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Mirrors > Home > MPE Home > Th. List > subgpgp | Structured version Visualization version GIF version |
Description: A subgroup of a p-group is a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.) |
Ref | Expression |
---|---|
subgpgp | ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾s 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pgpprm 18837 | . . 3 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ) | |
2 | 1 | adantr 484 | . 2 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑃 ∈ ℙ) |
3 | eqid 2738 | . . . 4 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆) | |
4 | 3 | subggrp 18401 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑆) ∈ Grp) |
5 | 4 | adantl 485 | . 2 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐺 ↾s 𝑆) ∈ Grp) |
6 | eqid 2738 | . . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
7 | eqid 2738 | . . . . . . 7 ⊢ (od‘𝐺) = (od‘𝐺) | |
8 | 6, 7 | ispgp 18836 | . . . . . 6 ⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛))) |
9 | 8 | simp3bi 1148 | . . . . 5 ⊢ (𝑃 pGrp 𝐺 → ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛)) |
10 | 9 | adantr 484 | . . . 4 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛)) |
11 | 6 | subgss 18399 | . . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
12 | 11 | adantl 485 | . . . . . 6 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺)) |
13 | ssralv 3944 | . . . . . 6 ⊢ (𝑆 ⊆ (Base‘𝐺) → (∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛) → ∀𝑥 ∈ 𝑆 ∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛))) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛) → ∀𝑥 ∈ 𝑆 ∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛))) |
15 | eqid 2738 | . . . . . . . . . 10 ⊢ (od‘(𝐺 ↾s 𝑆)) = (od‘(𝐺 ↾s 𝑆)) | |
16 | 3, 7, 15 | subgod 18814 | . . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆) → ((od‘𝐺)‘𝑥) = ((od‘(𝐺 ↾s 𝑆))‘𝑥)) |
17 | 16 | adantll 714 | . . . . . . . 8 ⊢ (((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑆) → ((od‘𝐺)‘𝑥) = ((od‘(𝐺 ↾s 𝑆))‘𝑥)) |
18 | 17 | eqeq1d 2740 | . . . . . . 7 ⊢ (((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑆) → (((od‘𝐺)‘𝑥) = (𝑃↑𝑛) ↔ ((od‘(𝐺 ↾s 𝑆))‘𝑥) = (𝑃↑𝑛))) |
19 | 18 | rexbidv 3207 | . . . . . 6 ⊢ (((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑆) → (∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛) ↔ ∃𝑛 ∈ ℕ0 ((od‘(𝐺 ↾s 𝑆))‘𝑥) = (𝑃↑𝑛))) |
20 | 19 | ralbidva 3108 | . . . . 5 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (∀𝑥 ∈ 𝑆 ∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛) ↔ ∀𝑥 ∈ 𝑆 ∃𝑛 ∈ ℕ0 ((od‘(𝐺 ↾s 𝑆))‘𝑥) = (𝑃↑𝑛))) |
21 | 14, 20 | sylibd 242 | . . . 4 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃↑𝑛) → ∀𝑥 ∈ 𝑆 ∃𝑛 ∈ ℕ0 ((od‘(𝐺 ↾s 𝑆))‘𝑥) = (𝑃↑𝑛))) |
22 | 10, 21 | mpd 15 | . . 3 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ 𝑆 ∃𝑛 ∈ ℕ0 ((od‘(𝐺 ↾s 𝑆))‘𝑥) = (𝑃↑𝑛)) |
23 | 3 | subgbas 18402 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
24 | 23 | adantl 485 | . . . 4 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
25 | 24 | raleqdv 3316 | . . 3 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (∀𝑥 ∈ 𝑆 ∃𝑛 ∈ ℕ0 ((od‘(𝐺 ↾s 𝑆))‘𝑥) = (𝑃↑𝑛) ↔ ∀𝑥 ∈ (Base‘(𝐺 ↾s 𝑆))∃𝑛 ∈ ℕ0 ((od‘(𝐺 ↾s 𝑆))‘𝑥) = (𝑃↑𝑛))) |
26 | 22, 25 | mpbid 235 | . 2 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ (Base‘(𝐺 ↾s 𝑆))∃𝑛 ∈ ℕ0 ((od‘(𝐺 ↾s 𝑆))‘𝑥) = (𝑃↑𝑛)) |
27 | eqid 2738 | . . 3 ⊢ (Base‘(𝐺 ↾s 𝑆)) = (Base‘(𝐺 ↾s 𝑆)) | |
28 | 27, 15 | ispgp 18836 | . 2 ⊢ (𝑃 pGrp (𝐺 ↾s 𝑆) ↔ (𝑃 ∈ ℙ ∧ (𝐺 ↾s 𝑆) ∈ Grp ∧ ∀𝑥 ∈ (Base‘(𝐺 ↾s 𝑆))∃𝑛 ∈ ℕ0 ((od‘(𝐺 ↾s 𝑆))‘𝑥) = (𝑃↑𝑛))) |
29 | 2, 5, 26, 28 | syl3anbrc 1344 | 1 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾s 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ∀wral 3053 ∃wrex 3054 ⊆ wss 3844 class class class wbr 5031 ‘cfv 6340 (class class class)co 7171 ℕ0cn0 11977 ↑cexp 13522 ℙcprime 16113 Basecbs 16587 ↾s cress 16588 Grpcgrp 18220 SubGrpcsubg 18392 odcod 18771 pGrp cpgp 18773 |
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-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-om 7601 df-1st 7715 df-2nd 7716 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-er 8321 df-en 8557 df-dom 8558 df-sdom 8559 df-sup 8980 df-inf 8981 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-nn 11718 df-2 11780 df-n0 11978 df-z 12064 df-uz 12326 df-seq 13462 df-ndx 16590 df-slot 16591 df-base 16593 df-sets 16594 df-ress 16595 df-plusg 16682 df-0g 16819 df-mgm 17969 df-sgrp 18018 df-mnd 18029 df-submnd 18074 df-grp 18223 df-minusg 18224 df-mulg 18344 df-subg 18395 df-od 18775 df-pgp 18777 |
This theorem is referenced by: pgpfaclem1 19323 pgpfaclem3 19325 |
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