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Mirrors > Home > MPE Home > Th. List > slwpgp | Structured version Visualization version GIF version |
Description: A Sylow 𝑃-subgroup is a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Ref | Expression |
---|---|
slwpgp.1 | ⊢ 𝑆 = (𝐺 ↾s 𝐻) |
Ref | Expression |
---|---|
slwpgp | ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ 𝐻 = 𝐻 | |
2 | slwsubg 19351 | . . . 4 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺)) | |
3 | slwpgp.1 | . . . . 5 ⊢ 𝑆 = (𝐺 ↾s 𝐻) | |
4 | 3 | slwispgp 19352 | . . . 4 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝐻 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐻)) |
5 | 2, 4 | mpdan 686 | . . 3 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → ((𝐻 ⊆ 𝐻 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐻)) |
6 | 1, 5 | mpbiri 258 | . 2 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → (𝐻 ⊆ 𝐻 ∧ 𝑃 pGrp 𝑆)) |
7 | 6 | simprd 497 | 1 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3909 class class class wbr 5104 ‘cfv 6494 (class class class)co 7352 ↾s cress 17072 SubGrpcsubg 18881 pGrp cpgp 19267 pSyl cslw 19268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fv 6502 df-ov 7355 df-oprab 7356 df-mpo 7357 df-subg 18884 df-slw 19272 |
This theorem is referenced by: slwhash 19365 sylow2 19367 sylow3lem6 19373 |
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