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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 2818 | . . 3 ⊢ 𝐻 = 𝐻 | |
2 | slwsubg 18664 | . . . 4 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺)) | |
3 | slwpgp.1 | . . . . 5 ⊢ 𝑆 = (𝐺 ↾s 𝐻) | |
4 | 3 | slwispgp 18665 | . . . 4 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝐻 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐻)) |
5 | 2, 4 | mpdan 683 | . . 3 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → ((𝐻 ⊆ 𝐻 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐻)) |
6 | 1, 5 | mpbiri 259 | . 2 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → (𝐻 ⊆ 𝐻 ∧ 𝑃 pGrp 𝑆)) |
7 | 6 | simprd 496 | 1 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ↾s cress 16472 SubGrpcsubg 18211 pGrp cpgp 18583 pSyl cslw 18584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-subg 18214 df-slw 18588 |
This theorem is referenced by: slwhash 18678 sylow2 18680 sylow3lem6 18686 |
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