Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2821 | . . 3 ⊢ 𝐻 = 𝐻 | |
2 | slwsubg 18735 | . . . 4 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺)) | |
3 | slwpgp.1 | . . . . 5 ⊢ 𝑆 = (𝐺 ↾s 𝐻) | |
4 | 3 | slwispgp 18736 | . . . 4 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝐻 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐻)) |
5 | 2, 4 | mpdan 685 | . . 3 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → ((𝐻 ⊆ 𝐻 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐻)) |
6 | 1, 5 | mpbiri 260 | . 2 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → (𝐻 ⊆ 𝐻 ∧ 𝑃 pGrp 𝑆)) |
7 | 6 | simprd 498 | 1 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ↾s cress 16484 SubGrpcsubg 18273 pGrp cpgp 18654 pSyl cslw 18655 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-subg 18276 df-slw 18659 |
This theorem is referenced by: slwhash 18749 sylow2 18751 sylow3lem6 18757 |
Copyright terms: Public domain | W3C validator |