Theorem slwprm 19628

Description: Reverse closure for the first argument of a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 2-May-2015.)

Assertion

Ref	Expression
slwprm	⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)

Proof of Theorem slwprm

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isslw 19627	. 2 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)))
2	1	simp1bi 1145	1 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060   ⊆ wss 3950   class class class wbr 5142  ‘cfv 6560  (class class class)co 7432  ℙcprime 16709   ↾_s cress 17275  SubGrpcsubg 19139   pGrp cpgp 19545   pSyl cslw 19546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-subg 19142  df-slw 19550

This theorem is referenced by:  subgslw  19635  slwhash  19643