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Theorem slwispgp 17942
Description: Defining property of a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypothesis
Ref Expression
slwispgp.1 𝑆 = (𝐺s 𝐾)
Assertion
Ref Expression
slwispgp ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))

Proof of Theorem slwispgp
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 isslw 17939 . . 3 (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
21simp3bi 1076 . 2 (𝐻 ∈ (𝑃 pSyl 𝐺) → ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘))
3 sseq2 3611 . . . . 5 (𝑘 = 𝐾 → (𝐻𝑘𝐻𝐾))
4 oveq2 6613 . . . . . . 7 (𝑘 = 𝐾 → (𝐺s 𝑘) = (𝐺s 𝐾))
5 slwispgp.1 . . . . . . 7 𝑆 = (𝐺s 𝐾)
64, 5syl6eqr 2678 . . . . . 6 (𝑘 = 𝐾 → (𝐺s 𝑘) = 𝑆)
76breq2d 4630 . . . . 5 (𝑘 = 𝐾 → (𝑃 pGrp (𝐺s 𝑘) ↔ 𝑃 pGrp 𝑆))
83, 7anbi12d 746 . . . 4 (𝑘 = 𝐾 → ((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ (𝐻𝐾𝑃 pGrp 𝑆)))
9 eqeq2 2637 . . . 4 (𝑘 = 𝐾 → (𝐻 = 𝑘𝐻 = 𝐾))
108, 9bibi12d 335 . . 3 (𝑘 = 𝐾 → (((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘) ↔ ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾)))
1110rspccva 3299 . 2 ((∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
122, 11sylan 488 1 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wss 3560   class class class wbr 4618  cfv 5850  (class class class)co 6605  cprime 15304  s cress 15777  SubGrpcsubg 17504   pGrp cpgp 17862   pSyl cslw 17863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-subg 17507  df-slw 17867
This theorem is referenced by:  slwpss  17943  slwpgp  17944  subgslw  17947  slwhash  17955
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