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Theorem slwispgp 19630
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 19627 . . 3 (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
21simp3bi 1147 . 2 (𝐻 ∈ (𝑃 pSyl 𝐺) → ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘))
3 sseq2 4009 . . . . 5 (𝑘 = 𝐾 → (𝐻𝑘𝐻𝐾))
4 oveq2 7440 . . . . . . 7 (𝑘 = 𝐾 → (𝐺s 𝑘) = (𝐺s 𝐾))
5 slwispgp.1 . . . . . . 7 𝑆 = (𝐺s 𝐾)
64, 5eqtr4di 2794 . . . . . 6 (𝑘 = 𝐾 → (𝐺s 𝑘) = 𝑆)
76breq2d 5154 . . . . 5 (𝑘 = 𝐾 → (𝑃 pGrp (𝐺s 𝑘) ↔ 𝑃 pGrp 𝑆))
83, 7anbi12d 632 . . . 4 (𝑘 = 𝐾 → ((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ (𝐻𝐾𝑃 pGrp 𝑆)))
9 eqeq2 2748 . . . 4 (𝑘 = 𝐾 → (𝐻 = 𝑘𝐻 = 𝐾))
108, 9bibi12d 345 . . 3 (𝑘 = 𝐾 → (((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘) ↔ ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾)))
1110rspccva 3620 . 2 ((∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
122, 11sylan 580 1 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
Colors of variables: wff setvar class
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  cprime 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:  slwpss  19631  slwpgp  19632  subgslw  19635  slwhash  19643
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