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Theorem slwispgp 18233
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 18230 . . 3 (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
21simp3bi 1141 . 2 (𝐻 ∈ (𝑃 pSyl 𝐺) → ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘))
3 sseq2 3776 . . . . 5 (𝑘 = 𝐾 → (𝐻𝑘𝐻𝐾))
4 oveq2 6801 . . . . . . 7 (𝑘 = 𝐾 → (𝐺s 𝑘) = (𝐺s 𝐾))
5 slwispgp.1 . . . . . . 7 𝑆 = (𝐺s 𝐾)
64, 5syl6eqr 2823 . . . . . 6 (𝑘 = 𝐾 → (𝐺s 𝑘) = 𝑆)
76breq2d 4798 . . . . 5 (𝑘 = 𝐾 → (𝑃 pGrp (𝐺s 𝑘) ↔ 𝑃 pGrp 𝑆))
83, 7anbi12d 616 . . . 4 (𝑘 = 𝐾 → ((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ (𝐻𝐾𝑃 pGrp 𝑆)))
9 eqeq2 2782 . . . 4 (𝑘 = 𝐾 → (𝐻 = 𝑘𝐻 = 𝐾))
108, 9bibi12d 334 . . 3 (𝑘 = 𝐾 → (((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘) ↔ ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾)))
1110rspccva 3459 . 2 ((∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
122, 11sylan 569 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 382   = wceq 1631  wcel 2145  wral 3061  wss 3723   class class class wbr 4786  cfv 6031  (class class class)co 6793  cprime 15592  s cress 16065  SubGrpcsubg 17796   pGrp cpgp 18153   pSyl cslw 18154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-subg 17799  df-slw 18158
This theorem is referenced by:  slwpss  18234  slwpgp  18235  subgslw  18238  slwhash  18246
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