MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  slwispgp Structured version   Visualization version   GIF version

Theorem slwispgp 18728
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 18725 . . 3 (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
21simp3bi 1144 . 2 (𝐻 ∈ (𝑃 pSyl 𝐺) → ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘))
3 sseq2 3941 . . . . 5 (𝑘 = 𝐾 → (𝐻𝑘𝐻𝐾))
4 oveq2 7143 . . . . . . 7 (𝑘 = 𝐾 → (𝐺s 𝑘) = (𝐺s 𝐾))
5 slwispgp.1 . . . . . . 7 𝑆 = (𝐺s 𝐾)
64, 5eqtr4di 2851 . . . . . 6 (𝑘 = 𝐾 → (𝐺s 𝑘) = 𝑆)
76breq2d 5042 . . . . 5 (𝑘 = 𝐾 → (𝑃 pGrp (𝐺s 𝑘) ↔ 𝑃 pGrp 𝑆))
83, 7anbi12d 633 . . . 4 (𝑘 = 𝐾 → ((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ (𝐻𝐾𝑃 pGrp 𝑆)))
9 eqeq2 2810 . . . 4 (𝑘 = 𝐾 → (𝐻 = 𝑘𝐻 = 𝐾))
108, 9bibi12d 349 . . 3 (𝑘 = 𝐾 → (((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘) ↔ ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾)))
1110rspccva 3570 . 2 ((∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
122, 11sylan 583 1 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wral 3106  wss 3881   class class class wbr 5030  cfv 6324  (class class class)co 7135  cprime 16005  s cress 16476  SubGrpcsubg 18265   pGrp cpgp 18646   pSyl cslw 18647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-subg 18268  df-slw 18651
This theorem is referenced by:  slwpss  18729  slwpgp  18730  subgslw  18733  slwhash  18741
  Copyright terms: Public domain W3C validator