Theorem slwispgp 19490

Description: Defining property of a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)

Hypothesis

Ref	Expression
slwispgp.1	⊢ 𝑆 = (𝐺 ↾s 𝐾)

Assertion

Ref	Expression
slwispgp	⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))

Proof of Theorem slwispgp

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isslw 19487	. . 3 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)))
2	1	simp3bi 1147	. 2 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘))
3		sseq2 3962	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝐻 ⊆ 𝑘 ↔ 𝐻 ⊆ 𝐾))
4		oveq2 7357	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝐺 ↾s 𝑘) = (𝐺 ↾s 𝐾))
5		slwispgp.1	. . . . . . 7 ⊢ 𝑆 = (𝐺 ↾s 𝐾)
6	4, 5	eqtr4di 2782	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝐺 ↾s 𝑘) = 𝑆)
7	6	breq2d 5104	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑃 pGrp (𝐺 ↾s 𝑘) ↔ 𝑃 pGrp 𝑆))
8	3, 7	anbi12d 632	. . . 4 ⊢ (𝑘 = 𝐾 → ((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ (𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆)))
9		eqeq2 2741	. . . 4 ⊢ (𝑘 = 𝐾 → (𝐻 = 𝑘 ↔ 𝐻 = 𝐾))
10	8, 9	bibi12d 345	. . 3 ⊢ (𝑘 = 𝐾 → (((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘) ↔ ((𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾)))
11	10	rspccva 3576	. 2 ⊢ ((∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
12	2, 11	sylan 580	1 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3903   class class class wbr 5092  ‘cfv 6482  (class class class)co 7349  ℙcprime 16582   ↾_s cress 17141  SubGrpcsubg 18999   pGrp cpgp 19405   pSyl cslw 19406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-subg 19002  df-slw 19410

This theorem is referenced by:  slwpss  19491  slwpgp  19492  subgslw  19495  slwhash  19503