Theorem slwispgp 19541

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 19538	. . 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 3973	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝐻 ⊆ 𝑘 ↔ 𝐻 ⊆ 𝐾))
4		oveq2 7395	. . . . . . 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 5119	. . . . 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 3587	. 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 3914   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  ℙcprime 16641   ↾_s cress 17200  SubGrpcsubg 19052   pGrp cpgp 19456   pSyl cslw 19457

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 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387

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 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-subg 19055  df-slw 19461

This theorem is referenced by:  slwpss  19542  slwpgp  19543  subgslw  19546  slwhash  19554