Theorem slwispgp 19644

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 19641	. . 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 1146	. 2 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘))
3		sseq2 4022	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝐻 ⊆ 𝑘 ↔ 𝐻 ⊆ 𝐾))
4		oveq2 7439	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝐺 ↾s 𝑘) = (𝐺 ↾s 𝐾))
5		slwispgp.1	. . . . . . 7 ⊢ 𝑆 = (𝐺 ↾s 𝐾)
6	4, 5	eqtr4di 2793	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝐺 ↾s 𝑘) = 𝑆)
7	6	breq2d 5160	. . . . 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 2747	. . . 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 3621	. 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 1537   ∈ wcel 2106  ∀wral 3059   ⊆ wss 3963   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  ℙcprime 16705   ↾_s cress 17274  SubGrpcsubg 19151   pGrp cpgp 19559   pSyl cslw 19560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-subg 19154  df-slw 19564

This theorem is referenced by:  slwpss  19645  slwpgp  19646  subgslw  19649  slwhash  19657