Theorem slwispgp 19834

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 19831	. . 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 1165	. 2 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘))
3		sseq2 3935	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝐻 ⊆ 𝑘 ↔ 𝐻 ⊆ 𝐾))
4		oveq2 7432	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝐺 ↾s 𝑘) = (𝐺 ↾s 𝐾))
5		slwispgp.1	. . . . . . 7 ⊢ 𝑆 = (𝐺 ↾s 𝐾)
6	4, 5	eqtr4di 2813	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝐺 ↾s 𝑘) = 𝑆)
7	6	breq2d 5114	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑃 pGrp (𝐺 ↾s 𝑘) ↔ 𝑃 pGrp 𝑆))
8	3, 7	anbi12d 644	. . . 4 ⊢ (𝑘 = 𝐾 → ((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ (𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆)))
9		eqeq2 2772	. . . 4 ⊢ (𝑘 = 𝐾 → (𝐻 = 𝑘 ↔ 𝐻 = 𝐾))
10	8, 9	bibi12d 348	. . 3 ⊢ (𝑘 = 𝐾 → (((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘) ↔ ((𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾)))
11	10	rspccva 3575	. 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 592	1 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3896   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424  ℙcprime 16855   ↾_s cress 17417  SubGrpcsubg 19339   pGrp cpgp 19749   pSyl cslw 19750

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-subg 19342  df-slw 19754

This theorem is used by:  slwpss  19835  slwpgp  19836  subgslw  19839  slwhash  19847