Theorem isslw 18854

Description: The property of being a Sylow subgroup. A Sylow 𝑃-subgroup is a 𝑃-group which has no proper supersets that are also 𝑃-groups. (Contributed by Mario Carneiro, 16-Jan-2015.)

Assertion

Ref	Expression
isslw	⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)))

Distinct variable groups:   𝑘,𝐺   𝑘,𝐻   𝑃,𝑘

Proof of Theorem isslw

Dummy variables 𝑔 ℎ 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-slw 18780	. . 3 ⊢ pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ {ℎ ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾s 𝑘)) ↔ ℎ = 𝑘)})
2	1	elmpocl 7406	. 2 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp))
3		simp1 1137	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)) → 𝑃 ∈ ℙ)
4		subgrcl 18405	. . . 4 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5	4	3ad2ant2 1135	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)) → 𝐺 ∈ Grp)
6	3, 5	jca 515	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)) → (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp))
7		simpr 488	. . . . . . . . 9 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
8	7	fveq2d 6681	. . . . . . . 8 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (SubGrp‘𝑔) = (SubGrp‘𝐺))
9		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → 𝑝 = 𝑃)
10	7	oveq1d 7188	. . . . . . . . . . . 12 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (𝑔 ↾s 𝑘) = (𝐺 ↾s 𝑘))
11	9, 10	breq12d 5044	. . . . . . . . . . 11 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (𝑝 pGrp (𝑔 ↾s 𝑘) ↔ 𝑃 pGrp (𝐺 ↾s 𝑘)))
12	11	anbi2d 632	. . . . . . . . . 10 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → ((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾s 𝑘)) ↔ (ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘))))
13	12	bibi1d 347	. . . . . . . . 9 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾s 𝑘)) ↔ ℎ = 𝑘) ↔ ((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ ℎ = 𝑘)))
14	8, 13	raleqbidv 3305	. . . . . . . 8 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (∀𝑘 ∈ (SubGrp‘𝑔)((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾s 𝑘)) ↔ ℎ = 𝑘) ↔ ∀𝑘 ∈ (SubGrp‘𝐺)((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ ℎ = 𝑘)))
15	8, 14	rabeqbidv 3388	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → {ℎ ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾s 𝑘)) ↔ ℎ = 𝑘)} = {ℎ ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ ℎ = 𝑘)})
16		fvex 6690	. . . . . . . 8 ⊢ (SubGrp‘𝐺) ∈ V
17	16	rabex 5201	. . . . . . 7 ⊢ {ℎ ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ ℎ = 𝑘)} ∈ V
18	15, 1, 17	ovmpoa 7323	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝑃 pSyl 𝐺) = {ℎ ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ ℎ = 𝑘)})
19	18	eleq2d 2819	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ 𝐻 ∈ {ℎ ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ ℎ = 𝑘)}))
20		cleq1lem 14434	. . . . . . . 8 ⊢ (ℎ = 𝐻 → ((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘))))
21		eqeq1 2743	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (ℎ = 𝑘 ↔ 𝐻 = 𝑘))
22	20, 21	bibi12d 349	. . . . . . 7 ⊢ (ℎ = 𝐻 → (((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ ℎ = 𝑘) ↔ ((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)))
23	22	ralbidv 3110	. . . . . 6 ⊢ (ℎ = 𝐻 → (∀𝑘 ∈ (SubGrp‘𝐺)((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ ℎ = 𝑘) ↔ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)))
24	23	elrab 3589	. . . . 5 ⊢ (𝐻 ∈ {ℎ ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ ℎ = 𝑘)} ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)))
25	19, 24	bitrdi 290	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘))))
26		simpl 486	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → 𝑃 ∈ ℙ)
27	26	biantrurd 536	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → ((𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)) ↔ (𝑃 ∈ ℙ ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)))))
28	25, 27	bitrd 282	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)))))
29		3anass 1096	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)) ↔ (𝑃 ∈ ℙ ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘))))
30	28, 29	bitr4di 292	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘))))
31	2, 6, 30	pm5.21nii 383	1 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∀wral 3054  {crab 3058   ⊆ wss 3844   class class class wbr 5031  ‘cfv 6340  (class class class)co 7173  ℙcprime 16115   ↾_s cress 16590  Grpcgrp 18222  SubGrpcsubg 18394   pGrp cpgp 18775   pSyl cslw 18776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-subg 18397  df-slw 18780

This theorem is referenced by:  slwprm  18855  slwsubg  18856  slwispgp  18857  pgpssslw  18860  subgslw  18862  fislw  18871