Theorem isslw 18023

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 17951	. . 3 ⊢ pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ {ℎ ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾s 𝑘)) ↔ ℎ = 𝑘)})
2	1	elmpt2cl 6876	. 2 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp))
3		simp1 1061	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)) → 𝑃 ∈ ℙ)
4		subgrcl 17599	. . . 4 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5	4	3ad2ant2 1083	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)) → 𝐺 ∈ Grp)
6	3, 5	jca 554	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)) → (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp))
7		simpr 477	. . . . . . . . 9 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
8	7	fveq2d 6195	. . . . . . . 8 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (SubGrp‘𝑔) = (SubGrp‘𝐺))
9		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → 𝑝 = 𝑃)
10	7	oveq1d 6665	. . . . . . . . . . . 12 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (𝑔 ↾s 𝑘) = (𝐺 ↾s 𝑘))
11	9, 10	breq12d 4666	. . . . . . . . . . 11 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (𝑝 pGrp (𝑔 ↾s 𝑘) ↔ 𝑃 pGrp (𝐺 ↾s 𝑘)))
12	11	anbi2d 740	. . . . . . . . . 10 ⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → ((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾s 𝑘)) ↔ (ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘))))
13	12	bibi1d 333	. . . . . . . . 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 3152	. . . . . . . 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 3195	. . . . . . 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 6201	. . . . . . . 8 ⊢ (SubGrp‘𝐺) ∈ V
17	16	rabex 4813	. . . . . . 7 ⊢ {ℎ ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ ℎ = 𝑘)} ∈ V
18	15, 1, 17	ovmpt2a 6791	. . . . . 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 2687	. . . . 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		sseq1 3626	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → (ℎ ⊆ 𝑘 ↔ 𝐻 ⊆ 𝑘))
21	20	anbi1d 741	. . . . . . . 8 ⊢ (ℎ = 𝐻 → ((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘))))
22		eqeq1 2626	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (ℎ = 𝑘 ↔ 𝐻 = 𝑘))
23	21, 22	bibi12d 335	. . . . . . 7 ⊢ (ℎ = 𝐻 → (((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ ℎ = 𝑘) ↔ ((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)))
24	23	ralbidv 2986	. . . . . 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)))
25	24	elrab 3363	. . . . 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)))
26	19, 25	syl6bb 276	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘))))
27		simpl 473	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → 𝑃 ∈ ℙ)
28	27	biantrurd 529	. . . 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)))))
29	26, 28	bitrd 268	. . 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)))))
30		3anass 1042	. . 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))))
31	29, 30	syl6bbr 278	. 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))))
32	2, 6, 31	pm5.21nii 368	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916   ⊆ wss 3574   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  ℙcprime 15385   ↾_s cress 15858  Grpcgrp 17422  SubGrpcsubg 17588   pGrp cpgp 17946   pSyl cslw 17947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-subg 17591  df-slw 17951

This theorem is referenced by:  slwprm  18024  slwsubg  18025  slwispgp  18026  pgpssslw  18029  subgslw  18031  fislw  18040