Theorem sylow2blem2 19550

Description: Lemma for sylow2b 19552. Left multiplication in a subgroup 𝐻 is a group action on the set of all left cosets of 𝐾. (Contributed by Mario Carneiro, 17-Jan-2015.)

Hypotheses

Ref	Expression
sylow2b.x	⊢ 𝑋 = (Base‘𝐺)
sylow2b.xf	⊢ (𝜑 → 𝑋 ∈ Fin)
sylow2b.h	⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
sylow2b.k	⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
sylow2b.a	⊢ + = (+g‘𝐺)
sylow2b.r	⊢ ∼ = (𝐺 ~QG 𝐾)
sylow2b.m	⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))

Assertion

Ref	Expression
sylow2blem2	⊢ (𝜑 → · ∈ ((𝐺 ↾s 𝐻) GrpAct (𝑋 / ∼ )))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝑥,𝐾,𝑦,𝑧   𝑥, · ,𝑦,𝑧   𝑥, + ,𝑦,𝑧   𝑥, ∼ ,𝑦,𝑧   𝜑,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sylow2blem2

Dummy variables 𝑎 𝑏 𝑠 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sylow2b.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
2		eqid 2736	. . . . 5 ⊢ (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝐻)
3	2	subggrp 19059	. . . 4 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝐻) ∈ Grp)
4	1, 3	syl 17	. . 3 ⊢ (𝜑 → (𝐺 ↾s 𝐻) ∈ Grp)
5		sylow2b.xf	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
6		pwfi 9219	. . . . 5 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
7	5, 6	sylib 218	. . . 4 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
8		sylow2b.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
9		sylow2b.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐺)
10		sylow2b.r	. . . . . . 7 ⊢ ∼ = (𝐺 ~QG 𝐾)
11	9, 10	eqger 19107	. . . . . 6 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
12	8, 11	syl 17	. . . . 5 ⊢ (𝜑 → ∼ Er 𝑋)
13	12	qsss 8713	. . . 4 ⊢ (𝜑 → (𝑋 / ∼ ) ⊆ 𝒫 𝑋)
14	7, 13	ssexd 5269	. . 3 ⊢ (𝜑 → (𝑋 / ∼ ) ∈ V)
15	4, 14	jca 511	. 2 ⊢ (𝜑 → ((𝐺 ↾s 𝐻) ∈ Grp ∧ (𝑋 / ∼ ) ∈ V))
16		sylow2b.m	. . . . . . 7 ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
17		vex 3444	. . . . . . . . 9 ⊢ 𝑦 ∈ V
18	17	mptex 7169	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ V
19	18	rnex 7852	. . . . . . 7 ⊢ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ V
20	16, 19	fnmpoi 8014	. . . . . 6 ⊢ · Fn (𝐻 × (𝑋 / ∼ ))
21	20	a1i 11	. . . . 5 ⊢ (𝜑 → · Fn (𝐻 × (𝑋 / ∼ )))
22		eqid 2736	. . . . . . . 8 ⊢ (𝑋 / ∼ ) = (𝑋 / ∼ )
23		oveq2 7366	. . . . . . . . 9 ⊢ ([𝑠] ∼ = 𝑣 → (𝑢 · [𝑠] ∼ ) = (𝑢 · 𝑣))
24	23	eleq1d 2821	. . . . . . . 8 ⊢ ([𝑠] ∼ = 𝑣 → ((𝑢 · [𝑠] ∼ ) ∈ (𝑋 / ∼ ) ↔ (𝑢 · 𝑣) ∈ (𝑋 / ∼ )))
25		sylow2b.a	. . . . . . . . . . 11 ⊢ + = (+g‘𝐺)
26	9, 5, 1, 8, 25, 10, 16	sylow2blem1 19549	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → (𝑢 · [𝑠] ∼ ) = [(𝑢 + 𝑠)] ∼ )
27	10	ovexi 7392	. . . . . . . . . . 11 ⊢ ∼ ∈ V
28		subgrcl 19061	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
29	1, 28	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ Grp)
30	29	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → 𝐺 ∈ Grp)
31	9	subgss 19057	. . . . . . . . . . . . . . 15 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋)
32	1, 31	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐻 ⊆ 𝑋)
33	32	sselda 3933	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻) → 𝑢 ∈ 𝑋)
34	33	3adant3 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → 𝑢 ∈ 𝑋)
35		simp3 1138	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋)
36	9, 25	grpcl 18871	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋) → (𝑢 + 𝑠) ∈ 𝑋)
37	30, 34, 35, 36	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → (𝑢 + 𝑠) ∈ 𝑋)
38		ecelqsw 8706	. . . . . . . . . . 11 ⊢ (( ∼ ∈ V ∧ (𝑢 + 𝑠) ∈ 𝑋) → [(𝑢 + 𝑠)] ∼ ∈ (𝑋 / ∼ ))
39	27, 37, 38	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → [(𝑢 + 𝑠)] ∼ ∈ (𝑋 / ∼ ))
40	26, 39	eqeltrd 2836	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → (𝑢 · [𝑠] ∼ ) ∈ (𝑋 / ∼ ))
41	40	3expa 1118	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐻) ∧ 𝑠 ∈ 𝑋) → (𝑢 · [𝑠] ∼ ) ∈ (𝑋 / ∼ ))
42	22, 24, 41	ectocld 8719	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐻) ∧ 𝑣 ∈ (𝑋 / ∼ )) → (𝑢 · 𝑣) ∈ (𝑋 / ∼ ))
43	42	ralrimiva 3128	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻) → ∀𝑣 ∈ (𝑋 / ∼ )(𝑢 · 𝑣) ∈ (𝑋 / ∼ ))
44	43	ralrimiva 3128	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐻 ∀𝑣 ∈ (𝑋 / ∼ )(𝑢 · 𝑣) ∈ (𝑋 / ∼ ))
45		ffnov 7484	. . . . 5 ⊢ ( · :(𝐻 × (𝑋 / ∼ ))⟶(𝑋 / ∼ ) ↔ ( · Fn (𝐻 × (𝑋 / ∼ )) ∧ ∀𝑢 ∈ 𝐻 ∀𝑣 ∈ (𝑋 / ∼ )(𝑢 · 𝑣) ∈ (𝑋 / ∼ )))
46	21, 44, 45	sylanbrc 583	. . . 4 ⊢ (𝜑 → · :(𝐻 × (𝑋 / ∼ ))⟶(𝑋 / ∼ ))
47	2	subgbas 19060	. . . . . . 7 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 = (Base‘(𝐺 ↾s 𝐻)))
48	1, 47	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻 = (Base‘(𝐺 ↾s 𝐻)))
49	48	xpeq1d 5653	. . . . 5 ⊢ (𝜑 → (𝐻 × (𝑋 / ∼ )) = ((Base‘(𝐺 ↾s 𝐻)) × (𝑋 / ∼ )))
50	49	feq2d 6646	. . . 4 ⊢ (𝜑 → ( · :(𝐻 × (𝑋 / ∼ ))⟶(𝑋 / ∼ ) ↔ · :((Base‘(𝐺 ↾s 𝐻)) × (𝑋 / ∼ ))⟶(𝑋 / ∼ )))
51	46, 50	mpbid 232	. . 3 ⊢ (𝜑 → · :((Base‘(𝐺 ↾s 𝐻)) × (𝑋 / ∼ ))⟶(𝑋 / ∼ ))
52		oveq2 7366	. . . . . . 7 ⊢ ([𝑠] ∼ = 𝑢 → ((0g‘(𝐺 ↾s 𝐻)) · [𝑠] ∼ ) = ((0g‘(𝐺 ↾s 𝐻)) · 𝑢))
53		id 22	. . . . . . 7 ⊢ ([𝑠] ∼ = 𝑢 → [𝑠] ∼ = 𝑢)
54	52, 53	eqeq12d 2752	. . . . . 6 ⊢ ([𝑠] ∼ = 𝑢 → (((0g‘(𝐺 ↾s 𝐻)) · [𝑠] ∼ ) = [𝑠] ∼ ↔ ((0g‘(𝐺 ↾s 𝐻)) · 𝑢) = 𝑢))
55		oveq2 7366	. . . . . . . 8 ⊢ ([𝑠] ∼ = 𝑢 → ((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = ((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢))
56		oveq2 7366	. . . . . . . . 9 ⊢ ([𝑠] ∼ = 𝑢 → (𝑏 · [𝑠] ∼ ) = (𝑏 · 𝑢))
57	56	oveq2d 7374	. . . . . . . 8 ⊢ ([𝑠] ∼ = 𝑢 → (𝑎 · (𝑏 · [𝑠] ∼ )) = (𝑎 · (𝑏 · 𝑢)))
58	55, 57	eqeq12d 2752	. . . . . . 7 ⊢ ([𝑠] ∼ = 𝑢 → (((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ )) ↔ ((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢) = (𝑎 · (𝑏 · 𝑢))))
59	58	2ralbidv 3200	. . . . . 6 ⊢ ([𝑠] ∼ = 𝑢 → (∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ )) ↔ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢) = (𝑎 · (𝑏 · 𝑢))))
60	54, 59	anbi12d 632	. . . . 5 ⊢ ([𝑠] ∼ = 𝑢 → ((((0g‘(𝐺 ↾s 𝐻)) · [𝑠] ∼ ) = [𝑠] ∼ ∧ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ ))) ↔ (((0g‘(𝐺 ↾s 𝐻)) · 𝑢) = 𝑢 ∧ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢) = (𝑎 · (𝑏 · 𝑢)))))
61		simpl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝜑)
62	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝐻 ∈ (SubGrp‘𝐺))
63		eqid 2736	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
64	63	subg0cl 19064	. . . . . . . . 9 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝐻)
65	62, 64	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (0g‘𝐺) ∈ 𝐻)
66		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋)
67	9, 5, 1, 8, 25, 10, 16	sylow2blem1 19549	. . . . . . . 8 ⊢ ((𝜑 ∧ (0g‘𝐺) ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → ((0g‘𝐺) · [𝑠] ∼ ) = [((0g‘𝐺) + 𝑠)] ∼ )
68	61, 65, 66, 67	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((0g‘𝐺) · [𝑠] ∼ ) = [((0g‘𝐺) + 𝑠)] ∼ )
69	2, 63	subg0 19062	. . . . . . . . 9 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → (0g‘𝐺) = (0g‘(𝐺 ↾s 𝐻)))
70	62, 69	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (0g‘𝐺) = (0g‘(𝐺 ↾s 𝐻)))
71	70	oveq1d 7373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((0g‘𝐺) · [𝑠] ∼ ) = ((0g‘(𝐺 ↾s 𝐻)) · [𝑠] ∼ ))
72	9, 25, 63	grplid 18897	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑠 ∈ 𝑋) → ((0g‘𝐺) + 𝑠) = 𝑠)
73	29, 72	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((0g‘𝐺) + 𝑠) = 𝑠)
74	73	eceq1d 8675	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → [((0g‘𝐺) + 𝑠)] ∼ = [𝑠] ∼ )
75	68, 71, 74	3eqtr3d 2779	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((0g‘(𝐺 ↾s 𝐻)) · [𝑠] ∼ ) = [𝑠] ∼ )
76	62	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝐻 ∈ (SubGrp‘𝐺))
77	76, 28	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝐺 ∈ Grp)
78	76, 31	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝐻 ⊆ 𝑋)
79		simprl 770	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝑎 ∈ 𝐻)
80	78, 79	sseldd 3934	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝑎 ∈ 𝑋)
81		simprr 772	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝑏 ∈ 𝐻)
82	78, 81	sseldd 3934	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝑏 ∈ 𝑋)
83	66	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝑠 ∈ 𝑋)
84	9, 25	grpass 18872	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋)) → ((𝑎 + 𝑏) + 𝑠) = (𝑎 + (𝑏 + 𝑠)))
85	77, 80, 82, 83, 84	syl13anc 1374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → ((𝑎 + 𝑏) + 𝑠) = (𝑎 + (𝑏 + 𝑠)))
86	85	eceq1d 8675	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → [((𝑎 + 𝑏) + 𝑠)] ∼ = [(𝑎 + (𝑏 + 𝑠))] ∼ )
87	61	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝜑)
88	9, 25	grpcl 18871	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋) → (𝑏 + 𝑠) ∈ 𝑋)
89	77, 82, 83, 88	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑏 + 𝑠) ∈ 𝑋)
90	9, 5, 1, 8, 25, 10, 16	sylow2blem1 19549	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐻 ∧ (𝑏 + 𝑠) ∈ 𝑋) → (𝑎 · [(𝑏 + 𝑠)] ∼ ) = [(𝑎 + (𝑏 + 𝑠))] ∼ )
91	87, 79, 89, 90	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑎 · [(𝑏 + 𝑠)] ∼ ) = [(𝑎 + (𝑏 + 𝑠))] ∼ )
92	86, 91	eqtr4d 2774	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → [((𝑎 + 𝑏) + 𝑠)] ∼ = (𝑎 · [(𝑏 + 𝑠)] ∼ ))
93	25	subgcl 19066	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻) → (𝑎 + 𝑏) ∈ 𝐻)
94	76, 79, 81, 93	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑎 + 𝑏) ∈ 𝐻)
95	9, 5, 1, 8, 25, 10, 16	sylow2blem1 19549	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 + 𝑏) ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → ((𝑎 + 𝑏) · [𝑠] ∼ ) = [((𝑎 + 𝑏) + 𝑠)] ∼ )
96	87, 94, 83, 95	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → ((𝑎 + 𝑏) · [𝑠] ∼ ) = [((𝑎 + 𝑏) + 𝑠)] ∼ )
97	9, 5, 1, 8, 25, 10, 16	sylow2blem1 19549	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → (𝑏 · [𝑠] ∼ ) = [(𝑏 + 𝑠)] ∼ )
98	87, 81, 83, 97	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑏 · [𝑠] ∼ ) = [(𝑏 + 𝑠)] ∼ )
99	98	oveq2d 7374	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑎 · (𝑏 · [𝑠] ∼ )) = (𝑎 · [(𝑏 + 𝑠)] ∼ ))
100	92, 96, 99	3eqtr4d 2781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → ((𝑎 + 𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ )))
101	100	ralrimivva 3179	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ∀𝑎 ∈ 𝐻 ∀𝑏 ∈ 𝐻 ((𝑎 + 𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ )))
102	62, 47	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝐻 = (Base‘(𝐺 ↾s 𝐻)))
103	2, 25	ressplusg 17211	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → + = (+g‘(𝐺 ↾s 𝐻)))
104	1, 103	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → + = (+g‘(𝐺 ↾s 𝐻)))
105	104	oveqdr 7386	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑎 + 𝑏) = (𝑎(+g‘(𝐺 ↾s 𝐻))𝑏))
106	105	oveq1d 7373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑎 + 𝑏) · [𝑠] ∼ ) = ((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ))
107	106	eqeq1d 2738	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (((𝑎 + 𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ )) ↔ ((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ ))))
108	102, 107	raleqbidv 3316	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (∀𝑏 ∈ 𝐻 ((𝑎 + 𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ )) ↔ ∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ ))))
109	102, 108	raleqbidv 3316	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (∀𝑎 ∈ 𝐻 ∀𝑏 ∈ 𝐻 ((𝑎 + 𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ )) ↔ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ ))))
110	101, 109	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ )))
111	75, 110	jca 511	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (((0g‘(𝐺 ↾s 𝐻)) · [𝑠] ∼ ) = [𝑠] ∼ ∧ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ ))))
112	22, 60, 111	ectocld 8719	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑋 / ∼ )) → (((0g‘(𝐺 ↾s 𝐻)) · 𝑢) = 𝑢 ∧ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢) = (𝑎 · (𝑏 · 𝑢))))
113	112	ralrimiva 3128	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ (𝑋 / ∼ )(((0g‘(𝐺 ↾s 𝐻)) · 𝑢) = 𝑢 ∧ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢) = (𝑎 · (𝑏 · 𝑢))))
114	51, 113	jca 511	. 2 ⊢ (𝜑 → ( · :((Base‘(𝐺 ↾s 𝐻)) × (𝑋 / ∼ ))⟶(𝑋 / ∼ ) ∧ ∀𝑢 ∈ (𝑋 / ∼ )(((0g‘(𝐺 ↾s 𝐻)) · 𝑢) = 𝑢 ∧ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢) = (𝑎 · (𝑏 · 𝑢)))))
115		eqid 2736	. . 3 ⊢ (Base‘(𝐺 ↾s 𝐻)) = (Base‘(𝐺 ↾s 𝐻))
116		eqid 2736	. . 3 ⊢ (+g‘(𝐺 ↾s 𝐻)) = (+g‘(𝐺 ↾s 𝐻))
117		eqid 2736	. . 3 ⊢ (0g‘(𝐺 ↾s 𝐻)) = (0g‘(𝐺 ↾s 𝐻))
118	115, 116, 117	isga 19220	. 2 ⊢ ( · ∈ ((𝐺 ↾s 𝐻) GrpAct (𝑋 / ∼ )) ↔ (((𝐺 ↾s 𝐻) ∈ Grp ∧ (𝑋 / ∼ ) ∈ V) ∧ ( · :((Base‘(𝐺 ↾s 𝐻)) × (𝑋 / ∼ ))⟶(𝑋 / ∼ ) ∧ ∀𝑢 ∈ (𝑋 / ∼ )(((0g‘(𝐺 ↾s 𝐻)) · 𝑢) = 𝑢 ∧ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢) = (𝑎 · (𝑏 · 𝑢))))))
119	15, 114, 118	sylanbrc 583	1 ⊢ (𝜑 → · ∈ ((𝐺 ↾s 𝐻) GrpAct (𝑋 / ∼ )))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  Vcvv 3440   ⊆ wss 3901  𝒫 cpw 4554   ↦ cmpt 5179   × cxp 5622  ran crn 5625   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360   Er wer 8632  [cec 8633   / cqs 8634  Fincfn 8883  Basecbs 17136   ↾_s cress 17157  +_gcplusg 17177  0_gc0g 17359  Grpcgrp 18863  SubGrpcsubg 19050   ~_QG cqg 19052   GrpAct cga 19218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-ec 8637  df-qs 8641  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-0g 17361  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18866  df-minusg 18867  df-sbg 18868  df-subg 19053  df-eqg 19055  df-ga 19219

This theorem is referenced by:  sylow2blem3  19551