Theorem sylow2blem1 19527

Description: Lemma for sylow2b 19530. Evaluate the group action on a left coset. (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
sylow2blem1	⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝐵 · [𝐶] ∼ ) = [(𝐵 + 𝐶)] ∼ )

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

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sylow2blem1

Step	Hyp	Ref	Expression
1		simp2 1137	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐵 ∈ 𝐻)
2		sylow2b.r	. . . . 5 ⊢ ∼ = (𝐺 ~QG 𝐾)
3	2	ovexi 7375	. . . 4 ⊢ ∼ ∈ V
4		simp3 1138	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ 𝑋)
5		ecelqsw 8688	. . . 4 ⊢ (( ∼ ∈ V ∧ 𝐶 ∈ 𝑋) → [𝐶] ∼ ∈ (𝑋 / ∼ ))
6	3, 4, 5	sylancr 587	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [𝐶] ∼ ∈ (𝑋 / ∼ ))
7		simpr 484	. . . . . 6 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = [𝐶] ∼ ) → 𝑦 = [𝐶] ∼ )
8		simpl 482	. . . . . . 7 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = [𝐶] ∼ ) → 𝑥 = 𝐵)
9	8	oveq1d 7356	. . . . . 6 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = [𝐶] ∼ ) → (𝑥 + 𝑧) = (𝐵 + 𝑧))
10	7, 9	mpteq12dv 5173	. . . . 5 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = [𝐶] ∼ ) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
11	10	rneqd 5873	. . . 4 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = [𝐶] ∼ ) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
12		sylow2b.m	. . . 4 ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
13		ecexg 8621	. . . . . . 7 ⊢ ( ∼ ∈ V → [𝐶] ∼ ∈ V)
14	3, 13	ax-mp 5	. . . . . 6 ⊢ [𝐶] ∼ ∈ V
15	14	mptex 7152	. . . . 5 ⊢ (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ∈ V
16	15	rnex 7835	. . . 4 ⊢ ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ∈ V
17	11, 12, 16	ovmpoa 7496	. . 3 ⊢ ((𝐵 ∈ 𝐻 ∧ [𝐶] ∼ ∈ (𝑋 / ∼ )) → (𝐵 · [𝐶] ∼ ) = ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
18	1, 6, 17	syl2anc 584	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝐵 · [𝐶] ∼ ) = ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
19		sylow2b.xf	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
20		sylow2b.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
21		sylow2b.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
22	21, 2	eqger 19085	. . . . . . 7 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
23	20, 22	syl 17	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝑋)
24	23	ecss 8668	. . . . 5 ⊢ (𝜑 → [(𝐵 + 𝐶)] ∼ ⊆ 𝑋)
25	19, 24	ssfid 9148	. . . 4 ⊢ (𝜑 → [(𝐵 + 𝐶)] ∼ ∈ Fin)
26	25	3ad2ant1 1133	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [(𝐵 + 𝐶)] ∼ ∈ Fin)
27		vex 3440	. . . . . . . 8 ⊢ 𝑧 ∈ V
28		elecg 8661	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ [𝐶] ∼ ↔ 𝐶 ∼ 𝑧))
29	27, 4, 28	sylancr 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ [𝐶] ∼ ↔ 𝐶 ∼ 𝑧))
30	29	biimpa 476	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝑧 ∈ [𝐶] ∼ ) → 𝐶 ∼ 𝑧)
31		sylow2b.h	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
32		subgrcl 19039	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
33	31, 32	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Grp)
34	33	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐺 ∈ Grp)
35	21	subgss 19035	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋)
36	31, 35	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ⊆ 𝑋)
37	36	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐻 ⊆ 𝑋)
38	37, 1	sseldd 3930	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐵 ∈ 𝑋)
39		sylow2b.a	. . . . . . . . . . 11 ⊢ + = (+g‘𝐺)
40	21, 39	grpcl 18849	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵 + 𝐶) ∈ 𝑋)
41	34, 38, 4, 40	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝐵 + 𝐶) ∈ 𝑋)
42	41	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (𝐵 + 𝐶) ∈ 𝑋)
43	34	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → 𝐺 ∈ Grp)
44	38	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → 𝐵 ∈ 𝑋)
45	21	subgss 19035	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋)
46	20, 45	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ⊆ 𝑋)
47		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (invg‘𝐺) = (invg‘𝐺)
48	21, 47, 39, 2	eqgval 19084	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝐾 ⊆ 𝑋) → (𝐶 ∼ 𝑧 ↔ (𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
49	33, 46, 48	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∼ 𝑧 ↔ (𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
50	49	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝐶 ∼ 𝑧 ↔ (𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
51	50	biimpa 476	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐶) + 𝑧) ∈ 𝐾))
52	51	simp2d 1143	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → 𝑧 ∈ 𝑋)
53	21, 39	grpcl 18849	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝐵 + 𝑧) ∈ 𝑋)
54	43, 44, 52, 53	syl3anc 1373	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (𝐵 + 𝑧) ∈ 𝑋)
55	21, 47	grpinvcl 18895	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (𝐵 + 𝐶) ∈ 𝑋) → ((invg‘𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
56	34, 41, 55	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((invg‘𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
57	56	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → ((invg‘𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
58	21, 39	grpass 18850	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((((invg‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)))
59	43, 57, 44, 52, 58	syl13anc 1374	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → ((((invg‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)))
60	21, 39, 47	grpinvadd 18926	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((invg‘𝐺)‘(𝐵 + 𝐶)) = (((invg‘𝐺)‘𝐶) + ((invg‘𝐺)‘𝐵)))
61	34, 38, 4, 60	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((invg‘𝐺)‘(𝐵 + 𝐶)) = (((invg‘𝐺)‘𝐶) + ((invg‘𝐺)‘𝐵)))
62	21, 47	grpinvcl 18895	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝐶 ∈ 𝑋) → ((invg‘𝐺)‘𝐶) ∈ 𝑋)
63	34, 4, 62	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((invg‘𝐺)‘𝐶) ∈ 𝑋)
64		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ (-g‘𝐺) = (-g‘𝐺)
65	21, 39, 47, 64	grpsubval 18893	. . . . . . . . . . . . . . . 16 ⊢ ((((invg‘𝐺)‘𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((invg‘𝐺)‘𝐶)(-g‘𝐺)𝐵) = (((invg‘𝐺)‘𝐶) + ((invg‘𝐺)‘𝐵)))
66	63, 38, 65	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (((invg‘𝐺)‘𝐶)(-g‘𝐺)𝐵) = (((invg‘𝐺)‘𝐶) + ((invg‘𝐺)‘𝐵)))
67	61, 66	eqtr4d 2769	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((invg‘𝐺)‘(𝐵 + 𝐶)) = (((invg‘𝐺)‘𝐶)(-g‘𝐺)𝐵))
68	67	oveq1d 7356	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (((invg‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) = ((((invg‘𝐺)‘𝐶)(-g‘𝐺)𝐵) + 𝐵))
69	21, 39, 64	grpnpcan 18940	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((invg‘𝐺)‘𝐶)(-g‘𝐺)𝐵) + 𝐵) = ((invg‘𝐺)‘𝐶))
70	34, 63, 38, 69	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((((invg‘𝐺)‘𝐶)(-g‘𝐺)𝐵) + 𝐵) = ((invg‘𝐺)‘𝐶))
71	68, 70	eqtrd 2766	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (((invg‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) = ((invg‘𝐺)‘𝐶))
72	71	oveq1d 7356	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((((invg‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg‘𝐺)‘𝐶) + 𝑧))
73	72	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → ((((invg‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg‘𝐺)‘𝐶) + 𝑧))
74	59, 73	eqtr3d 2768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (((invg‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) = (((invg‘𝐺)‘𝐶) + 𝑧))
75	51	simp3d 1144	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (((invg‘𝐺)‘𝐶) + 𝑧) ∈ 𝐾)
76	74, 75	eqeltrd 2831	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (((invg‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)
77	21, 47, 39, 2	eqgval 19084	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝐾 ⊆ 𝑋) → ((𝐵 + 𝐶) ∼ (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
78	33, 46, 77	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 + 𝐶) ∼ (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
79	78	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((𝐵 + 𝐶) ∼ (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
80	79	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → ((𝐵 + 𝐶) ∼ (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
81	42, 54, 76, 80	mpbir3and 1343	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (𝐵 + 𝐶) ∼ (𝐵 + 𝑧))
82		ovex 7374	. . . . . . . 8 ⊢ (𝐵 + 𝑧) ∈ V
83		ovex 7374	. . . . . . . 8 ⊢ (𝐵 + 𝐶) ∈ V
84	82, 83	elec 8663	. . . . . . 7 ⊢ ((𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] ∼ ↔ (𝐵 + 𝐶) ∼ (𝐵 + 𝑧))
85	81, 84	sylibr 234	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] ∼ )
86	30, 85	syldan 591	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝑧 ∈ [𝐶] ∼ ) → (𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] ∼ )
87	86	fmpttd 7043	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ ⟶[(𝐵 + 𝐶)] ∼ )
88	87	frnd 6654	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ⊆ [(𝐵 + 𝐶)] ∼ )
89		eqid 2731	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧))
90	21, 39, 89	grplmulf1o 18921	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)):𝑋–1-1-onto→𝑋)
91	34, 38, 90	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)):𝑋–1-1-onto→𝑋)
92		f1of1 6757	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)):𝑋–1-1-onto→𝑋 → (𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)):𝑋–1-1→𝑋)
93	91, 92	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)):𝑋–1-1→𝑋)
94	23	ecss 8668	. . . . . . . . 9 ⊢ (𝜑 → [𝐶] ∼ ⊆ 𝑋)
95	94	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [𝐶] ∼ ⊆ 𝑋)
96		f1ssres 6721	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)):𝑋–1-1→𝑋 ∧ [𝐶] ∼ ⊆ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ∼ ):[𝐶] ∼ –1-1→𝑋)
97	93, 95, 96	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ∼ ):[𝐶] ∼ –1-1→𝑋)
98		resmpt 5981	. . . . . . . 8 ⊢ ([𝐶] ∼ ⊆ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ∼ ) = (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
99		f1eq1 6709	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ∼ ) = (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) → (((𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ∼ ):[𝐶] ∼ –1-1→𝑋 ↔ (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ –1-1→𝑋))
100	95, 98, 99	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (((𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ∼ ):[𝐶] ∼ –1-1→𝑋 ↔ (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ –1-1→𝑋))
101	97, 100	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ –1-1→𝑋)
102		f1f1orn 6769	. . . . . 6 ⊢ ((𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ –1-1→𝑋 → (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ –1-1-onto→ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
103	101, 102	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ –1-1-onto→ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
104	14	f1oen 8890	. . . . 5 ⊢ ((𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ –1-1-onto→ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) → [𝐶] ∼ ≈ ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
105		ensym 8920	. . . . 5 ⊢ ([𝐶] ∼ ≈ ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ≈ [𝐶] ∼ )
106	103, 104, 105	3syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ≈ [𝐶] ∼ )
107	20	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐾 ∈ (SubGrp‘𝐺))
108	21, 2	eqgen 19088	. . . . . . 7 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ [𝐶] ∼ ∈ (𝑋 / ∼ )) → 𝐾 ≈ [𝐶] ∼ )
109	107, 6, 108	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐾 ≈ [𝐶] ∼ )
110		ensym 8920	. . . . . 6 ⊢ (𝐾 ≈ [𝐶] ∼ → [𝐶] ∼ ≈ 𝐾)
111	109, 110	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [𝐶] ∼ ≈ 𝐾)
112		ecelqsw 8688	. . . . . . 7 ⊢ (( ∼ ∈ V ∧ (𝐵 + 𝐶) ∈ 𝑋) → [(𝐵 + 𝐶)] ∼ ∈ (𝑋 / ∼ ))
113	3, 41, 112	sylancr 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [(𝐵 + 𝐶)] ∼ ∈ (𝑋 / ∼ ))
114	21, 2	eqgen 19088	. . . . . 6 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ [(𝐵 + 𝐶)] ∼ ∈ (𝑋 / ∼ )) → 𝐾 ≈ [(𝐵 + 𝐶)] ∼ )
115	107, 113, 114	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐾 ≈ [(𝐵 + 𝐶)] ∼ )
116		entr 8923	. . . . 5 ⊢ (([𝐶] ∼ ≈ 𝐾 ∧ 𝐾 ≈ [(𝐵 + 𝐶)] ∼ ) → [𝐶] ∼ ≈ [(𝐵 + 𝐶)] ∼ )
117	111, 115, 116	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [𝐶] ∼ ≈ [(𝐵 + 𝐶)] ∼ )
118		entr 8923	. . . 4 ⊢ ((ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ≈ [𝐶] ∼ ∧ [𝐶] ∼ ≈ [(𝐵 + 𝐶)] ∼ ) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ≈ [(𝐵 + 𝐶)] ∼ )
119	106, 117, 118	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ≈ [(𝐵 + 𝐶)] ∼ )
120		fisseneq 9142	. . 3 ⊢ (([(𝐵 + 𝐶)] ∼ ∈ Fin ∧ ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ⊆ [(𝐵 + 𝐶)] ∼ ∧ ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ≈ [(𝐵 + 𝐶)] ∼ ) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) = [(𝐵 + 𝐶)] ∼ )
121	26, 88, 119, 120	syl3anc 1373	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) = [(𝐵 + 𝐶)] ∼ )
122	18, 121	eqtrd 2766	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝐵 · [𝐶] ∼ ) = [(𝐵 + 𝐶)] ∼ )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3897   class class class wbr 5086   ↦ cmpt 5167  ran crn 5612   ↾ cres 5613  –1-1→wf1 6473  –1-1-onto→wf1o 6475  ‘cfv 6476  (class class class)co 7341   ∈ cmpo 7343   Er wer 8614  [cec 8615   / cqs 8616   ≈ cen 8861  Fincfn 8864  Basecbs 17115  +_gcplusg 17156  Grpcgrp 18841  inv_gcminusg 18842  -_gcsg 18843  SubGrpcsubg 19028   ~_QG cqg 19030

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-ec 8619  df-qs 8623  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-2 12183  df-sets 17070  df-slot 17088  df-ndx 17100  df-base 17116  df-ress 17137  df-plusg 17169  df-0g 17340  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-grp 18844  df-minusg 18845  df-sbg 18846  df-subg 19031  df-eqg 19033

This theorem is referenced by:  sylow2blem2  19528  sylow2blem3  19529