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Theorem sylow2blem1 19530

Description: Lemma for sylow2b 19533. 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 1136	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐵 ∈ 𝐻)
2		sylow2b.r	. . . . 5 ⊢ ∼ = (𝐺 ~_QG 𝐾)
3	2	ovexi 7446	. . . 4 ⊢ ∼ ∈ V
4		simp3 1137	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ 𝑋)
5		ecelqsg 8769	. . . 4 ⊢ (( ∼ ∈ V ∧ 𝐶 ∈ 𝑋) → [𝐶] ∼ ∈ (𝑋 / ∼ ))
6	3, 4, 5	sylancr 586	. . 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 7427	. . . . . 6 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = [𝐶] ∼ ) → (𝑥 + 𝑧) = (𝐵 + 𝑧))
10	7, 9	mpteq12dv 5240	. . . . 5 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = [𝐶] ∼ ) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
11	10	rneqd 5938	. . . 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 8710	. . . . . . 7 ⊢ ( ∼ ∈ V → [𝐶] ∼ ∈ V)
14	3, 13	ax-mp 5	. . . . . 6 ⊢ [𝐶] ∼ ∈ V
15	14	mptex 7228	. . . . 5 ⊢ (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ∈ V
16	15	rnex 7906	. . . 4 ⊢ ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ∈ V
17	11, 12, 16	ovmpoa 7566	. . 3 ⊢ ((𝐵 ∈ 𝐻 ∧ [𝐶] ∼ ∈ (𝑋 / ∼ )) → (𝐵 · [𝐶] ∼ ) = ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
18	1, 6, 17	syl2anc 583	. 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 19095	. . . . . . 7 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
23	20, 22	syl 17	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝑋)
24	23	ecss 8752	. . . . 5 ⊢ (𝜑 → [(𝐵 + 𝐶)] ∼ ⊆ 𝑋)
25	19, 24	ssfid 9270	. . . 4 ⊢ (𝜑 → [(𝐵 + 𝐶)] ∼ ∈ Fin)
26	25	3ad2ant1 1132	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [(𝐵 + 𝐶)] ∼ ∈ Fin)
27		vex 3477	. . . . . . . 8 ⊢ 𝑧 ∈ V
28		elecg 8749	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ [𝐶] ∼ ↔ 𝐶 ∼ 𝑧))
29	27, 4, 28	sylancr 586	. . . . . . 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 19048	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
33	31, 32	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Grp)
34	33	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐺 ∈ Grp)
35	21	subgss 19044	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋)
36	31, 35	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ⊆ 𝑋)
37	36	3ad2ant1 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐻 ⊆ 𝑋)
38	37, 1	sseldd 3984	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐵 ∈ 𝑋)
39		sylow2b.a	. . . . . . . . . . 11 ⊢ + = (+_g‘𝐺)
40	21, 39	grpcl 18864	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵 + 𝐶) ∈ 𝑋)
41	34, 38, 4, 40	syl3anc 1370	. . . . . . . . 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 19044	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋)
46	20, 45	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ⊆ 𝑋)
47		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
48	21, 47, 39, 2	eqgval 19094	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝐾 ⊆ 𝑋) → (𝐶 ∼ 𝑧 ↔ (𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((inv_g‘𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
49	33, 46, 48	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∼ 𝑧 ↔ (𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((inv_g‘𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
50	49	3ad2ant1 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝐶 ∼ 𝑧 ↔ (𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((inv_g‘𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
51	50	biimpa 476	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((inv_g‘𝐺)‘𝐶) + 𝑧) ∈ 𝐾))
52	51	simp2d 1142	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → 𝑧 ∈ 𝑋)
53	21, 39	grpcl 18864	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝐵 + 𝑧) ∈ 𝑋)
54	43, 44, 52, 53	syl3anc 1370	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (𝐵 + 𝑧) ∈ 𝑋)
55	21, 47	grpinvcl 18909	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (𝐵 + 𝐶) ∈ 𝑋) → ((inv_g‘𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
56	34, 41, 55	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((inv_g‘𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
57	56	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → ((inv_g‘𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
58	21, 39	grpass 18865	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (((inv_g‘𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((((inv_g‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((inv_g‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)))
59	43, 57, 44, 52, 58	syl13anc 1371	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → ((((inv_g‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((inv_g‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)))
60	21, 39, 47	grpinvadd 18938	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((inv_g‘𝐺)‘(𝐵 + 𝐶)) = (((inv_g‘𝐺)‘𝐶) + ((inv_g‘𝐺)‘𝐵)))
61	34, 38, 4, 60	syl3anc 1370	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((inv_g‘𝐺)‘(𝐵 + 𝐶)) = (((inv_g‘𝐺)‘𝐶) + ((inv_g‘𝐺)‘𝐵)))
62	21, 47	grpinvcl 18909	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝐶 ∈ 𝑋) → ((inv_g‘𝐺)‘𝐶) ∈ 𝑋)
63	34, 4, 62	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((inv_g‘𝐺)‘𝐶) ∈ 𝑋)
64		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ (-_g‘𝐺) = (-_g‘𝐺)
65	21, 39, 47, 64	grpsubval 18907	. . . . . . . . . . . . . . . 16 ⊢ ((((inv_g‘𝐺)‘𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((inv_g‘𝐺)‘𝐶)(-_g‘𝐺)𝐵) = (((inv_g‘𝐺)‘𝐶) + ((inv_g‘𝐺)‘𝐵)))
66	63, 38, 65	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (((inv_g‘𝐺)‘𝐶)(-_g‘𝐺)𝐵) = (((inv_g‘𝐺)‘𝐶) + ((inv_g‘𝐺)‘𝐵)))
67	61, 66	eqtr4d 2774	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((inv_g‘𝐺)‘(𝐵 + 𝐶)) = (((inv_g‘𝐺)‘𝐶)(-_g‘𝐺)𝐵))
68	67	oveq1d 7427	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (((inv_g‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) = ((((inv_g‘𝐺)‘𝐶)(-_g‘𝐺)𝐵) + 𝐵))
69	21, 39, 64	grpnpcan 18952	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ ((inv_g‘𝐺)‘𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((inv_g‘𝐺)‘𝐶)(-_g‘𝐺)𝐵) + 𝐵) = ((inv_g‘𝐺)‘𝐶))
70	34, 63, 38, 69	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((((inv_g‘𝐺)‘𝐶)(-_g‘𝐺)𝐵) + 𝐵) = ((inv_g‘𝐺)‘𝐶))
71	68, 70	eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (((inv_g‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) = ((inv_g‘𝐺)‘𝐶))
72	71	oveq1d 7427	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((((inv_g‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((inv_g‘𝐺)‘𝐶) + 𝑧))
73	72	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → ((((inv_g‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((inv_g‘𝐺)‘𝐶) + 𝑧))
74	59, 73	eqtr3d 2773	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (((inv_g‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) = (((inv_g‘𝐺)‘𝐶) + 𝑧))
75	51	simp3d 1143	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (((inv_g‘𝐺)‘𝐶) + 𝑧) ∈ 𝐾)
76	74, 75	eqeltrd 2832	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (((inv_g‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)
77	21, 47, 39, 2	eqgval 19094	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝐾 ⊆ 𝑋) → ((𝐵 + 𝐶) ∼ (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((inv_g‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
78	33, 46, 77	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 + 𝐶) ∼ (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((inv_g‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
79	78	3ad2ant1 1132	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((𝐵 + 𝐶) ∼ (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((inv_g‘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 ∧ (((inv_g‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
81	42, 54, 76, 80	mpbir3and 1341	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (𝐵 + 𝐶) ∼ (𝐵 + 𝑧))
82		ovex 7445	. . . . . . . 8 ⊢ (𝐵 + 𝑧) ∈ V
83		ovex 7445	. . . . . . . 8 ⊢ (𝐵 + 𝐶) ∈ V
84	82, 83	elec 8750	. . . . . . 7 ⊢ ((𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] ∼ ↔ (𝐵 + 𝐶) ∼ (𝐵 + 𝑧))
85	81, 84	sylibr 233	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] ∼ )
86	30, 85	syldan 590	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝑧 ∈ [𝐶] ∼ ) → (𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] ∼ )
87	86	fmpttd 7117	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ ⟶[(𝐵 + 𝐶)] ∼ )
88	87	frnd 6726	. . 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 18934	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)):𝑋–1-1-onto→𝑋)
91	34, 38, 90	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)):𝑋–1-1-onto→𝑋)
92		f1of1 6833	. . . . . . . . 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 8752	. . . . . . . . 9 ⊢ (𝜑 → [𝐶] ∼ ⊆ 𝑋)
95	94	3ad2ant1 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [𝐶] ∼ ⊆ 𝑋)
96		f1ssres 6796	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ∼ ):[𝐶] ∼ –1-1→𝑋)
98		resmpt 6038	. . . . . . . 8 ⊢ ([𝐶] ∼ ⊆ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ∼ ) = (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
99		f1eq1 6783	. . . . . . . 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 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ –1-1→𝑋)
102		f1f1orn 6845	. . . . . 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 8972	. . . . 5 ⊢ ((𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ –1-1-onto→ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) → [𝐶] ∼ ≈ ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
105		ensym 9002	. . . . 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 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐾 ∈ (SubGrp‘𝐺))
108	21, 2	eqgen 19098	. . . . . . 7 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ [𝐶] ∼ ∈ (𝑋 / ∼ )) → 𝐾 ≈ [𝐶] ∼ )
109	107, 6, 108	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐾 ≈ [𝐶] ∼ )
110		ensym 9002	. . . . . 6 ⊢ (𝐾 ≈ [𝐶] ∼ → [𝐶] ∼ ≈ 𝐾)
111	109, 110	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [𝐶] ∼ ≈ 𝐾)
112		ecelqsg 8769	. . . . . . 7 ⊢ (( ∼ ∈ V ∧ (𝐵 + 𝐶) ∈ 𝑋) → [(𝐵 + 𝐶)] ∼ ∈ (𝑋 / ∼ ))
113	3, 41, 112	sylancr 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [(𝐵 + 𝐶)] ∼ ∈ (𝑋 / ∼ ))
114	21, 2	eqgen 19098	. . . . . 6 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ [(𝐵 + 𝐶)] ∼ ∈ (𝑋 / ∼ )) → 𝐾 ≈ [(𝐵 + 𝐶)] ∼ )
115	107, 113, 114	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐾 ≈ [(𝐵 + 𝐶)] ∼ )
116		entr 9005	. . . . 5 ⊢ (([𝐶] ∼ ≈ 𝐾 ∧ 𝐾 ≈ [(𝐵 + 𝐶)] ∼ ) → [𝐶] ∼ ≈ [(𝐵 + 𝐶)] ∼ )
117	111, 115, 116	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [𝐶] ∼ ≈ [(𝐵 + 𝐶)] ∼ )
118		entr 9005	. . . 4 ⊢ ((ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ≈ [𝐶] ∼ ∧ [𝐶] ∼ ≈ [(𝐵 + 𝐶)] ∼ ) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ≈ [(𝐵 + 𝐶)] ∼ )
119	106, 117, 118	syl2anc 583	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ≈ [(𝐵 + 𝐶)] ∼ )
120		fisseneq 9260	. . 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 1370	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) = [(𝐵 + 𝐶)] ∼ )
122	18, 121	eqtrd 2771	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝐵 · [𝐶] ∼ ) = [(𝐵 + 𝐶)] ∼ )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ⊆ wss 3949 class class class wbr 5149 ↦ cmpt 5232 ran crn 5678 ↾ cres 5679 –1-1→wf1 6541 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7412 ∈ cmpo 7414 Er wer 8703 [cec 8704 / cqs 8705 ≈ cen 8939 Fincfn 8942 Basecbs 17149 +_gcplusg 17202 Grpcgrp 18856 inv_gcminusg 18857 -_gcsg 18858 SubGrpcsubg 19037 ~_QG cqg 19039

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-ec 8708 df-qs 8712 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-eqg 19042

This theorem is referenced by: sylow2blem2 19531 sylow2blem3 19532

W3C validator