Theorem sylow2blem1 17858

Description: Lemma for sylow2b 17861. 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 1055	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐵 ∈ 𝐻)
2		sylow2b.r	. . . . 5 ⊢ ∼ = (𝐺 ~QG 𝐾)
3		ovex 6577	. . . . 5 ⊢ (𝐺 ~QG 𝐾) ∈ V
4	2, 3	eqeltri 2684	. . . 4 ⊢ ∼ ∈ V
5		simp3 1056	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ 𝑋)
6		ecelqsg 7689	. . . 4 ⊢ (( ∼ ∈ V ∧ 𝐶 ∈ 𝑋) → [𝐶] ∼ ∈ (𝑋 / ∼ ))
7	4, 5, 6	sylancr 694	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [𝐶] ∼ ∈ (𝑋 / ∼ ))
8		simpr 476	. . . . . 6 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = [𝐶] ∼ ) → 𝑦 = [𝐶] ∼ )
9		simpl 472	. . . . . . 7 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = [𝐶] ∼ ) → 𝑥 = 𝐵)
10	9	oveq1d 6564	. . . . . 6 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = [𝐶] ∼ ) → (𝑥 + 𝑧) = (𝐵 + 𝑧))
11	8, 10	mpteq12dv 4663	. . . . 5 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = [𝐶] ∼ ) → (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
12	11	rneqd 5274	. . . 4 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = [𝐶] ∼ ) → ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) = ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
13		sylow2b.m	. . . 4 ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
14		ecexg 7633	. . . . . . 7 ⊢ ( ∼ ∈ V → [𝐶] ∼ ∈ V)
15	4, 14	ax-mp 5	. . . . . 6 ⊢ [𝐶] ∼ ∈ V
16	15	mptex 6390	. . . . 5 ⊢ (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ∈ V
17	16	rnex 6992	. . . 4 ⊢ ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ∈ V
18	12, 13, 17	ovmpt2a 6689	. . 3 ⊢ ((𝐵 ∈ 𝐻 ∧ [𝐶] ∼ ∈ (𝑋 / ∼ )) → (𝐵 · [𝐶] ∼ ) = ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
19	1, 7, 18	syl2anc 691	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝐵 · [𝐶] ∼ ) = ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
20		sylow2b.xf	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
21		sylow2b.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
22		sylow2b.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
23	22, 2	eqger 17467	. . . . . . 7 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
24	21, 23	syl 17	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝑋)
25	24	ecss 7675	. . . . 5 ⊢ (𝜑 → [(𝐵 + 𝐶)] ∼ ⊆ 𝑋)
26		ssfi 8065	. . . . 5 ⊢ ((𝑋 ∈ Fin ∧ [(𝐵 + 𝐶)] ∼ ⊆ 𝑋) → [(𝐵 + 𝐶)] ∼ ∈ Fin)
27	20, 25, 26	syl2anc 691	. . . 4 ⊢ (𝜑 → [(𝐵 + 𝐶)] ∼ ∈ Fin)
28	27	3ad2ant1 1075	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [(𝐵 + 𝐶)] ∼ ∈ Fin)
29		vex 3176	. . . . . . . 8 ⊢ 𝑧 ∈ V
30		elecg 7672	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ [𝐶] ∼ ↔ 𝐶 ∼ 𝑧))
31	29, 5, 30	sylancr 694	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ [𝐶] ∼ ↔ 𝐶 ∼ 𝑧))
32	31	biimpa 500	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝑧 ∈ [𝐶] ∼ ) → 𝐶 ∼ 𝑧)
33		sylow2b.h	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
34		subgrcl 17422	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
35	33, 34	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Grp)
36	35	3ad2ant1 1075	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐺 ∈ Grp)
37	22	subgss 17418	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋)
38	33, 37	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ⊆ 𝑋)
39	38	3ad2ant1 1075	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐻 ⊆ 𝑋)
40	39, 1	sseldd 3569	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐵 ∈ 𝑋)
41		sylow2b.a	. . . . . . . . . . 11 ⊢ + = (+g‘𝐺)
42	22, 41	grpcl 17253	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵 + 𝐶) ∈ 𝑋)
43	36, 40, 5, 42	syl3anc 1318	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝐵 + 𝐶) ∈ 𝑋)
44	43	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (𝐵 + 𝐶) ∈ 𝑋)
45	36	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → 𝐺 ∈ Grp)
46	40	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → 𝐵 ∈ 𝑋)
47	22	subgss 17418	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋)
48	21, 47	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ⊆ 𝑋)
49		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (invg‘𝐺) = (invg‘𝐺)
50	22, 49, 41, 2	eqgval 17466	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝐾 ⊆ 𝑋) → (𝐶 ∼ 𝑧 ↔ (𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
51	35, 48, 50	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∼ 𝑧 ↔ (𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
52	51	3ad2ant1 1075	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝐶 ∼ 𝑧 ↔ (𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐶) + 𝑧) ∈ 𝐾)))
53	52	biimpa 500	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐶) + 𝑧) ∈ 𝐾))
54	53	simp2d 1067	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → 𝑧 ∈ 𝑋)
55	22, 41	grpcl 17253	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝐵 + 𝑧) ∈ 𝑋)
56	45, 46, 54, 55	syl3anc 1318	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (𝐵 + 𝑧) ∈ 𝑋)
57	22, 49	grpinvcl 17290	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (𝐵 + 𝐶) ∈ 𝑋) → ((invg‘𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
58	36, 43, 57	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((invg‘𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
59	58	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → ((invg‘𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋)
60	22, 41	grpass 17254	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘(𝐵 + 𝐶)) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((((invg‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)))
61	45, 59, 46, 54, 60	syl13anc 1320	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → ((((invg‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)))
62	22, 41, 49	grpinvadd 17316	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((invg‘𝐺)‘(𝐵 + 𝐶)) = (((invg‘𝐺)‘𝐶) + ((invg‘𝐺)‘𝐵)))
63	36, 40, 5, 62	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((invg‘𝐺)‘(𝐵 + 𝐶)) = (((invg‘𝐺)‘𝐶) + ((invg‘𝐺)‘𝐵)))
64	22, 49	grpinvcl 17290	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝐶 ∈ 𝑋) → ((invg‘𝐺)‘𝐶) ∈ 𝑋)
65	36, 5, 64	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((invg‘𝐺)‘𝐶) ∈ 𝑋)
66		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (-g‘𝐺) = (-g‘𝐺)
67	22, 41, 49, 66	grpsubval 17288	. . . . . . . . . . . . . . . 16 ⊢ ((((invg‘𝐺)‘𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((invg‘𝐺)‘𝐶)(-g‘𝐺)𝐵) = (((invg‘𝐺)‘𝐶) + ((invg‘𝐺)‘𝐵)))
68	65, 40, 67	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (((invg‘𝐺)‘𝐶)(-g‘𝐺)𝐵) = (((invg‘𝐺)‘𝐶) + ((invg‘𝐺)‘𝐵)))
69	63, 68	eqtr4d 2647	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((invg‘𝐺)‘(𝐵 + 𝐶)) = (((invg‘𝐺)‘𝐶)(-g‘𝐺)𝐵))
70	69	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (((invg‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) = ((((invg‘𝐺)‘𝐶)(-g‘𝐺)𝐵) + 𝐵))
71	22, 41, 66	grpnpcan 17330	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((invg‘𝐺)‘𝐶)(-g‘𝐺)𝐵) + 𝐵) = ((invg‘𝐺)‘𝐶))
72	36, 65, 40, 71	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((((invg‘𝐺)‘𝐶)(-g‘𝐺)𝐵) + 𝐵) = ((invg‘𝐺)‘𝐶))
73	70, 72	eqtrd 2644	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (((invg‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) = ((invg‘𝐺)‘𝐶))
74	73	oveq1d 6564	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((((invg‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg‘𝐺)‘𝐶) + 𝑧))
75	74	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → ((((invg‘𝐺)‘(𝐵 + 𝐶)) + 𝐵) + 𝑧) = (((invg‘𝐺)‘𝐶) + 𝑧))
76	61, 75	eqtr3d 2646	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (((invg‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) = (((invg‘𝐺)‘𝐶) + 𝑧))
77	53	simp3d 1068	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (((invg‘𝐺)‘𝐶) + 𝑧) ∈ 𝐾)
78	76, 77	eqeltrd 2688	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (((invg‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)
79	22, 49, 41, 2	eqgval 17466	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝐾 ⊆ 𝑋) → ((𝐵 + 𝐶) ∼ (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
80	35, 48, 79	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 + 𝐶) ∼ (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
81	80	3ad2ant1 1075	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((𝐵 + 𝐶) ∼ (𝐵 + 𝑧) ↔ ((𝐵 + 𝐶) ∈ 𝑋 ∧ (𝐵 + 𝑧) ∈ 𝑋 ∧ (((invg‘𝐺)‘(𝐵 + 𝐶)) + (𝐵 + 𝑧)) ∈ 𝐾)))
82	81	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)))
83	44, 56, 78, 82	mpbir3and 1238	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (𝐵 + 𝐶) ∼ (𝐵 + 𝑧))
84		ovex 6577	. . . . . . . 8 ⊢ (𝐵 + 𝑧) ∈ V
85		ovex 6577	. . . . . . . 8 ⊢ (𝐵 + 𝐶) ∈ V
86	84, 85	elec 7673	. . . . . . 7 ⊢ ((𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] ∼ ↔ (𝐵 + 𝐶) ∼ (𝐵 + 𝑧))
87	83, 86	sylibr 223	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝐶 ∼ 𝑧) → (𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] ∼ )
88	32, 87	syldan 486	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) ∧ 𝑧 ∈ [𝐶] ∼ ) → (𝐵 + 𝑧) ∈ [(𝐵 + 𝐶)] ∼ )
89		eqid 2610	. . . . 5 ⊢ (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) = (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧))
90	88, 89	fmptd 6292	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ ⟶[(𝐵 + 𝐶)] ∼ )
91		frn 5966	. . . 4 ⊢ ((𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ ⟶[(𝐵 + 𝐶)] ∼ → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ⊆ [(𝐵 + 𝐶)] ∼ )
92	90, 91	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ⊆ [(𝐵 + 𝐶)] ∼ )
93		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧))
94	22, 41, 93	grplmulf1o 17312	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)):𝑋–1-1-onto→𝑋)
95	36, 40, 94	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)):𝑋–1-1-onto→𝑋)
96		f1of1 6049	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)):𝑋–1-1-onto→𝑋 → (𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)):𝑋–1-1→𝑋)
97	95, 96	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)):𝑋–1-1→𝑋)
98	24	ecss 7675	. . . . . . . . 9 ⊢ (𝜑 → [𝐶] ∼ ⊆ 𝑋)
99	98	3ad2ant1 1075	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [𝐶] ∼ ⊆ 𝑋)
100		f1ssres 6021	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)):𝑋–1-1→𝑋 ∧ [𝐶] ∼ ⊆ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ∼ ):[𝐶] ∼ –1-1→𝑋)
101	97, 99, 100	syl2anc 691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ∼ ):[𝐶] ∼ –1-1→𝑋)
102		resmpt 5369	. . . . . . . 8 ⊢ ([𝐶] ∼ ⊆ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (𝐵 + 𝑧)) ↾ [𝐶] ∼ ) = (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
103		f1eq1 6009	. . . . . . . 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))
104	99, 102, 103	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))
105	101, 104	mpbid 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ –1-1→𝑋)
106		f1f1orn 6061	. . . . . 6 ⊢ ((𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ –1-1→𝑋 → (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ –1-1-onto→ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
107	105, 106	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ –1-1-onto→ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
108	15	f1oen 7862	. . . . 5 ⊢ ((𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)):[𝐶] ∼ –1-1-onto→ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) → [𝐶] ∼ ≈ ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)))
109		ensym 7891	. . . . 5 ⊢ ([𝐶] ∼ ≈ ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ≈ [𝐶] ∼ )
110	107, 108, 109	3syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ≈ [𝐶] ∼ )
111	21	3ad2ant1 1075	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐾 ∈ (SubGrp‘𝐺))
112	22, 2	eqgen 17470	. . . . . . 7 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ [𝐶] ∼ ∈ (𝑋 / ∼ )) → 𝐾 ≈ [𝐶] ∼ )
113	111, 7, 112	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐾 ≈ [𝐶] ∼ )
114		ensym 7891	. . . . . 6 ⊢ (𝐾 ≈ [𝐶] ∼ → [𝐶] ∼ ≈ 𝐾)
115	113, 114	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [𝐶] ∼ ≈ 𝐾)
116		ecelqsg 7689	. . . . . . 7 ⊢ (( ∼ ∈ V ∧ (𝐵 + 𝐶) ∈ 𝑋) → [(𝐵 + 𝐶)] ∼ ∈ (𝑋 / ∼ ))
117	4, 43, 116	sylancr 694	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [(𝐵 + 𝐶)] ∼ ∈ (𝑋 / ∼ ))
118	22, 2	eqgen 17470	. . . . . 6 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ [(𝐵 + 𝐶)] ∼ ∈ (𝑋 / ∼ )) → 𝐾 ≈ [(𝐵 + 𝐶)] ∼ )
119	111, 117, 118	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → 𝐾 ≈ [(𝐵 + 𝐶)] ∼ )
120		entr 7894	. . . . 5 ⊢ (([𝐶] ∼ ≈ 𝐾 ∧ 𝐾 ≈ [(𝐵 + 𝐶)] ∼ ) → [𝐶] ∼ ≈ [(𝐵 + 𝐶)] ∼ )
121	115, 119, 120	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → [𝐶] ∼ ≈ [(𝐵 + 𝐶)] ∼ )
122		entr 7894	. . . 4 ⊢ ((ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ≈ [𝐶] ∼ ∧ [𝐶] ∼ ≈ [(𝐵 + 𝐶)] ∼ ) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ≈ [(𝐵 + 𝐶)] ∼ )
123	110, 121, 122	syl2anc 691	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) ≈ [(𝐵 + 𝐶)] ∼ )
124		fisseneq 8056	. . 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)] ∼ )
125	28, 92, 123, 124	syl3anc 1318	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → ran (𝑧 ∈ [𝐶] ∼ ↦ (𝐵 + 𝑧)) = [(𝐵 + 𝐶)] ∼ )
126	19, 125	eqtrd 2644	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝐵 · [𝐶] ∼ ) = [(𝐵 + 𝐶)] ∼ )

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039   ↾ cres 5040  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   Er wer 7626  [cec 7627   / cqs 7628   ≈ cen 7838  Fincfn 7841  Basecbs 15695  +_gcplusg 15768  Grpcgrp 17245  inv_gcminusg 17246  -_gcsg 17247  SubGrpcsubg 17411   ~_QG cqg 17413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-ec 7631  df-qs 7635  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-eqg 17416

This theorem is referenced by:  sylow2blem2  17859  sylow2blem3  17860