Theorem dprd2dlem1 20114

Description: The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)

Hypotheses

Ref	Expression
dprd2d.1	⊢ (𝜑 → Rel 𝐴)
dprd2d.2	⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
dprd2d.3	⊢ (𝜑 → dom 𝐴 ⊆ 𝐼)
dprd2d.4	⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
dprd2d.5	⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
dprd2d.k	⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
dprd2d.6	⊢ (𝜑 → 𝐶 ⊆ 𝐼)

Assertion

Ref	Expression
dprd2dlem1	⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) = (𝐺 DProd (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))

Distinct variable groups:   𝑖,𝑗,𝐴   𝐶,𝑖   𝑖,𝐺,𝑗   𝑖,𝐼   𝑖,𝐾   𝜑,𝑖,𝑗   𝑆,𝑖,𝑗

Allowed substitution hints:   𝐶(𝑗)   𝐼(𝑗)   𝐾(𝑗)

Proof of Theorem dprd2dlem1

Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dprd2d.5	. . . . . 6 ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
2		dprdgrp 20078	. . . . . 6 ⊢ (𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → 𝐺 ∈ Grp)
3	1, 2	syl 18	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		eqid 2763	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgacs 19228	. . . . 5 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
6		acsmre 17709	. . . . 5 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
7	3, 5, 6	3syl 19	. . . 4 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
8		dprd2d.k	. . . 4 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
9		dprd2d.2	. . . . . 6 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
10		ffun 6710	. . . . . 6 ⊢ (𝑆:𝐴⟶(SubGrp‘𝐺) → Fun 𝑆)
11		funiunfv 7248	. . . . . 6 ⊢ (Fun 𝑆 → ∪ 𝑥 ∈ (𝐴 ↾ 𝐶)(𝑆‘𝑥) = ∪ (𝑆 “ (𝐴 ↾ 𝐶)))
12	9, 10, 11	3syl 19	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ (𝐴 ↾ 𝐶)(𝑆‘𝑥) = ∪ (𝑆 “ (𝐴 ↾ 𝐶)))
13		resss 6002	. . . . . . . . . 10 ⊢ (𝐴 ↾ 𝐶) ⊆ 𝐴
14	13	sseli 3934	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ↾ 𝐶) → 𝑥 ∈ 𝐴)
15		dprd2d.1	. . . . . . . . . 10 ⊢ (𝜑 → Rel 𝐴)
16		dprd2d.3	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐴 ⊆ 𝐼)
17		dprd2d.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
18	15, 9, 16, 17, 1, 8	dprd2dlem2 20113	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))))
19	14, 18	sylan2 604	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))))
20		1st2nd 8037	. . . . . . . . . . . . 13 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
21	15, 14, 20	syl2an 607	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
22		simpr 489	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → 𝑥 ∈ (𝐴 ↾ 𝐶))
23	21, 22	eqeltrrd 2864	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ (𝐴 ↾ 𝐶))
24		fvex 6896	. . . . . . . . . . . . 13 ⊢ (2nd ‘𝑥) ∈ V
25	24	opelresi 5988	. . . . . . . . . . . 12 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ (𝐴 ↾ 𝐶) ↔ ((1st ‘𝑥) ∈ 𝐶 ∧ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐴))
26	25	simplbi 501	. . . . . . . . . . 11 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ (𝐴 ↾ 𝐶) → (1st ‘𝑥) ∈ 𝐶)
27	23, 26	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → (1st ‘𝑥) ∈ 𝐶)
28		ovex 7445	. . . . . . . . . 10 ⊢ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))) ∈ V
29		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
30		sneq 4600	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (1st ‘𝑥) → {𝑖} = {(1st ‘𝑥)})
31	30	imaeq2d 6064	. . . . . . . . . . . . 13 ⊢ (𝑖 = (1st ‘𝑥) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st ‘𝑥)}))
32		oveq1 7419	. . . . . . . . . . . . 13 ⊢ (𝑖 = (1st ‘𝑥) → (𝑖𝑆𝑗) = ((1st ‘𝑥)𝑆𝑗))
33	31, 32	mpteq12dv 5199	. . . . . . . . . . . 12 ⊢ (𝑖 = (1st ‘𝑥) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗)))
34	33	oveq2d 7428	. . . . . . . . . . 11 ⊢ (𝑖 = (1st ‘𝑥) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))))
35	29, 34	elrnmpt1s 5951	. . . . . . . . . 10 ⊢ (((1st ‘𝑥) ∈ 𝐶 ∧ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))) ∈ V) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))) ∈ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
36	27, 28, 35	sylancl 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))) ∈ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
37		elssuni 4905	. . . . . . . . 9 ⊢ ((𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))) ∈ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
38	36, 37	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
39	19, 38	sstrd 3948	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → (𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
40	39	ralrimiva 3157	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ↾ 𝐶)(𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
41		iunss 5010	. . . . . 6 ⊢ (∪ 𝑥 ∈ (𝐴 ↾ 𝐶)(𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↔ ∀𝑥 ∈ (𝐴 ↾ 𝐶)(𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
42	40, 41	sylibr 237	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ (𝐴 ↾ 𝐶)(𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
43	12, 42	eqsstrrd 3973	. . . 4 ⊢ (𝜑 → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
44		dprd2d.6	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ⊆ 𝐼)
45	44	sselda 3938	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → 𝑖 ∈ 𝐼)
46	45, 17	syldan 602	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
47		ovex 7445	. . . . . . . . . . . 12 ⊢ (𝑖𝑆𝑗) ∈ V
48		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))
49	47, 48	dmmpti 6681	. . . . . . . . . . 11 ⊢ dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝐴 “ {𝑖})
50	49	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝐴 “ {𝑖}))
51		imassrn 6075	. . . . . . . . . . . . . 14 ⊢ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ ran 𝑆
52	9	frnd 6716	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
53		mresspw 17645	. . . . . . . . . . . . . . . 16 ⊢ ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
54	7, 53	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
55	52, 54	sstrd 3948	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
56	51, 55	sstrid 3949	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ 𝒫 (Base‘𝐺))
57		sspwuni 5067	. . . . . . . . . . . . 13 ⊢ ((𝑆 “ (𝐴 ↾ 𝐶)) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (Base‘𝐺))
58	56, 57	sylib 221	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (Base‘𝐺))
59	8	mrccl 17668	. . . . . . . . . . . 12 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (Base‘𝐺)) → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺))
60	7, 58, 59	syl2anc 595	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺))
61	60	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺))
62		oveq2 7420	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (𝑖𝑆𝑗) = (𝑖𝑆𝑘))
63	62, 48, 47	fvmpt3i 6997	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝐴 “ {𝑖}) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
64	63	adantl 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
65		df-ov 7415	. . . . . . . . . . . . . 14 ⊢ (𝑖𝑆𝑘) = (𝑆‘⟨𝑖, 𝑘⟩)
66	9	ffnd 6708	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 Fn 𝐴)
67	66	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑆 Fn 𝐴)
68	13	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝐴 ↾ 𝐶) ⊆ 𝐴)
69		simplr 780	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖 ∈ 𝐶)
70		elrelimasn 6090	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Rel 𝐴 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
71	15, 70	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
72	71	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
73	72	biimpa 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖𝐴𝑘)
74		df-br 5111	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖𝐴𝑘 ↔ ⟨𝑖, 𝑘⟩ ∈ 𝐴)
75	73, 74	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ 𝐴)
76		vex 3459	. . . . . . . . . . . . . . . . 17 ⊢ 𝑘 ∈ V
77	76	opelresi 5988	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑖, 𝑘⟩ ∈ (𝐴 ↾ 𝐶) ↔ (𝑖 ∈ 𝐶 ∧ ⟨𝑖, 𝑘⟩ ∈ 𝐴))
78	69, 75, 77	sylanbrc 594	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ (𝐴 ↾ 𝐶))
79		fnfvima 7233	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 Fn 𝐴 ∧ (𝐴 ↾ 𝐶) ⊆ 𝐴 ∧ ⟨𝑖, 𝑘⟩ ∈ (𝐴 ↾ 𝐶)) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴 ↾ 𝐶)))
80	67, 68, 78, 79	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴 ↾ 𝐶)))
81	65, 80	eqeltrid 2867	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴 ↾ 𝐶)))
82		elssuni 4905	. . . . . . . . . . . . 13 ⊢ ((𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴 ↾ 𝐶)) → (𝑖𝑆𝑘) ⊆ ∪ (𝑆 “ (𝐴 ↾ 𝐶)))
83	81, 82	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ ∪ (𝑆 “ (𝐴 ↾ 𝐶)))
84	7, 8, 58	mrcssidd 17682	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
85	84	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
86	83, 85	sstrd 3948	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
87	64, 86	eqsstrd 3972	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
88	46, 50, 61, 87	dprdlub 20099	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
89		ovex 7445	. . . . . . . . . 10 ⊢ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ V
90	89	elpw 4567	. . . . . . . . 9 ⊢ ((𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ 𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ↔ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
91	88, 90	sylibr 237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ 𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
92	91	fmpttd 7112	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))):𝐶⟶𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
93	92	frnd 6716	. . . . . 6 ⊢ (𝜑 → ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ 𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
94		sspwuni 5067	. . . . . 6 ⊢ (ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ 𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ↔ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
95	93, 94	sylib 221	. . . . 5 ⊢ (𝜑 → ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
96	7, 8	mrcssvd 17680	. . . . 5 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ⊆ (Base‘𝐺))
97	95, 96	sstrd 3948	. . . 4 ⊢ (𝜑 → ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (Base‘𝐺))
98	7, 8, 43, 97	mrcssd 17681	. . 3 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ⊆ (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
99	8	mrcsscl 17677	. . . 4 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∧ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺)) → (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
100	7, 95, 60, 99	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
101	98, 100	eqssd 3955	. 2 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) = (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
102		eqid 2763	. . . . . . . 8 ⊢ (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
103	89, 102	dmmpti 6681	. . . . . . 7 ⊢ dom (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = 𝐼
104	103	a1i 11	. . . . . 6 ⊢ (𝜑 → dom (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = 𝐼)
105	1, 104, 44	dprdres 20101	. . . . 5 ⊢ (𝜑 → (𝐺dom DProd ((𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶) ∧ (𝐺 DProd ((𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶)) ⊆ (𝐺 DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))))
106	105	simpld 499	. . . 4 ⊢ (𝜑 → 𝐺dom DProd ((𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶))
107	44	resmptd 6044	. . . 4 ⊢ (𝜑 → ((𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶) = (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
108	106, 107	breqtrd 5138	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
109	8	dprdspan 20100	. . 3 ⊢ (𝐺dom DProd (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → (𝐺 DProd (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) = (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
110	108, 109	syl 18	. 2 ⊢ (𝜑 → (𝐺 DProd (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) = (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
111	101, 110	eqtr4d 2801	1 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) = (𝐺 DProd (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ⊆ wss 3906  𝒫 cpw 4563  {csn 4590  ⟨cop 4596  ∪ cuni 4873  ∪ ciun 4957   class class class wbr 5110   ↦ cmpt 5193  dom cdm 5663  ran crn 5664   ↾ cres 5665   “ cima 5666  Rel wrel 5668  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412  1^st c1st 7985  2^nd c2nd 7986  Basecbs 17270  Moorecmre 17635  mrClscmrc 17636  ACScacs 17638  Grpcgrp 19001  SubGrpcsubg 19187   DProd cdprd 20066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8158  df-tpos 8223  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-er 8695  df-map 8827  df-ixp 8897  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-fsupp 9323  df-oi 9473  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-n0 12506  df-z 12593  df-uz 12864  df-fz 13537  df-fzo 13685  df-seq 14040  df-hash 14369  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-0g 17495  df-gsum 17496  df-mre 17639  df-mrc 17640  df-acs 17642  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-mhm 18842  df-submnd 18843  df-grp 19004  df-minusg 19005  df-sbg 19006  df-mulg 19135  df-subg 19190  df-ghm 19285  df-gim 19330  df-cntz 19388  df-oppg 19417  df-cmn 19853  df-dprd 20068

This theorem is referenced by:  dprd2da  20115  dprd2db  20116