Theorem dprd2dlem1 20015

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 19979	. . . . . 6 ⊢ (𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → 𝐺 ∈ Grp)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		eqid 2725	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgacs 19129	. . . . 5 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
6		acsmre 17640	. . . . 5 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
7	3, 5, 6	3syl 18	. . . 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 6726	. . . . . 6 ⊢ (𝑆:𝐴⟶(SubGrp‘𝐺) → Fun 𝑆)
11		funiunfv 7258	. . . . . 6 ⊢ (Fun 𝑆 → ∪ 𝑥 ∈ (𝐴 ↾ 𝐶)(𝑆‘𝑥) = ∪ (𝑆 “ (𝐴 ↾ 𝐶)))
12	9, 10, 11	3syl 18	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ (𝐴 ↾ 𝐶)(𝑆‘𝑥) = ∪ (𝑆 “ (𝐴 ↾ 𝐶)))
13		resss 6007	. . . . . . . . . 10 ⊢ (𝐴 ↾ 𝐶) ⊆ 𝐴
14	13	sseli 3972	. . . . . . . . 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 20014	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))))
19	14, 18	sylan2 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))))
20		1st2nd 8044	. . . . . . . . . . . . 13 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
21	15, 14, 20	syl2an 594	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
22		simpr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → 𝑥 ∈ (𝐴 ↾ 𝐶))
23	21, 22	eqeltrrd 2826	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ (𝐴 ↾ 𝐶))
24		fvex 6909	. . . . . . . . . . . . 13 ⊢ (2nd ‘𝑥) ∈ V
25	24	opelresi 5993	. . . . . . . . . . . 12 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ (𝐴 ↾ 𝐶) ↔ ((1st ‘𝑥) ∈ 𝐶 ∧ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐴))
26	25	simplbi 496	. . . . . . . . . . 11 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ (𝐴 ↾ 𝐶) → (1st ‘𝑥) ∈ 𝐶)
27	23, 26	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → (1st ‘𝑥) ∈ 𝐶)
28		ovex 7452	. . . . . . . . . 10 ⊢ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))) ∈ V
29		eqid 2725	. . . . . . . . . . 11 ⊢ (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
30		sneq 4640	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (1st ‘𝑥) → {𝑖} = {(1st ‘𝑥)})
31	30	imaeq2d 6064	. . . . . . . . . . . . 13 ⊢ (𝑖 = (1st ‘𝑥) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st ‘𝑥)}))
32		oveq1 7426	. . . . . . . . . . . . 13 ⊢ (𝑖 = (1st ‘𝑥) → (𝑖𝑆𝑗) = ((1st ‘𝑥)𝑆𝑗))
33	31, 32	mpteq12dv 5240	. . . . . . . . . . . 12 ⊢ (𝑖 = (1st ‘𝑥) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗)))
34	33	oveq2d 7435	. . . . . . . . . . 11 ⊢ (𝑖 = (1st ‘𝑥) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))))
35	29, 34	elrnmpt1s 5959	. . . . . . . . . 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 584	. . . . . . . . 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 4941	. . . . . . . . 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 17	. . . . . . . 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 3987	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → (𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
40	39	ralrimiva 3135	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ↾ 𝐶)(𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
41		iunss 5049	. . . . . 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 233	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ (𝐴 ↾ 𝐶)(𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
43	12, 42	eqsstrrd 4016	. . . 4 ⊢ (𝜑 → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
44		dprd2d.6	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ⊆ 𝐼)
45	44	sselda 3976	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → 𝑖 ∈ 𝐼)
46	45, 17	syldan 589	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
47		ovex 7452	. . . . . . . . . . . 12 ⊢ (𝑖𝑆𝑗) ∈ V
48		eqid 2725	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))
49	47, 48	dmmpti 6700	. . . . . . . . . . 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 6731	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
53		mresspw 17580	. . . . . . . . . . . . . . . 16 ⊢ ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
54	7, 53	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
55	52, 54	sstrd 3987	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
56	51, 55	sstrid 3988	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ 𝒫 (Base‘𝐺))
57		sspwuni 5104	. . . . . . . . . . . . 13 ⊢ ((𝑆 “ (𝐴 ↾ 𝐶)) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (Base‘𝐺))
58	56, 57	sylib 217	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (Base‘𝐺))
59	8	mrccl 17599	. . . . . . . . . . . 12 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (Base‘𝐺)) → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺))
60	7, 58, 59	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺))
61	60	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺))
62		oveq2 7427	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (𝑖𝑆𝑗) = (𝑖𝑆𝑘))
63	62, 48, 47	fvmpt3i 7009	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝐴 “ {𝑖}) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
64	63	adantl 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
65		df-ov 7422	. . . . . . . . . . . . . 14 ⊢ (𝑖𝑆𝑘) = (𝑆‘⟨𝑖, 𝑘⟩)
66	9	ffnd 6724	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 Fn 𝐴)
67	66	ad2antrr 724	. . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖 ∈ 𝐶)
70		elrelimasn 6090	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Rel 𝐴 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
71	15, 70	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
72	71	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
73	72	biimpa 475	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖𝐴𝑘)
74		df-br 5150	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖𝐴𝑘 ↔ ⟨𝑖, 𝑘⟩ ∈ 𝐴)
75	73, 74	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ 𝐴)
76		vex 3465	. . . . . . . . . . . . . . . . 17 ⊢ 𝑘 ∈ V
77	76	opelresi 5993	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑖, 𝑘⟩ ∈ (𝐴 ↾ 𝐶) ↔ (𝑖 ∈ 𝐶 ∧ ⟨𝑖, 𝑘⟩ ∈ 𝐴))
78	69, 75, 77	sylanbrc 581	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ (𝐴 ↾ 𝐶))
79		fnfvima 7245	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 Fn 𝐴 ∧ (𝐴 ↾ 𝐶) ⊆ 𝐴 ∧ ⟨𝑖, 𝑘⟩ ∈ (𝐴 ↾ 𝐶)) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴 ↾ 𝐶)))
80	67, 68, 78, 79	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴 ↾ 𝐶)))
81	65, 80	eqeltrid 2829	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴 ↾ 𝐶)))
82		elssuni 4941	. . . . . . . . . . . . 13 ⊢ ((𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴 ↾ 𝐶)) → (𝑖𝑆𝑘) ⊆ ∪ (𝑆 “ (𝐴 ↾ 𝐶)))
83	81, 82	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ ∪ (𝑆 “ (𝐴 ↾ 𝐶)))
84	7, 8, 58	mrcssidd 17613	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
85	84	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
86	83, 85	sstrd 3987	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
87	64, 86	eqsstrd 4015	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
88	46, 50, 61, 87	dprdlub 20000	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
89		ovex 7452	. . . . . . . . . 10 ⊢ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ V
90	89	elpw 4608	. . . . . . . . 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 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ 𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
92	91	fmpttd 7124	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))):𝐶⟶𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
93	92	frnd 6731	. . . . . 6 ⊢ (𝜑 → ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ 𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
94		sspwuni 5104	. . . . . 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 217	. . . . 5 ⊢ (𝜑 → ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
96	7, 8	mrcssvd 17611	. . . . 5 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ⊆ (Base‘𝐺))
97	95, 96	sstrd 3987	. . . 4 ⊢ (𝜑 → ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (Base‘𝐺))
98	7, 8, 43, 97	mrcssd 17612	. . 3 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ⊆ (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
99	8	mrcsscl 17608	. . . 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 1368	. . 3 ⊢ (𝜑 → (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
101	98, 100	eqssd 3994	. 2 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) = (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
102		eqid 2725	. . . . . . . 8 ⊢ (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
103	89, 102	dmmpti 6700	. . . . . . 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 20002	. . . . 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 493	. . . 4 ⊢ (𝜑 → 𝐺dom DProd ((𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶))
107	44	resmptd 6045	. . . 4 ⊢ (𝜑 → ((𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶) = (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
108	106, 107	breqtrd 5175	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
109	8	dprdspan 20001	. . 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 17	. 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 2768	1 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) = (𝐺 DProd (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3050  Vcvv 3461   ⊆ wss 3944  𝒫 cpw 4604  {csn 4630  ⟨cop 4636  ∪ cuni 4909  ∪ ciun 4997   class class class wbr 5149   ↦ cmpt 5232  dom cdm 5678  ran crn 5679   ↾ cres 5680   “ cima 5681  Rel wrel 5683  Fun wfun 6543   Fn wfn 6544  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419  1^st c1st 7992  2^nd c2nd 7993  Basecbs 17188  Moorecmre 17570  mrClscmrc 17571  ACScacs 17573  Grpcgrp 18903  SubGrpcsubg 19088   DProd cdprd 19967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-supp 8166  df-tpos 8232  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9393  df-oi 9540  df-card 9969  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-nn 12251  df-2 12313  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-fzo 13668  df-seq 14008  df-hash 14331  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17189  df-ress 17218  df-plusg 17254  df-0g 17431  df-gsum 17432  df-mre 17574  df-mrc 17575  df-acs 17577  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18748  df-submnd 18749  df-grp 18906  df-minusg 18907  df-sbg 18908  df-mulg 19037  df-subg 19091  df-ghm 19181  df-gim 19227  df-cntz 19285  df-oppg 19314  df-cmn 19754  df-dprd 19969

This theorem is referenced by:  dprd2da  20016  dprd2db  20017