Theorem dprd2dlem1 19400

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 19364	. . . . . 6 ⊢ (𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → 𝐺 ∈ Grp)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		eqid 2734	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgacs 18549	. . . . 5 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
6		acsmre 17127	. . . . 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 6537	. . . . . 6 ⊢ (𝑆:𝐴⟶(SubGrp‘𝐺) → Fun 𝑆)
11		funiunfv 7050	. . . . . 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 5865	. . . . . . . . . 10 ⊢ (𝐴 ↾ 𝐶) ⊆ 𝐴
14	13	sseli 3887	. . . . . . . . 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 19399	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))))
19	14, 18	sylan2 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))))
20		1st2nd 7799	. . . . . . . . . . . . 13 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
21	15, 14, 20	syl2an 599	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
22		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → 𝑥 ∈ (𝐴 ↾ 𝐶))
23	21, 22	eqeltrrd 2835	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ (𝐴 ↾ 𝐶))
24		fvex 6719	. . . . . . . . . . . . 13 ⊢ (2nd ‘𝑥) ∈ V
25	24	opelresi 5848	. . . . . . . . . . . 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 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → (1st ‘𝑥) ∈ 𝐶)
28		ovex 7235	. . . . . . . . . 10 ⊢ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))) ∈ V
29		eqid 2734	. . . . . . . . . . 11 ⊢ (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
30		sneq 4541	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (1st ‘𝑥) → {𝑖} = {(1st ‘𝑥)})
31	30	imaeq2d 5918	. . . . . . . . . . . . 13 ⊢ (𝑖 = (1st ‘𝑥) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st ‘𝑥)}))
32		oveq1 7209	. . . . . . . . . . . . 13 ⊢ (𝑖 = (1st ‘𝑥) → (𝑖𝑆𝑗) = ((1st ‘𝑥)𝑆𝑗))
33	31, 32	mpteq12dv 5129	. . . . . . . . . . . 12 ⊢ (𝑖 = (1st ‘𝑥) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗)))
34	33	oveq2d 7218	. . . . . . . . . . 11 ⊢ (𝑖 = (1st ‘𝑥) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))))
35	29, 34	elrnmpt1s 5815	. . . . . . . . . 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 589	. . . . . . . . 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 4841	. . . . . . . . 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 3901	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → (𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
40	39	ralrimiva 3098	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ↾ 𝐶)(𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
41		iunss 4944	. . . . . 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 3930	. . . 4 ⊢ (𝜑 → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
44		dprd2d.6	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ⊆ 𝐼)
45	44	sselda 3891	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → 𝑖 ∈ 𝐼)
46	45, 17	syldan 594	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
47		ovex 7235	. . . . . . . . . . . 12 ⊢ (𝑖𝑆𝑗) ∈ V
48		eqid 2734	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))
49	47, 48	dmmpti 6511	. . . . . . . . . . 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 5929	. . . . . . . . . . . . . 14 ⊢ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ ran 𝑆
52	9	frnd 6542	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
53		mresspw 17067	. . . . . . . . . . . . . . . 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 3901	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
56	51, 55	sstrid 3902	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ 𝒫 (Base‘𝐺))
57		sspwuni 4998	. . . . . . . . . . . . 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 17086	. . . . . . . . . . . 12 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (Base‘𝐺)) → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺))
60	7, 58, 59	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺))
61	60	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺))
62		oveq2 7210	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (𝑖𝑆𝑗) = (𝑖𝑆𝑘))
63	62, 48, 47	fvmpt3i 6812	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝐴 “ {𝑖}) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
64	63	adantl 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
65		df-ov 7205	. . . . . . . . . . . . . 14 ⊢ (𝑖𝑆𝑘) = (𝑆‘⟨𝑖, 𝑘⟩)
66	9	ffnd 6535	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 Fn 𝐴)
67	66	ad2antrr 726	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖 ∈ 𝐶)
70		elrelimasn 5942	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Rel 𝐴 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
71	15, 70	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
72	71	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
73	72	biimpa 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖𝐴𝑘)
74		df-br 5044	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖𝐴𝑘 ↔ ⟨𝑖, 𝑘⟩ ∈ 𝐴)
75	73, 74	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ 𝐴)
76		vex 3405	. . . . . . . . . . . . . . . . 17 ⊢ 𝑘 ∈ V
77	76	opelresi 5848	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑖, 𝑘⟩ ∈ (𝐴 ↾ 𝐶) ↔ (𝑖 ∈ 𝐶 ∧ ⟨𝑖, 𝑘⟩ ∈ 𝐴))
78	69, 75, 77	sylanbrc 586	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ (𝐴 ↾ 𝐶))
79		fnfvima 7038	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 Fn 𝐴 ∧ (𝐴 ↾ 𝐶) ⊆ 𝐴 ∧ ⟨𝑖, 𝑘⟩ ∈ (𝐴 ↾ 𝐶)) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴 ↾ 𝐶)))
80	67, 68, 78, 79	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴 ↾ 𝐶)))
81	65, 80	eqeltrid 2838	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴 ↾ 𝐶)))
82		elssuni 4841	. . . . . . . . . . . . 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 17100	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
85	84	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
86	83, 85	sstrd 3901	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
87	64, 86	eqsstrd 3929	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
88	46, 50, 61, 87	dprdlub 19385	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
89		ovex 7235	. . . . . . . . . 10 ⊢ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ V
90	89	elpw 4507	. . . . . . . . 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 6921	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))):𝐶⟶𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
93	92	frnd 6542	. . . . . 6 ⊢ (𝜑 → ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ 𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
94		sspwuni 4998	. . . . . 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 17098	. . . . 5 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ⊆ (Base‘𝐺))
97	95, 96	sstrd 3901	. . . 4 ⊢ (𝜑 → ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (Base‘𝐺))
98	7, 8, 43, 97	mrcssd 17099	. . 3 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ⊆ (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
99	8	mrcsscl 17095	. . . 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 1373	. . 3 ⊢ (𝜑 → (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
101	98, 100	eqssd 3908	. 2 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) = (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
102		eqid 2734	. . . . . . . 8 ⊢ (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
103	89, 102	dmmpti 6511	. . . . . . 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 19387	. . . . 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 498	. . . 4 ⊢ (𝜑 → 𝐺dom DProd ((𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶))
107	44	resmptd 5897	. . . 4 ⊢ (𝜑 → ((𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶) = (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
108	106, 107	breqtrd 5069	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
109	8	dprdspan 19386	. . 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 2777	1 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) = (𝐺 DProd (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3054  Vcvv 3401   ⊆ wss 3857  𝒫 cpw 4503  {csn 4531  ⟨cop 4537  ∪ cuni 4809  ∪ ciun 4894   class class class wbr 5043   ↦ cmpt 5124  dom cdm 5540  ran crn 5541   ↾ cres 5542   “ cima 5543  Rel wrel 5545  Fun wfun 6363   Fn wfn 6364  ⟶wf 6365  ‘cfv 6369  (class class class)co 7202  1^st c1st 7748  2^nd c2nd 7749  Basecbs 16684  Moorecmre 17057  mrClscmrc 17058  ACScacs 17060  Grpcgrp 18337  SubGrpcsubg 18509   DProd cdprd 19352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-iun 4896  df-iin 4897  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-se 5499  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-isom 6378  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-of 7458  df-om 7634  df-1st 7750  df-2nd 7751  df-supp 7893  df-tpos 7957  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-er 8380  df-map 8499  df-ixp 8568  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-fsupp 8975  df-oi 9115  df-card 9538  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-nn 11814  df-2 11876  df-n0 12074  df-z 12160  df-uz 12422  df-fz 13079  df-fzo 13222  df-seq 13558  df-hash 13880  df-ndx 16687  df-slot 16688  df-base 16690  df-sets 16691  df-ress 16692  df-plusg 16780  df-0g 16918  df-gsum 16919  df-mre 17061  df-mrc 17062  df-acs 17064  df-mgm 18086  df-sgrp 18135  df-mnd 18146  df-mhm 18190  df-submnd 18191  df-grp 18340  df-minusg 18341  df-sbg 18342  df-mulg 18461  df-subg 18512  df-ghm 18592  df-gim 18635  df-cntz 18683  df-oppg 18710  df-cmn 19144  df-dprd 19354

This theorem is referenced by:  dprd2da  19401  dprd2db  19402