Theorem dprd2dlem1 19950

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 19914	. . . . . 6 ⊢ (𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → 𝐺 ∈ Grp)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		eqid 2731	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgacs 19068	. . . . 5 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
6		acsmre 17553	. . . . 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 6649	. . . . . 6 ⊢ (𝑆:𝐴⟶(SubGrp‘𝐺) → Fun 𝑆)
11		funiunfv 7177	. . . . . 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 5945	. . . . . . . . . 10 ⊢ (𝐴 ↾ 𝐶) ⊆ 𝐴
14	13	sseli 3925	. . . . . . . . 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 19949	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))))
19	14, 18	sylan2 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))))
20		1st2nd 7966	. . . . . . . . . . . . 13 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
21	15, 14, 20	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
22		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → 𝑥 ∈ (𝐴 ↾ 𝐶))
23	21, 22	eqeltrrd 2832	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ (𝐴 ↾ 𝐶))
24		fvex 6830	. . . . . . . . . . . . 13 ⊢ (2nd ‘𝑥) ∈ V
25	24	opelresi 5931	. . . . . . . . . . . 12 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ (𝐴 ↾ 𝐶) ↔ ((1st ‘𝑥) ∈ 𝐶 ∧ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐴))
26	25	simplbi 497	. . . . . . . . . . 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 7374	. . . . . . . . . 10 ⊢ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))) ∈ V
29		eqid 2731	. . . . . . . . . . 11 ⊢ (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
30		sneq 4581	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (1st ‘𝑥) → {𝑖} = {(1st ‘𝑥)})
31	30	imaeq2d 6004	. . . . . . . . . . . . 13 ⊢ (𝑖 = (1st ‘𝑥) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st ‘𝑥)}))
32		oveq1 7348	. . . . . . . . . . . . 13 ⊢ (𝑖 = (1st ‘𝑥) → (𝑖𝑆𝑗) = ((1st ‘𝑥)𝑆𝑗))
33	31, 32	mpteq12dv 5173	. . . . . . . . . . . 12 ⊢ (𝑖 = (1st ‘𝑥) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗)))
34	33	oveq2d 7357	. . . . . . . . . . 11 ⊢ (𝑖 = (1st ‘𝑥) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))))
35	29, 34	elrnmpt1s 5894	. . . . . . . . . 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 586	. . . . . . . . 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 4884	. . . . . . . . 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 3940	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → (𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
40	39	ralrimiva 3124	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ↾ 𝐶)(𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
41		iunss 4989	. . . . . 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 234	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ (𝐴 ↾ 𝐶)(𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
43	12, 42	eqsstrrd 3965	. . . 4 ⊢ (𝜑 → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
44		dprd2d.6	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ⊆ 𝐼)
45	44	sselda 3929	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → 𝑖 ∈ 𝐼)
46	45, 17	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
47		ovex 7374	. . . . . . . . . . . 12 ⊢ (𝑖𝑆𝑗) ∈ V
48		eqid 2731	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))
49	47, 48	dmmpti 6620	. . . . . . . . . . 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 6015	. . . . . . . . . . . . . 14 ⊢ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ ran 𝑆
52	9	frnd 6654	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
53		mresspw 17489	. . . . . . . . . . . . . . . 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 3940	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
56	51, 55	sstrid 3941	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ 𝒫 (Base‘𝐺))
57		sspwuni 5043	. . . . . . . . . . . . 13 ⊢ ((𝑆 “ (𝐴 ↾ 𝐶)) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (Base‘𝐺))
58	56, 57	sylib 218	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (Base‘𝐺))
59	8	mrccl 17512	. . . . . . . . . . . 12 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (Base‘𝐺)) → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺))
60	7, 58, 59	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺))
61	60	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺))
62		oveq2 7349	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (𝑖𝑆𝑗) = (𝑖𝑆𝑘))
63	62, 48, 47	fvmpt3i 6929	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝐴 “ {𝑖}) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
64	63	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
65		df-ov 7344	. . . . . . . . . . . . . 14 ⊢ (𝑖𝑆𝑘) = (𝑆‘⟨𝑖, 𝑘⟩)
66	9	ffnd 6647	. . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖 ∈ 𝐶)
70		elrelimasn 6030	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Rel 𝐴 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
71	15, 70	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
72	71	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
73	72	biimpa 476	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖𝐴𝑘)
74		df-br 5087	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖𝐴𝑘 ↔ ⟨𝑖, 𝑘⟩ ∈ 𝐴)
75	73, 74	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ 𝐴)
76		vex 3440	. . . . . . . . . . . . . . . . 17 ⊢ 𝑘 ∈ V
77	76	opelresi 5931	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑖, 𝑘⟩ ∈ (𝐴 ↾ 𝐶) ↔ (𝑖 ∈ 𝐶 ∧ ⟨𝑖, 𝑘⟩ ∈ 𝐴))
78	69, 75, 77	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ (𝐴 ↾ 𝐶))
79		fnfvima 7162	. . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴 ↾ 𝐶)))
82		elssuni 4884	. . . . . . . . . . . . 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 17526	. . . . . . . . . . . . 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 3940	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
87	64, 86	eqsstrd 3964	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
88	46, 50, 61, 87	dprdlub 19935	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
89		ovex 7374	. . . . . . . . . 10 ⊢ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ V
90	89	elpw 4549	. . . . . . . . 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 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ 𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
92	91	fmpttd 7043	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))):𝐶⟶𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
93	92	frnd 6654	. . . . . 6 ⊢ (𝜑 → ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ 𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
94		sspwuni 5043	. . . . . 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 218	. . . . 5 ⊢ (𝜑 → ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
96	7, 8	mrcssvd 17524	. . . . 5 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ⊆ (Base‘𝐺))
97	95, 96	sstrd 3940	. . . 4 ⊢ (𝜑 → ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (Base‘𝐺))
98	7, 8, 43, 97	mrcssd 17525	. . 3 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ⊆ (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
99	8	mrcsscl 17521	. . . 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 3947	. 2 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) = (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
102		eqid 2731	. . . . . . . 8 ⊢ (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
103	89, 102	dmmpti 6620	. . . . . . 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 19937	. . . . 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 494	. . . 4 ⊢ (𝜑 → 𝐺dom DProd ((𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶))
107	44	resmptd 5984	. . . 4 ⊢ (𝜑 → ((𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶) = (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
108	106, 107	breqtrd 5112	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
109	8	dprdspan 19936	. . 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 2769	1 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) = (𝐺 DProd (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ⊆ wss 3897  𝒫 cpw 4545  {csn 4571  ⟨cop 4577  ∪ cuni 4854  ∪ ciun 4936   class class class wbr 5086   ↦ cmpt 5167  dom cdm 5611  ran crn 5612   ↾ cres 5613   “ cima 5614  Rel wrel 5616  Fun wfun 6470   Fn wfn 6471  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341  1^st c1st 7914  2^nd c2nd 7915  Basecbs 17115  Moorecmre 17479  mrClscmrc 17480  ACScacs 17482  Grpcgrp 18841  SubGrpcsubg 19028   DProd cdprd 19902

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-iin 4939  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-se 5565  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-isom 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-om 7792  df-1st 7916  df-2nd 7917  df-supp 8086  df-tpos 8151  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-ixp 8817  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-fsupp 9241  df-oi 9391  df-card 9827  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-2 12183  df-n0 12377  df-z 12464  df-uz 12728  df-fz 13403  df-fzo 13550  df-seq 13904  df-hash 14233  df-sets 17070  df-slot 17088  df-ndx 17100  df-base 17116  df-ress 17137  df-plusg 17169  df-0g 17340  df-gsum 17341  df-mre 17483  df-mrc 17484  df-acs 17486  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-mhm 18686  df-submnd 18687  df-grp 18844  df-minusg 18845  df-sbg 18846  df-mulg 18976  df-subg 19031  df-ghm 19120  df-gim 19166  df-cntz 19224  df-oppg 19253  df-cmn 19689  df-dprd 19904

This theorem is referenced by:  dprd2da  19951  dprd2db  19952