Theorem dprd2dlem1 18903

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 18867	. . . . . 6 ⊢ (𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → 𝐺 ∈ Grp)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		eqid 2772	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgacs 18088	. . . . 5 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
6		acsmre 16771	. . . . 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 6341	. . . . . 6 ⊢ (𝑆:𝐴⟶(SubGrp‘𝐺) → Fun 𝑆)
11		funiunfv 6826	. . . . . 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 5717	. . . . . . . . . 10 ⊢ (𝐴 ↾ 𝐶) ⊆ 𝐴
14	13	sseli 3850	. . . . . . . . 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 18902	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))))
19	14, 18	sylan2 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → (𝑆‘𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))))
20		1st2nd 7543	. . . . . . . . . . . . 13 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
21	15, 14, 20	syl2an 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
22		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → 𝑥 ∈ (𝐴 ↾ 𝐶))
23	21, 22	eqeltrrd 2861	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ (𝐴 ↾ 𝐶))
24		fvex 6506	. . . . . . . . . . . . 13 ⊢ (2nd ‘𝑥) ∈ V
25	24	opelresi 5696	. . . . . . . . . . . 12 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ (𝐴 ↾ 𝐶) ↔ ((1st ‘𝑥) ∈ 𝐶 ∧ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐴))
26	25	simplbi 490	. . . . . . . . . . 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 7002	. . . . . . . . . 10 ⊢ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))) ∈ V
29		eqid 2772	. . . . . . . . . . 11 ⊢ (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
30		sneq 4445	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (1st ‘𝑥) → {𝑖} = {(1st ‘𝑥)})
31	30	imaeq2d 5764	. . . . . . . . . . . . 13 ⊢ (𝑖 = (1st ‘𝑥) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st ‘𝑥)}))
32		oveq1 6977	. . . . . . . . . . . . 13 ⊢ (𝑖 = (1st ‘𝑥) → (𝑖𝑆𝑗) = ((1st ‘𝑥)𝑆𝑗))
33	31, 32	mpteq12dv 5006	. . . . . . . . . . . 12 ⊢ (𝑖 = (1st ‘𝑥) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗)))
34	33	oveq2d 6986	. . . . . . . . . . 11 ⊢ (𝑖 = (1st ‘𝑥) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑥)}) ↦ ((1st ‘𝑥)𝑆𝑗))))
35	29, 34	elrnmpt1s 5665	. . . . . . . . . 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 577	. . . . . . . . 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 4735	. . . . . . . . 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 3864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ↾ 𝐶)) → (𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
40	39	ralrimiva 3126	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ↾ 𝐶)(𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
41		iunss 4829	. . . . . 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 226	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ (𝐴 ↾ 𝐶)(𝑆‘𝑥) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
43	12, 42	eqsstr3d 3892	. . . 4 ⊢ (𝜑 → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
44		dprd2d.6	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ⊆ 𝐼)
45	44	sselda 3854	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → 𝑖 ∈ 𝐼)
46	45, 17	syldan 582	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
47		ovex 7002	. . . . . . . . . . . 12 ⊢ (𝑖𝑆𝑗) ∈ V
48		eqid 2772	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))
49	47, 48	dmmpti 6316	. . . . . . . . . . 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 5775	. . . . . . . . . . . . . 14 ⊢ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ ran 𝑆
52	9	frnd 6345	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
53		mresspw 16711	. . . . . . . . . . . . . . . 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 3864	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
56	51, 55	syl5ss 3865	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ 𝒫 (Base‘𝐺))
57		sspwuni 4882	. . . . . . . . . . . . 13 ⊢ ((𝑆 “ (𝐴 ↾ 𝐶)) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (Base‘𝐺))
58	56, 57	sylib 210	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (Base‘𝐺))
59	8	mrccl 16730	. . . . . . . . . . . 12 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (Base‘𝐺)) → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺))
60	7, 58, 59	syl2anc 576	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺))
61	60	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ∈ (SubGrp‘𝐺))
62		oveq2 6978	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (𝑖𝑆𝑗) = (𝑖𝑆𝑘))
63	62, 48, 47	fvmpt3i 6594	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝐴 “ {𝑖}) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
64	63	adantl 474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
65		df-ov 6973	. . . . . . . . . . . . . 14 ⊢ (𝑖𝑆𝑘) = (𝑆‘⟨𝑖, 𝑘⟩)
66	9	ffnd 6339	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 Fn 𝐴)
67	66	ad2antrr 713	. . . . . . . . . . . . . . 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 756	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖 ∈ 𝐶)
70		elrelimasn 5787	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Rel 𝐴 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
71	15, 70	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
72	71	adantr 473	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
73	72	biimpa 469	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖𝐴𝑘)
74		df-br 4924	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖𝐴𝑘 ↔ ⟨𝑖, 𝑘⟩ ∈ 𝐴)
75	73, 74	sylib 210	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ 𝐴)
76		vex 3412	. . . . . . . . . . . . . . . . 17 ⊢ 𝑘 ∈ V
77	76	opelresi 5696	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑖, 𝑘⟩ ∈ (𝐴 ↾ 𝐶) ↔ (𝑖 ∈ 𝐶 ∧ ⟨𝑖, 𝑘⟩ ∈ 𝐴))
78	69, 75, 77	sylanbrc 575	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ (𝐴 ↾ 𝐶))
79		fnfvima 6814	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 Fn 𝐴 ∧ (𝐴 ↾ 𝐶) ⊆ 𝐴 ∧ ⟨𝑖, 𝑘⟩ ∈ (𝐴 ↾ 𝐶)) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴 ↾ 𝐶)))
80	67, 68, 78, 79	syl3anc 1351	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴 ↾ 𝐶)))
81	65, 80	syl5eqel 2864	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴 ↾ 𝐶)))
82		elssuni 4735	. . . . . . . . . . . . 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 16744	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
85	84	ad2antrr 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ∪ (𝑆 “ (𝐴 ↾ 𝐶)) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
86	83, 85	sstrd 3864	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
87	64, 86	eqsstrd 3891	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
88	46, 50, 61, 87	dprdlub 18888	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
89		ovex 7002	. . . . . . . . . 10 ⊢ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ V
90	89	elpw 4422	. . . . . . . . 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 226	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ 𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
92	91	fmpttd 6696	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))):𝐶⟶𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
93	92	frnd 6345	. . . . . 6 ⊢ (𝜑 → ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ 𝒫 (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
94		sspwuni 4882	. . . . . 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 210	. . . . 5 ⊢ (𝜑 → ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
96	7, 8	mrcssvd 16742	. . . . 5 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ⊆ (Base‘𝐺))
97	95, 96	sstrd 3864	. . . 4 ⊢ (𝜑 → ∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (Base‘𝐺))
98	7, 8, 43, 97	mrcssd 16743	. . 3 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) ⊆ (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
99	8	mrcsscl 16739	. . . 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 1351	. . 3 ⊢ (𝜑 → (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) ⊆ (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))))
101	98, 100	eqssd 3871	. 2 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) = (𝐾‘∪ ran (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
102		eqid 2772	. . . . . . . 8 ⊢ (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
103	89, 102	dmmpti 6316	. . . . . . 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 18890	. . . . 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 487	. . . 4 ⊢ (𝜑 → 𝐺dom DProd ((𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶))
107	44	resmptd 5747	. . . 4 ⊢ (𝜑 → ((𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶) = (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
108	106, 107	breqtrd 4949	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
109	8	dprdspan 18889	. . 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 2811	1 ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) = (𝐺 DProd (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2048  ∀wral 3082  Vcvv 3409   ⊆ wss 3825  𝒫 cpw 4416  {csn 4435  ⟨cop 4441  ∪ cuni 4706  ∪ ciun 4786   class class class wbr 4923   ↦ cmpt 5002  dom cdm 5400  ran crn 5401   ↾ cres 5402   “ cima 5403  Rel wrel 5405  Fun wfun 6176   Fn wfn 6177  ⟶wf 6178  ‘cfv 6182  (class class class)co 6970  1^st c1st 7492  2^nd c2nd 7493  Basecbs 16329  Moorecmre 16701  mrClscmrc 16702  ACScacs 16704  Grpcgrp 17881  SubGrpcsubg 18047   DProd cdprd 18855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-se 5360  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-of 7221  df-om 7391  df-1st 7494  df-2nd 7495  df-supp 7627  df-tpos 7688  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-oadd 7901  df-er 8081  df-map 8200  df-ixp 8252  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-fsupp 8621  df-oi 8761  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-nn 11432  df-2 11496  df-n0 11701  df-z 11787  df-uz 12052  df-fz 12702  df-fzo 12843  df-seq 13178  df-hash 13499  df-ndx 16332  df-slot 16333  df-base 16335  df-sets 16336  df-ress 16337  df-plusg 16424  df-0g 16561  df-gsum 16562  df-mre 16705  df-mrc 16706  df-acs 16708  df-mgm 17700  df-sgrp 17742  df-mnd 17753  df-mhm 17793  df-submnd 17794  df-grp 17884  df-minusg 17885  df-sbg 17886  df-mulg 18002  df-subg 18050  df-ghm 18117  df-gim 18160  df-cntz 18208  df-oppg 18235  df-cmn 18658  df-dprd 18857

This theorem is referenced by:  dprd2da  18904  dprd2db  18905