Theorem dprd2d2 20107

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

Hypotheses

Ref	Expression
dprd2d2.1	⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)) → 𝑆 ∈ (SubGrp‘𝐺))
dprd2d2.2	⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ 𝐽 ↦ 𝑆))
dprd2d2.3	⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ 𝑆))))

Assertion

Ref	Expression
dprd2d2	⊢ (𝜑 → (𝐺dom DProd (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆) ∧ (𝐺 DProd (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)) = (𝐺 DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ 𝑆))))))

Distinct variable groups:   𝑖,𝑗,𝐺   𝑖,𝐼,𝑗   𝑗,𝐽   𝜑,𝑖,𝑗

Allowed substitution hints:   𝑆(𝑖,𝑗)   𝐽(𝑖)

Proof of Theorem dprd2d2

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relxp 5670	. . . . . 6 ⊢ Rel ({𝑖} × 𝐽)
2	1	rgenw 3083	. . . . 5 ⊢ ∀𝑖 ∈ 𝐼 Rel ({𝑖} × 𝐽)
3		reliun 5794	. . . . 5 ⊢ (Rel ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) ↔ ∀𝑖 ∈ 𝐼 Rel ({𝑖} × 𝐽))
4	2, 3	mpbir 234	. . . 4 ⊢ Rel ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽)
5	4	a1i 11	. . 3 ⊢ (𝜑 → Rel ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽))
6		dprd2d2.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)) → 𝑆 ∈ (SubGrp‘𝐺))
7	6	ralrimivva 3208	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐽 𝑆 ∈ (SubGrp‘𝐺))
8		eqid 2765	. . . . 5 ⊢ (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆) = (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)
9	8	fmpox 8052	. . . 4 ⊢ (∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐽 𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆):∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽)⟶(SubGrp‘𝐺))
10	7, 9	sylib 221	. . 3 ⊢ (𝜑 → (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆):∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽)⟶(SubGrp‘𝐺))
11		dmiun 5894	. . . 4 ⊢ dom ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) = ∪ 𝑖 ∈ 𝐼 dom ({𝑖} × 𝐽)
12		dmxpss 6161	. . . . . . 7 ⊢ dom ({𝑖} × 𝐽) ⊆ {𝑖}
13		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
14	13	snssd 4748	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → {𝑖} ⊆ 𝐼)
15	12, 14	sstrid 3950	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → dom ({𝑖} × 𝐽) ⊆ 𝐼)
16	15	ralrimiva 3157	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 dom ({𝑖} × 𝐽) ⊆ 𝐼)
17		iunss 5005	. . . . 5 ⊢ (∪ 𝑖 ∈ 𝐼 dom ({𝑖} × 𝐽) ⊆ 𝐼 ↔ ∀𝑖 ∈ 𝐼 dom ({𝑖} × 𝐽) ⊆ 𝐼)
18	16, 17	sylibr 237	. . . 4 ⊢ (𝜑 → ∪ 𝑖 ∈ 𝐼 dom ({𝑖} × 𝐽) ⊆ 𝐼)
19	11, 18	eqsstrid 3977	. . 3 ⊢ (𝜑 → dom ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) ⊆ 𝐼)
20		dprd2d2.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ 𝐽 ↦ 𝑆))
21		simprl 782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)) → 𝑖 ∈ 𝐼)
22		simprr 784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)) → 𝑗 ∈ 𝐽)
23	8	ovmpt4g 7547	. . . . . . . . . 10 ⊢ ((𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗) = 𝑆)
24	21, 22, 6, 23	syl3anc 1394	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)) → (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗) = 𝑆)
25	24	anassrs 472	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐽) → (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗) = 𝑆)
26	25	mpteq2dva 5198	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)) = (𝑗 ∈ 𝐽 ↦ 𝑆))
27	20, 26	breqtrrd 5133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
28	27	ralrimiva 3157	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
29		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑖𝐺
30		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑖dom DProd
31		nfcsb1v 3879	. . . . . . . 8 ⊢ Ⅎ𝑖⦋𝑥 / 𝑖⦌𝐽
32		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑖𝑥
33		nfmpo1 7480	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)
34		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑖𝑗
35	32, 33, 34	nfov 7430	. . . . . . . 8 ⊢ Ⅎ𝑖(𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)
36	31, 35	nfmpt 5203	. . . . . . 7 ⊢ Ⅎ𝑖(𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))
37	29, 30, 36	nfbr 5152	. . . . . 6 ⊢ Ⅎ𝑖 𝐺dom DProd (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))
38		csbeq1a 3869	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → 𝐽 = ⦋𝑥 / 𝑖⦌𝐽)
39		oveq1 7407	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗) = (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))
40	38, 39	mpteq12dv 5192	. . . . . . 7 ⊢ (𝑖 = 𝑥 → (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)) = (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
41	40	breq2d 5117	. . . . . 6 ⊢ (𝑖 = 𝑥 → (𝐺dom DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)) ↔ 𝐺dom DProd (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))))
42	37, 41	rspc 3572	. . . . 5 ⊢ (𝑥 ∈ 𝐼 → (∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)) → 𝐺dom DProd (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))))
43	28, 42	mpan9 515	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
44		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑦(𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)
45		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑗𝑥
46		nfmpo2 7481	. . . . . . 7 ⊢ Ⅎ𝑗(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)
47		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑗𝑦
48	45, 46, 47	nfov 7430	. . . . . 6 ⊢ Ⅎ𝑗(𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦)
49		oveq2 7408	. . . . . 6 ⊢ (𝑗 = 𝑦 → (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗) = (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦))
50	44, 48, 49	cbvmpt 5207	. . . . 5 ⊢ (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)) = (𝑦 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦))
51		nfv 1937	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝑗 = 𝑧
52	31	nfcri 2919	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽
53	51, 52	nfan 1922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽)
54	38	eleq2d 2851	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑥 → (𝑗 ∈ 𝐽 ↔ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽))
55	54	anbi2d 641	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑥 → ((𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽) ↔ (𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽)))
56	53, 55	equsexv 2306	. . . . . . . . . . 11 ⊢ (∃𝑖(𝑖 = 𝑥 ∧ (𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽)) ↔ (𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽))
57		simprl 782	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑧)) → 𝑖 = 𝑥)
58		simplr 780	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑧)) → 𝑥 ∈ 𝐼)
59	57, 58	eqeltrd 2865	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑧)) → 𝑖 ∈ 𝐼)
60	59	biantrurd 541	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑧)) → (𝑗 ∈ 𝐽 ↔ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)))
61	60	pm5.32da 589	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (((𝑖 = 𝑥 ∧ 𝑗 = 𝑧) ∧ 𝑗 ∈ 𝐽) ↔ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑧) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽))))
62		anass 473	. . . . . . . . . . . . 13 ⊢ (((𝑖 = 𝑥 ∧ 𝑗 = 𝑧) ∧ 𝑗 ∈ 𝐽) ↔ (𝑖 = 𝑥 ∧ (𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽)))
63		eqcom 2772	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ↔ ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑧⟩)
64		vex 3461	. . . . . . . . . . . . . . . 16 ⊢ 𝑖 ∈ V
65		vex 3461	. . . . . . . . . . . . . . . 16 ⊢ 𝑗 ∈ V
66	64, 65	opth 5449	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑧⟩ ↔ (𝑖 = 𝑥 ∧ 𝑗 = 𝑧))
67	63, 66	bitr2i 279	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑧) ↔ ⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩)
68	67	anbi1i 635	. . . . . . . . . . . . 13 ⊢ (((𝑖 = 𝑥 ∧ 𝑗 = 𝑧) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)) ↔ (⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)))
69	61, 62, 68	3bitr3g 316	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑖 = 𝑥 ∧ (𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽)) ↔ (⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽))))
70	69	exbidv 1944	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∃𝑖(𝑖 = 𝑥 ∧ (𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽)) ↔ ∃𝑖(⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽))))
71	56, 70	bitr3id 288	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽) ↔ ∃𝑖(⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽))))
72	71	exbidv 1944	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∃𝑗(𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽) ↔ ∃𝑗∃𝑖(⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽))))
73		vex 3461	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
74		eleq1w 2848	. . . . . . . . . 10 ⊢ (𝑗 = 𝑧 → (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↔ 𝑧 ∈ ⦋𝑥 / 𝑖⦌𝐽))
75	73, 74	ceqsexv 3505	. . . . . . . . 9 ⊢ (∃𝑗(𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽) ↔ 𝑧 ∈ ⦋𝑥 / 𝑖⦌𝐽)
76		excom 2199	. . . . . . . . 9 ⊢ (∃𝑗∃𝑖(⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)) ↔ ∃𝑖∃𝑗(⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)))
77	72, 75, 76	3bitr3g 316	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑧 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↔ ∃𝑖∃𝑗(⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽))))
78		elrelimasn 6079	. . . . . . . . . 10 ⊢ (Rel ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) → (𝑧 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↔ 𝑥∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽)𝑧))
79	4, 78	ax-mp 5	. . . . . . . . 9 ⊢ (𝑧 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↔ 𝑥∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽)𝑧)
80		df-br 5106	. . . . . . . . 9 ⊢ (𝑥∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽)𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽))
81		eliunxp 5814	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑧⟩ ∈ ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) ↔ ∃𝑖∃𝑗(⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)))
82	79, 80, 81	3bitri 300	. . . . . . . 8 ⊢ (𝑧 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↔ ∃𝑖∃𝑗(⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)))
83	77, 82	bitr4di 292	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑧 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↔ 𝑧 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥})))
84	83	eqrdv 2763	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ⦋𝑥 / 𝑖⦌𝐽 = (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}))
85	84	mpteq1d 5195	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦)) = (𝑦 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦)))
86	50, 85	eqtrid 2812	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)) = (𝑦 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦)))
87	43, 86	breqtrd 5131	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd (𝑦 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦)))
88		dprd2d2.3	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ 𝑆))))
89	26	oveq2d 7416	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝐺 DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))) = (𝐺 DProd (𝑗 ∈ 𝐽 ↦ 𝑆)))
90	89	mpteq2dva 5198	. . . . 5 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))) = (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ 𝑆))))
91	88, 90	breqtrrd 5133	. . . 4 ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))))
92		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑥(𝐺 DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
93		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑖 DProd
94	29, 93, 36	nfov 7430	. . . . . 6 ⊢ Ⅎ𝑖(𝐺 DProd (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
95	40	oveq2d 7416	. . . . . 6 ⊢ (𝑖 = 𝑥 → (𝐺 DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))) = (𝐺 DProd (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))))
96	92, 94, 95	cbvmpt 5207	. . . . 5 ⊢ (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))) = (𝑥 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))))
97	86	oveq2d 7416	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐺 DProd (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))) = (𝐺 DProd (𝑦 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦))))
98	97	mpteq2dva 5198	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))) = (𝑥 ∈ 𝐼 ↦ (𝐺 DProd (𝑦 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦)))))
99	96, 98	eqtrid 2812	. . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))) = (𝑥 ∈ 𝐼 ↦ (𝐺 DProd (𝑦 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦)))))
100	91, 99	breqtrd 5131	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑥 ∈ 𝐼 ↦ (𝐺 DProd (𝑦 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦)))))
101		eqid 2765	. . 3 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
102	5, 10, 19, 87, 100, 101	dprd2da 20105	. 2 ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆))
103	5, 10, 19, 87, 100, 101	dprd2db 20106	. . 3 ⊢ (𝜑 → (𝐺 DProd (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)) = (𝐺 DProd (𝑥 ∈ 𝐼 ↦ (𝐺 DProd (𝑦 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦))))))
104	99, 90	eqtr3d 2802	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐺 DProd (𝑦 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦)))) = (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ 𝑆))))
105	104	oveq2d 7416	. . 3 ⊢ (𝜑 → (𝐺 DProd (𝑥 ∈ 𝐼 ↦ (𝐺 DProd (𝑦 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦))))) = (𝐺 DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ 𝑆)))))
106	103, 105	eqtrd 2800	. 2 ⊢ (𝜑 → (𝐺 DProd (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)) = (𝐺 DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ 𝑆)))))
107	102, 106	jca 520	1 ⊢ (𝜑 → (𝐺dom DProd (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆) ∧ (𝐺 DProd (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)) = (𝐺 DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ 𝑆))))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∀wral 3079  ⦋csb 3855   ⊆ wss 3907  {csn 4585  ⟨cop 4591  ∪ ciun 4952   class class class wbr 5105   ↦ cmpt 5186   × cxp 5650  dom cdm 5652   “ cima 5655  Rel wrel 5657  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  mrClscmrc 17625  SubGrpcsubg 19177   DProd cdprd 20056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-tpos 8210  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-er 8682  df-map 8814  df-ixp 8884  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-n0 12496  df-z 12583  df-uz 12854  df-fz 13527  df-fzo 13674  df-seq 14029  df-hash 14358  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-0g 17484  df-gsum 17485  df-mre 17628  df-mrc 17629  df-acs 17631  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-mhm 18831  df-submnd 18832  df-grp 18993  df-minusg 18994  df-sbg 18995  df-mulg 19125  df-subg 19180  df-ghm 19275  df-gim 19320  df-cntz 19378  df-oppg 19407  df-lsm 19697  df-cmn 19843  df-dprd 20058

This theorem is referenced by:  ablfaclem2  20149