Theorem dprd2d2 19975

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 5642	. . . . . 6 ⊢ Rel ({𝑖} × 𝐽)
2	1	rgenw 3055	. . . . 5 ⊢ ∀𝑖 ∈ 𝐼 Rel ({𝑖} × 𝐽)
3		reliun 5765	. . . . 5 ⊢ (Rel ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) ↔ ∀𝑖 ∈ 𝐼 Rel ({𝑖} × 𝐽))
4	2, 3	mpbir 231	. . . 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 3179	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐽 𝑆 ∈ (SubGrp‘𝐺))
8		eqid 2736	. . . . 5 ⊢ (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆) = (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)
9	8	fmpox 8011	. . . 4 ⊢ (∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐽 𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆):∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽)⟶(SubGrp‘𝐺))
10	7, 9	sylib 218	. . 3 ⊢ (𝜑 → (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆):∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽)⟶(SubGrp‘𝐺))
11		dmiun 5862	. . . 4 ⊢ dom ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) = ∪ 𝑖 ∈ 𝐼 dom ({𝑖} × 𝐽)
12		dmxpss 6129	. . . . . . 7 ⊢ dom ({𝑖} × 𝐽) ⊆ {𝑖}
13		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
14	13	snssd 4765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → {𝑖} ⊆ 𝐼)
15	12, 14	sstrid 3945	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → dom ({𝑖} × 𝐽) ⊆ 𝐼)
16	15	ralrimiva 3128	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 dom ({𝑖} × 𝐽) ⊆ 𝐼)
17		iunss 5000	. . . . 5 ⊢ (∪ 𝑖 ∈ 𝐼 dom ({𝑖} × 𝐽) ⊆ 𝐼 ↔ ∀𝑖 ∈ 𝐼 dom ({𝑖} × 𝐽) ⊆ 𝐼)
18	16, 17	sylibr 234	. . . 4 ⊢ (𝜑 → ∪ 𝑖 ∈ 𝐼 dom ({𝑖} × 𝐽) ⊆ 𝐼)
19	11, 18	eqsstrid 3972	. . 3 ⊢ (𝜑 → dom ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) ⊆ 𝐼)
20		dprd2d2.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ 𝐽 ↦ 𝑆))
21		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)) → 𝑖 ∈ 𝐼)
22		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)) → 𝑗 ∈ 𝐽)
23	8	ovmpt4g 7505	. . . . . . . . . 10 ⊢ ((𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗) = 𝑆)
24	21, 22, 6, 23	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)) → (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗) = 𝑆)
25	24	anassrs 467	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐽) → (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗) = 𝑆)
26	25	mpteq2dva 5191	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)) = (𝑗 ∈ 𝐽 ↦ 𝑆))
27	20, 26	breqtrrd 5126	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
28	27	ralrimiva 3128	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
29		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑖𝐺
30		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑖dom DProd
31		nfcsb1v 3873	. . . . . . . 8 ⊢ Ⅎ𝑖⦋𝑥 / 𝑖⦌𝐽
32		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑖𝑥
33		nfmpo1 7438	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)
34		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑖𝑗
35	32, 33, 34	nfov 7388	. . . . . . . 8 ⊢ Ⅎ𝑖(𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)
36	31, 35	nfmpt 5196	. . . . . . 7 ⊢ Ⅎ𝑖(𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))
37	29, 30, 36	nfbr 5145	. . . . . 6 ⊢ Ⅎ𝑖 𝐺dom DProd (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))
38		csbeq1a 3863	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → 𝐽 = ⦋𝑥 / 𝑖⦌𝐽)
39		oveq1 7365	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗) = (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))
40	38, 39	mpteq12dv 5185	. . . . . . 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 5110	. . . . . 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 3564	. . . . 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 506	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
44		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑦(𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)
45		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑗𝑥
46		nfmpo2 7439	. . . . . . 7 ⊢ Ⅎ𝑗(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)
47		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑗𝑦
48	45, 46, 47	nfov 7388	. . . . . 6 ⊢ Ⅎ𝑗(𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦)
49		oveq2 7366	. . . . . 6 ⊢ (𝑗 = 𝑦 → (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗) = (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦))
50	44, 48, 49	cbvmpt 5200	. . . . 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 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝑗 = 𝑧
52	31	nfcri 2890	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽
53	51, 52	nfan 1900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽)
54	38	eleq2d 2822	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑥 → (𝑗 ∈ 𝐽 ↔ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽))
55	54	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑥 → ((𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽) ↔ (𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽)))
56	53, 55	equsexv 2275	. . . . . . . . . . 11 ⊢ (∃𝑖(𝑖 = 𝑥 ∧ (𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽)) ↔ (𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽))
57		simprl 770	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑧)) → 𝑖 = 𝑥)
58		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑧)) → 𝑥 ∈ 𝐼)
59	57, 58	eqeltrd 2836	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑧)) → 𝑖 ∈ 𝐼)
60	59	biantrurd 532	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑧)) → (𝑗 ∈ 𝐽 ↔ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)))
61	60	pm5.32da 579	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (((𝑖 = 𝑥 ∧ 𝑗 = 𝑧) ∧ 𝑗 ∈ 𝐽) ↔ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑧) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽))))
62		anass 468	. . . . . . . . . . . . 13 ⊢ (((𝑖 = 𝑥 ∧ 𝑗 = 𝑧) ∧ 𝑗 ∈ 𝐽) ↔ (𝑖 = 𝑥 ∧ (𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽)))
63		eqcom 2743	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ↔ ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑧⟩)
64		vex 3444	. . . . . . . . . . . . . . . 16 ⊢ 𝑖 ∈ V
65		vex 3444	. . . . . . . . . . . . . . . 16 ⊢ 𝑗 ∈ V
66	64, 65	opth 5424	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑧⟩ ↔ (𝑖 = 𝑥 ∧ 𝑗 = 𝑧))
67	63, 66	bitr2i 276	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑧) ↔ ⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩)
68	67	anbi1i 624	. . . . . . . . . . . . 13 ⊢ (((𝑖 = 𝑥 ∧ 𝑗 = 𝑧) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)) ↔ (⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)))
69	61, 62, 68	3bitr3g 313	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑖 = 𝑥 ∧ (𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽)) ↔ (⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽))))
70	69	exbidv 1922	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∃𝑖(𝑖 = 𝑥 ∧ (𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽)) ↔ ∃𝑖(⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽))))
71	56, 70	bitr3id 285	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽) ↔ ∃𝑖(⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽))))
72	71	exbidv 1922	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∃𝑗(𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽) ↔ ∃𝑗∃𝑖(⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽))))
73		vex 3444	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
74		eleq1w 2819	. . . . . . . . . 10 ⊢ (𝑗 = 𝑧 → (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↔ 𝑧 ∈ ⦋𝑥 / 𝑖⦌𝐽))
75	73, 74	ceqsexv 3490	. . . . . . . . 9 ⊢ (∃𝑗(𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽) ↔ 𝑧 ∈ ⦋𝑥 / 𝑖⦌𝐽)
76		excom 2167	. . . . . . . . 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 313	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑧 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↔ ∃𝑖∃𝑗(⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽))))
78		elrelimasn 6045	. . . . . . . . . 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 5099	. . . . . . . . 9 ⊢ (𝑥∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽)𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽))
81		eliunxp 5786	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑧⟩ ∈ ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) ↔ ∃𝑖∃𝑗(⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)))
82	79, 80, 81	3bitri 297	. . . . . . . 8 ⊢ (𝑧 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↔ ∃𝑖∃𝑗(⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)))
83	77, 82	bitr4di 289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑧 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↔ 𝑧 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥})))
84	83	eqrdv 2734	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ⦋𝑥 / 𝑖⦌𝐽 = (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}))
85	84	mpteq1d 5188	. . . . 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 2783	. . . 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 5124	. . 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 7374	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝐺 DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))) = (𝐺 DProd (𝑗 ∈ 𝐽 ↦ 𝑆)))
90	89	mpteq2dva 5191	. . . . 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 5126	. . . 4 ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))))
92		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑥(𝐺 DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
93		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑖 DProd
94	29, 93, 36	nfov 7388	. . . . . 6 ⊢ Ⅎ𝑖(𝐺 DProd (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
95	40	oveq2d 7374	. . . . . 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 5200	. . . . 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 7374	. . . . . 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 5191	. . . . 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 2783	. . . 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 5124	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑥 ∈ 𝐼 ↦ (𝐺 DProd (𝑦 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦)))))
101		eqid 2736	. . 3 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
102	5, 10, 19, 87, 100, 101	dprd2da 19973	. 2 ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆))
103	5, 10, 19, 87, 100, 101	dprd2db 19974	. . 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 2773	. . . 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 7374	. . 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 2771	. 2 ⊢ (𝜑 → (𝐺 DProd (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)) = (𝐺 DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ 𝑆)))))
107	102, 106	jca 511	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 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3051  ⦋csb 3849   ⊆ wss 3901  {csn 4580  ⟨cop 4586  ∪ ciun 4946   class class class wbr 5098   ↦ cmpt 5179   × cxp 5622  dom cdm 5624   “ cima 5627  Rel wrel 5629  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  mrClscmrc 17502  SubGrpcsubg 19050   DProd cdprd 19924

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-0g 17361  df-gsum 17362  df-mre 17505  df-mrc 17506  df-acs 17508  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18708  df-submnd 18709  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18998  df-subg 19053  df-ghm 19142  df-gim 19188  df-cntz 19246  df-oppg 19275  df-lsm 19565  df-cmn 19711  df-dprd 19926

This theorem is referenced by:  ablfaclem2  20017