Theorem dprd2d2 20290

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 5673	. . . . . 6 ⊢ Rel ({𝑖} × 𝐽)
2	1	rgenw 3084	. . . . 5 ⊢ ∀𝑖 ∈ 𝐼 Rel ({𝑖} × 𝐽)
3		reliun 5799	. . . . 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 3209	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐽 𝑆 ∈ (SubGrp‘𝐺))
8		eqid 2763	. . . . 5 ⊢ (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆) = (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)
9	8	fmpox 8103	. . . 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 5902	. . . 4 ⊢ dom ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) = ∪ 𝑖 ∈ 𝐼 dom ({𝑖} × 𝐽)
12		dmxpss 6175	. . . . . . 7 ⊢ dom ({𝑖} × 𝐽) ⊆ {𝑖}
13		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
14	13	snssd 4747	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → {𝑖} ⊆ 𝐼)
15	12, 14	sstrid 3924	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → dom ({𝑖} × 𝐽) ⊆ 𝐼)
16	15	ralrimiva 3158	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 dom ({𝑖} × 𝐽) ⊆ 𝐼)
17		iunss 5006	. . . . 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 3951	. . 3 ⊢ (𝜑 → dom ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) ⊆ 𝐼)
20		dprd2d2.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ 𝐽 ↦ 𝑆))
21		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)) → 𝑖 ∈ 𝐼)
22		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)) → 𝑗 ∈ 𝐽)
23	8	ovmpt4g 7591	. . . . . . . . . 10 ⊢ ((𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗) = 𝑆)
24	21, 22, 6, 23	syl3anc 1399	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)) → (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗) = 𝑆)
25	24	anassrs 473	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐽) → (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗) = 𝑆)
26	25	mpteq2dva 5201	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)) = (𝑗 ∈ 𝐽 ↦ 𝑆))
27	20, 26	breqtrrd 5136	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
28	27	ralrimiva 3158	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
29		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑖𝐺
30		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑖dom DProd
31		nfcsb1v 3875	. . . . . . . 8 ⊢ Ⅎ𝑖⦋𝑥 / 𝑖⦌𝐽
32		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑖𝑥
33		nfmpo1 7523	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)
34		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑖𝑗
35	32, 33, 34	nfov 7473	. . . . . . . 8 ⊢ Ⅎ𝑖(𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)
36	31, 35	nfmpt 5206	. . . . . . 7 ⊢ Ⅎ𝑖(𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))
37	29, 30, 36	nfbr 5155	. . . . . 6 ⊢ Ⅎ𝑖 𝐺dom DProd (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))
38		csbeq1a 3865	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → 𝐽 = ⦋𝑥 / 𝑖⦌𝐽)
39		oveq1 7450	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗) = (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))
40	38, 39	mpteq12dv 5195	. . . . . . 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 5118	. . . . . 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 3568	. . . . 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 516	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
44		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑦(𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)
45		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑗𝑥
46		nfmpo2 7524	. . . . . . 7 ⊢ Ⅎ𝑗(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)
47		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑗𝑦
48	45, 46, 47	nfov 7473	. . . . . 6 ⊢ Ⅎ𝑗(𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦)
49		oveq2 7451	. . . . . 6 ⊢ (𝑗 = 𝑦 → (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗) = (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦))
50	44, 48, 49	cbvmpt 5210	. . . . 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 1949	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝑗 = 𝑧
52	31	nfcri 2918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽
53	51, 52	nfan 1934	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽)
54	38	eleq2d 2850	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑥 → (𝑗 ∈ 𝐽 ↔ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽))
55	54	anbi2d 642	. . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑧)) → 𝑖 = 𝑥)
58		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑧)) → 𝑥 ∈ 𝐼)
59	57, 58	eqeltrd 2864	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑧)) → 𝑖 ∈ 𝐼)
60	59	biantrurd 542	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑧)) → (𝑗 ∈ 𝐽 ↔ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)))
61	60	pm5.32da 590	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (((𝑖 = 𝑥 ∧ 𝑗 = 𝑧) ∧ 𝑗 ∈ 𝐽) ↔ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑧) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽))))
62		anass 474	. . . . . . . . . . . . 13 ⊢ (((𝑖 = 𝑥 ∧ 𝑗 = 𝑧) ∧ 𝑗 ∈ 𝐽) ↔ (𝑖 = 𝑥 ∧ (𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽)))
63		eqcom 2770	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ↔ ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑧⟩)
64		vex 3458	. . . . . . . . . . . . . . . 16 ⊢ 𝑖 ∈ V
65		vex 3458	. . . . . . . . . . . . . . . 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 636	. . . . . . . . . . . . 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 1956	. . . . . . . . . . 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 1956	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (∃𝑗(𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽) ↔ ∃𝑗∃𝑖(⟨𝑥, 𝑧⟩ = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽))))
73		vex 3458	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
74		eleq1w 2847	. . . . . . . . . 10 ⊢ (𝑗 = 𝑧 → (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↔ 𝑧 ∈ ⦋𝑥 / 𝑖⦌𝐽))
75	73, 74	ceqsexv 3502	. . . . . . . . 9 ⊢ (∃𝑗(𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽) ↔ 𝑧 ∈ ⦋𝑥 / 𝑖⦌𝐽)
76		excom 2202	. . . . . . . . 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 6096	. . . . . . . . . 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 5107	. . . . . . . . 9 ⊢ (𝑥∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽)𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽))
81		eliunxp 5820	. . . . . . . . 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 2761	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ⦋𝑥 / 𝑖⦌𝐽 = (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}))
85	84	mpteq1d 5198	. . . . 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 2810	. . . 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 5134	. . 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 7459	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝐺 DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗))) = (𝐺 DProd (𝑗 ∈ 𝐽 ↦ 𝑆)))
90	89	mpteq2dva 5201	. . . . 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 5136	. . . 4 ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))))
92		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑥(𝐺 DProd (𝑗 ∈ 𝐽 ↦ (𝑖(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
93		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑖 DProd
94	29, 93, 36	nfov 7473	. . . . . 6 ⊢ Ⅎ𝑖(𝐺 DProd (𝑗 ∈ ⦋𝑥 / 𝑖⦌𝐽 ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑗)))
95	40	oveq2d 7459	. . . . . 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 5210	. . . . 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 7459	. . . . . 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 5201	. . . . 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 2810	. . . 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 5134	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑥 ∈ 𝐼 ↦ (𝐺 DProd (𝑦 ∈ (∪ 𝑖 ∈ 𝐼 ({𝑖} × 𝐽) “ {𝑥}) ↦ (𝑥(𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)𝑦)))))
101		eqid 2763	. . 3 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
102	5, 10, 19, 87, 100, 101	dprd2da 20288	. 2 ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆))
103	5, 10, 19, 87, 100, 101	dprd2db 20289	. . 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 2800	. . . 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 7459	. . 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 2798	. 2 ⊢ (𝜑 → (𝐺 DProd (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)) = (𝐺 DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ 𝑆)))))
107	102, 106	jca 521	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))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  ⦋csb 3851   ⊆ wss 3900  {csn 4588  ⟨cop 4838  ∪ ciun 4954   class class class wbr 5106   ↦ cmpt 5189   × cxp 5653  dom cdm 5655   “ cima 5658  Rel wrel 5660  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445  mrClscmrc 17782  SubGrpcsubg 19359   DProd cdprd 20239

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-om 7902  df-1st 8025  df-2nd 8026  df-supp 8205  df-tpos 8271  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-er 8745  df-map 8877  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-fsupp 9382  df-oi 9531  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-n0 12634  df-z 12721  df-uz 12993  df-fz 13667  df-fzo 13816  df-seq 14172  df-hash 14502  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-0g 17641  df-gsum 17642  df-mre 17785  df-mrc 17786  df-acs 17788  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-mhm 19007  df-submnd 19008  df-grp 19176  df-minusg 19177  df-sbg 19178  df-mulg 19307  df-subg 19362  df-ghm 19457  df-gim 19502  df-cntz 19561  df-oppg 19590  df-lsm 19880  df-cmn 20026  df-dprd 20241

This theorem is used by:  ablfaclem2  20332