Theorem gsumcom2 18374

Description: Two-dimensional commutation of a group sum. Note that while 𝐴 and 𝐷 are constants w.r.t. 𝑗, 𝑘, 𝐶(𝑗) and 𝐸(𝑘) are not. (Contributed by Mario Carneiro, 28-Dec-2014.)

Hypotheses

Ref	Expression
gsum2d2.b	⊢ 𝐵 = (Base‘𝐺)
gsum2d2.z	⊢ 0 = (0g‘𝐺)
gsum2d2.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
gsum2d2.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsum2d2.r	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑊)
gsum2d2.f	⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵)
gsum2d2.u	⊢ (𝜑 → 𝑈 ∈ Fin)
gsum2d2.n	⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )
gsumcom2.d	⊢ (𝜑 → 𝐷 ∈ 𝑌)
gsumcom2.c	⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ↔ (𝑘 ∈ 𝐷 ∧ 𝑗 ∈ 𝐸)))

Assertion

Ref	Expression
gsumcom2	⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ 𝑋)))

Distinct variable groups:   𝑗,𝑘,𝐵   𝐷,𝑗,𝑘   𝑗,𝐸   𝜑,𝑗,𝑘   𝐴,𝑗,𝑘   𝑗,𝐺,𝑘   𝑈,𝑗,𝑘   𝐶,𝑘   𝑗,𝑉   0 ,𝑗,𝑘

Allowed substitution hints:   𝐶(𝑗)   𝐸(𝑘)   𝑉(𝑘)   𝑊(𝑗,𝑘)   𝑋(𝑗,𝑘)   𝑌(𝑗,𝑘)

Proof of Theorem gsumcom2

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

Step	Hyp	Ref	Expression
1		gsum2d2.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		gsum2d2.z	. . 3 ⊢ 0 = (0g‘𝐺)
3		gsum2d2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd)
4		gsum2d2.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		snex 4908	. . . . . 6 ⊢ {𝑗} ∈ V
6		gsum2d2.r	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑊)
7		xpexg 6960	. . . . . 6 ⊢ (({𝑗} ∈ V ∧ 𝐶 ∈ 𝑊) → ({𝑗} × 𝐶) ∈ V)
8	5, 6, 7	sylancr 695	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ({𝑗} × 𝐶) ∈ V)
9	8	ralrimiva 2966	. . . 4 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∈ V)
10		iunexg 7143	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∈ V) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∈ V)
11	4, 9, 10	syl2anc 693	. . 3 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∈ V)
12		gsum2d2.f	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵)
13	12	ralrimivva 2971	. . . 4 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐶 𝑋 ∈ 𝐵)
14		eqid 2622	. . . . 5 ⊢ (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) = (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
15	14	fmpt2x 7236	. . . 4 ⊢ (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐶 𝑋 ∈ 𝐵 ↔ (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋):∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)⟶𝐵)
16	13, 15	sylib 208	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋):∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)⟶𝐵)
17		gsum2d2.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ Fin)
18		gsum2d2.n	. . . 4 ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )
19	1, 2, 3, 4, 6, 12, 17, 18	gsum2d2lem 18372	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) finSupp 0 )
20		relxp 5227	. . . . . . 7 ⊢ Rel ({𝑘} × 𝐸)
21	20	rgenw 2924	. . . . . 6 ⊢ ∀𝑘 ∈ 𝐷 Rel ({𝑘} × 𝐸)
22		reliun 5239	. . . . . 6 ⊢ (Rel ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↔ ∀𝑘 ∈ 𝐷 Rel ({𝑘} × 𝐸))
23	21, 22	mpbir 221	. . . . 5 ⊢ Rel ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)
24		cnvf1o 7276	. . . . 5 ⊢ (Rel ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) → (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}):∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)–1-1-onto→◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸))
25	23, 24	ax-mp 5	. . . 4 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}):∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)–1-1-onto→◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)
26		relxp 5227	. . . . . . . 8 ⊢ Rel ({𝑗} × 𝐶)
27	26	rgenw 2924	. . . . . . 7 ⊢ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐶)
28		reliun 5239	. . . . . . 7 ⊢ (Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐶))
29	27, 28	mpbir 221	. . . . . 6 ⊢ Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
30		relcnv 5503	. . . . . 6 ⊢ Rel ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)
31		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑘𝜑
32		nfv 1843	. . . . . . . . 9 ⊢ Ⅎ𝑘⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
33		nfiu1 4550	. . . . . . . . . . 11 ⊢ Ⅎ𝑘∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)
34	33	nfcnv 5301	. . . . . . . . . 10 ⊢ Ⅎ𝑘◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)
35	34	nfel2 2781	. . . . . . . . 9 ⊢ Ⅎ𝑘⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)
36	32, 35	nfbi 1833	. . . . . . . 8 ⊢ Ⅎ𝑘(⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸))
37	31, 36	nfim 1825	. . . . . . 7 ⊢ Ⅎ𝑘(𝜑 → (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
38		opeq2 4403	. . . . . . . . . 10 ⊢ (𝑘 = 𝑦 → ⟨𝑥, 𝑘⟩ = ⟨𝑥, 𝑦⟩)
39	38	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → (⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)))
40	38	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → (⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
41	39, 40	bibi12d 335	. . . . . . . 8 ⊢ (𝑘 = 𝑦 → ((⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)) ↔ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸))))
42	41	imbi2d 330	. . . . . . 7 ⊢ (𝑘 = 𝑦 → ((𝜑 → (⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸))) ↔ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))))
43		nfv 1843	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
44		nfiu1 4550	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
45	44	nfel2 2781	. . . . . . . . . 10 ⊢ Ⅎ𝑗⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
46		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑗⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)
47	45, 46	nfbi 1833	. . . . . . . . 9 ⊢ Ⅎ𝑗(⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸))
48	43, 47	nfim 1825	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 → (⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
49		opeq1 4402	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑥 → ⟨𝑗, 𝑘⟩ = ⟨𝑥, 𝑘⟩)
50	49	eleq1d 2686	. . . . . . . . . 10 ⊢ (𝑗 = 𝑥 → (⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)))
51	49	eleq1d 2686	. . . . . . . . . 10 ⊢ (𝑗 = 𝑥 → (⟨𝑗, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↔ ⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
52	50, 51	bibi12d 335	. . . . . . . . 9 ⊢ (𝑗 = 𝑥 → ((⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑗, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)) ↔ (⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸))))
53	52	imbi2d 330	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → ((𝜑 → (⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑗, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸))) ↔ (𝜑 → (⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))))
54		gsumcom2.c	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ↔ (𝑘 ∈ 𝐷 ∧ 𝑗 ∈ 𝐸)))
55		opeliunxp 5170	. . . . . . . . . 10 ⊢ (⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶))
56		opeliunxp 5170	. . . . . . . . . 10 ⊢ (⟨𝑘, 𝑗⟩ ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↔ (𝑘 ∈ 𝐷 ∧ 𝑗 ∈ 𝐸))
57	54, 55, 56	3bitr4g 303	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑘, 𝑗⟩ ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
58		vex 3203	. . . . . . . . . 10 ⊢ 𝑗 ∈ V
59		vex 3203	. . . . . . . . . 10 ⊢ 𝑘 ∈ V
60	58, 59	opelcnv 5304	. . . . . . . . 9 ⊢ (⟨𝑗, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↔ ⟨𝑘, 𝑗⟩ ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸))
61	57, 60	syl6bbr 278	. . . . . . . 8 ⊢ (𝜑 → (⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑗, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
62	48, 53, 61	chvar 2262	. . . . . . 7 ⊢ (𝜑 → (⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
63	37, 42, 62	chvar 2262	. . . . . 6 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
64	29, 30, 63	eqrelrdv 5216	. . . . 5 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) = ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸))
65		f1oeq3 6129	. . . . 5 ⊢ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) = ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) → ((𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}):∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)–1-1-onto→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}):∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)–1-1-onto→◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
66	64, 65	syl 17	. . . 4 ⊢ (𝜑 → ((𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}):∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)–1-1-onto→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}):∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)–1-1-onto→◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
67	25, 66	mpbiri 248	. . 3 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}):∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)–1-1-onto→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
68	1, 2, 3, 11, 16, 19, 67	gsumf1o 18317	. 2 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)) = (𝐺 Σg ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∘ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}))))
69		sneq 4187	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → {𝑧} = {⟨𝑥, 𝑦⟩})
70	69	cnveqd 5298	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ◡{𝑧} = ◡{⟨𝑥, 𝑦⟩})
71	70	unieqd 4446	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ∪ ◡{𝑧} = ∪ ◡{⟨𝑥, 𝑦⟩})
72		opswap 5622	. . . . . . . . 9 ⊢ ∪ ◡{⟨𝑥, 𝑦⟩} = ⟨𝑦, 𝑥⟩
73	71, 72	syl6eq 2672	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ∪ ◡{𝑧} = ⟨𝑦, 𝑥⟩)
74	73	fveq2d 6195	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘∪ ◡{𝑧}) = ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘⟨𝑦, 𝑥⟩))
75		df-ov 6653	. . . . . . 7 ⊢ (𝑦(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥) = ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘⟨𝑦, 𝑥⟩)
76	74, 75	syl6eqr 2674	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘∪ ◡{𝑧}) = (𝑦(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥))
77	76	mpt2mptx 6751	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐷 ({𝑥} × ⦋𝑥 / 𝑘⦌𝐸) ↦ ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘∪ ◡{𝑧})) = (𝑥 ∈ 𝐷, 𝑦 ∈ ⦋𝑥 / 𝑘⦌𝐸 ↦ (𝑦(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥))
78		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑥({𝑘} × 𝐸)
79		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑘{𝑥}
80		nfcsb1v 3549	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐸
81	79, 80	nfxp 5142	. . . . . . 7 ⊢ Ⅎ𝑘({𝑥} × ⦋𝑥 / 𝑘⦌𝐸)
82		sneq 4187	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → {𝑘} = {𝑥})
83		csbeq1a 3542	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → 𝐸 = ⦋𝑥 / 𝑘⦌𝐸)
84	82, 83	xpeq12d 5140	. . . . . . 7 ⊢ (𝑘 = 𝑥 → ({𝑘} × 𝐸) = ({𝑥} × ⦋𝑥 / 𝑘⦌𝐸))
85	78, 81, 84	cbviun 4557	. . . . . 6 ⊢ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) = ∪ 𝑥 ∈ 𝐷 ({𝑥} × ⦋𝑥 / 𝑘⦌𝐸)
86		mpteq1 4737	. . . . . 6 ⊢ (∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) = ∪ 𝑥 ∈ 𝐷 ({𝑥} × ⦋𝑥 / 𝑘⦌𝐸) → (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘∪ ◡{𝑧})) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐷 ({𝑥} × ⦋𝑥 / 𝑘⦌𝐸) ↦ ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘∪ ◡{𝑧})))
87	85, 86	ax-mp 5	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘∪ ◡{𝑧})) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐷 ({𝑥} × ⦋𝑥 / 𝑘⦌𝐸) ↦ ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘∪ ◡{𝑧}))
88		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑥𝐸
89		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑥(𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘)
90		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑦(𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘)
91		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑘𝑦
92		nfmpt22 6723	. . . . . . 7 ⊢ Ⅎ𝑘(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
93		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑘𝑥
94	91, 92, 93	nfov 6676	. . . . . 6 ⊢ Ⅎ𝑘(𝑦(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥)
95		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑗𝑦
96		nfmpt21 6722	. . . . . . 7 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
97		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑗𝑥
98	95, 96, 97	nfov 6676	. . . . . 6 ⊢ Ⅎ𝑗(𝑦(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥)
99		oveq2 6658	. . . . . . 7 ⊢ (𝑘 = 𝑥 → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥))
100		oveq1 6657	. . . . . . 7 ⊢ (𝑗 = 𝑦 → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥) = (𝑦(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥))
101	99, 100	sylan9eq 2676	. . . . . 6 ⊢ ((𝑘 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = (𝑦(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥))
102	88, 80, 89, 90, 94, 98, 83, 101	cbvmpt2x 6733	. . . . 5 ⊢ (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘)) = (𝑥 ∈ 𝐷, 𝑦 ∈ ⦋𝑥 / 𝑘⦌𝐸 ↦ (𝑦(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥))
103	77, 87, 102	3eqtr4i 2654	. . . 4 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘∪ ◡{𝑧})) = (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘))
104		f1of 6137	. . . . . . 7 ⊢ ((𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}):∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)–1-1-onto→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) → (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}):∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)⟶∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
105	67, 104	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}):∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)⟶∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
106		eqid 2622	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}) = (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧})
107	106	fmpt 6381	. . . . . 6 ⊢ (∀𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)∪ ◡{𝑧} ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}):∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)⟶∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
108	105, 107	sylibr 224	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)∪ ◡{𝑧} ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
109		eqidd 2623	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}) = (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}))
110	16	feqmptd 6249	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↦ ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑥)))
111		fveq2 6191	. . . . 5 ⊢ (𝑥 = ∪ ◡{𝑧} → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑥) = ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘∪ ◡{𝑧}))
112	108, 109, 110, 111	fmptcof 6397	. . . 4 ⊢ (𝜑 → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∘ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧})) = (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘∪ ◡{𝑧})))
113	12	ex 450	. . . . . . . . 9 ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) → 𝑋 ∈ 𝐵))
114	14	ovmpt4g 6783	. . . . . . . . . 10 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋)
115	114	3expia 1267	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) → (𝑋 ∈ 𝐵 → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋))
116	113, 115	sylcom 30	. . . . . . . 8 ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋))
117	54, 116	sylbird 250	. . . . . . 7 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ∧ 𝑗 ∈ 𝐸) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋))
118	117	3impib 1262	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷 ∧ 𝑗 ∈ 𝐸) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋)
119	118	eqcomd 2628	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷 ∧ 𝑗 ∈ 𝐸) → 𝑋 = (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘))
120	119	mpt2eq3dva 6719	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ 𝑋) = (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘)))
121	103, 112, 120	3eqtr4a 2682	. . 3 ⊢ (𝜑 → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∘ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧})) = (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ 𝑋))
122	121	oveq2d 6666	. 2 ⊢ (𝜑 → (𝐺 Σg ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∘ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}))) = (𝐺 Σg (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ 𝑋)))
123	68, 122	eqtrd 2656	1 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  ⦋csb 3533  {csn 4177  ⟨cop 4183  ∪ cuni 4436  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113   ∘ ccom 5118  Rel wrel 5119  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Fincfn 7955  Basecbs 15857  0_gc0g 16100   Σ_g cgsu 16101  CMndccmn 18193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-cntz 17750  df-cmn 18195

This theorem is referenced by:  gsumcom  18376  gsumbagdiag  19376