Theorem gsumcom2 19993

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		vsnex 5434	. . . . . 6 ⊢ {𝑗} ∈ V
6		gsum2d2.r	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑊)
7		xpexg 7770	. . . . . 6 ⊢ (({𝑗} ∈ V ∧ 𝐶 ∈ 𝑊) → ({𝑗} × 𝐶) ∈ V)
8	5, 6, 7	sylancr 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ({𝑗} × 𝐶) ∈ V)
9	8	ralrimiva 3146	. . . 4 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∈ V)
10		iunexg 7988	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∈ V) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∈ V)
11	4, 9, 10	syl2anc 584	. . 3 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∈ V)
12		gsum2d2.f	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵)
13	12	ralrimivva 3202	. . . 4 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐶 𝑋 ∈ 𝐵)
14		eqid 2737	. . . . 5 ⊢ (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) = (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
15	14	fmpox 8092	. . . 4 ⊢ (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐶 𝑋 ∈ 𝐵 ↔ (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋):∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)⟶𝐵)
16	13, 15	sylib 218	. . 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 19991	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) finSupp 0 )
20		relxp 5703	. . . . . . 7 ⊢ Rel ({𝑘} × 𝐸)
21	20	rgenw 3065	. . . . . 6 ⊢ ∀𝑘 ∈ 𝐷 Rel ({𝑘} × 𝐸)
22		reliun 5826	. . . . . 6 ⊢ (Rel ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↔ ∀𝑘 ∈ 𝐷 Rel ({𝑘} × 𝐸))
23	21, 22	mpbir 231	. . . . 5 ⊢ Rel ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)
24		cnvf1o 8136	. . . . 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 5703	. . . . . . . 8 ⊢ Rel ({𝑗} × 𝐶)
27	26	rgenw 3065	. . . . . . 7 ⊢ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐶)
28		reliun 5826	. . . . . . 7 ⊢ (Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐶))
29	27, 28	mpbir 231	. . . . . 6 ⊢ Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
30		relcnv 6122	. . . . . 6 ⊢ Rel ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)
31		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑘𝜑
32		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑘⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
33		nfiu1 5027	. . . . . . . . . . 11 ⊢ Ⅎ𝑘∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)
34	33	nfcnv 5889	. . . . . . . . . 10 ⊢ Ⅎ𝑘◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)
35	34	nfel2 2924	. . . . . . . . 9 ⊢ Ⅎ𝑘⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)
36	32, 35	nfbi 1903	. . . . . . . 8 ⊢ Ⅎ𝑘(⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸))
37	31, 36	nfim 1896	. . . . . . 7 ⊢ Ⅎ𝑘(𝜑 → (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
38		opeq2 4874	. . . . . . . . . 10 ⊢ (𝑘 = 𝑦 → ⟨𝑥, 𝑘⟩ = ⟨𝑥, 𝑦⟩)
39	38	eleq1d 2826	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → (⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)))
40	38	eleq1d 2826	. . . . . . . . 9 ⊢ (𝑘 = 𝑦 → (⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
41	39, 40	bibi12d 345	. . . . . . . 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 340	. . . . . . 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 1914	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
44		nfiu1 5027	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
45	44	nfel2 2924	. . . . . . . . . 10 ⊢ Ⅎ𝑗⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
46		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑗⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)
47	45, 46	nfbi 1903	. . . . . . . . 9 ⊢ Ⅎ𝑗(⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸))
48	43, 47	nfim 1896	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 → (⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
49		opeq1 4873	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑥 → ⟨𝑗, 𝑘⟩ = ⟨𝑥, 𝑘⟩)
50	49	eleq1d 2826	. . . . . . . . . 10 ⊢ (𝑗 = 𝑥 → (⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)))
51	49	eleq1d 2826	. . . . . . . . . 10 ⊢ (𝑗 = 𝑥 → (⟨𝑗, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↔ ⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
52	50, 51	bibi12d 345	. . . . . . . . 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 340	. . . . . . . 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 5752	. . . . . . . . . 10 ⊢ (⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶))
56		opeliunxp 5752	. . . . . . . . . 10 ⊢ (⟨𝑘, 𝑗⟩ ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↔ (𝑘 ∈ 𝐷 ∧ 𝑗 ∈ 𝐸))
57	54, 55, 56	3bitr4g 314	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑘, 𝑗⟩ ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
58		vex 3484	. . . . . . . . . 10 ⊢ 𝑗 ∈ V
59		vex 3484	. . . . . . . . . 10 ⊢ 𝑘 ∈ V
60	58, 59	opelcnv 5892	. . . . . . . . 9 ⊢ (⟨𝑗, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↔ ⟨𝑘, 𝑗⟩ ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸))
61	57, 60	bitr4di 289	. . . . . . . 8 ⊢ (𝜑 → (⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑗, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
62	48, 53, 61	chvarfv 2240	. . . . . . 7 ⊢ (𝜑 → (⟨𝑥, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑘⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
63	37, 42, 62	chvarfv 2240	. . . . . 6 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)))
64	29, 30, 63	eqrelrdv 5802	. . . . 5 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) = ◡∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸))
65	64	f1oeq3d 6845	. . . 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)))
66	25, 65	mpbiri 258	. . 3 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}):∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)–1-1-onto→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
67	1, 2, 3, 11, 16, 19, 66	gsumf1o 19934	. 2 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)) = (𝐺 Σg ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∘ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}))))
68		sneq 4636	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → {𝑧} = {⟨𝑥, 𝑦⟩})
69	68	cnveqd 5886	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ◡{𝑧} = ◡{⟨𝑥, 𝑦⟩})
70	69	unieqd 4920	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ∪ ◡{𝑧} = ∪ ◡{⟨𝑥, 𝑦⟩})
71		opswap 6249	. . . . . . . . 9 ⊢ ∪ ◡{⟨𝑥, 𝑦⟩} = ⟨𝑦, 𝑥⟩
72	70, 71	eqtrdi 2793	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ∪ ◡{𝑧} = ⟨𝑦, 𝑥⟩)
73	72	fveq2d 6910	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘∪ ◡{𝑧}) = ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘⟨𝑦, 𝑥⟩))
74		df-ov 7434	. . . . . . 7 ⊢ (𝑦(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥) = ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘⟨𝑦, 𝑥⟩)
75	73, 74	eqtr4di 2795	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘∪ ◡{𝑧}) = (𝑦(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥))
76	75	mpomptx 7546	. . . . 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))
77		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑥({𝑘} × 𝐸)
78		nfcv 2905	. . . . . . . 8 ⊢ Ⅎ𝑘{𝑥}
79		nfcsb1v 3923	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐸
80	78, 79	nfxp 5718	. . . . . . 7 ⊢ Ⅎ𝑘({𝑥} × ⦋𝑥 / 𝑘⦌𝐸)
81		sneq 4636	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → {𝑘} = {𝑥})
82		csbeq1a 3913	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → 𝐸 = ⦋𝑥 / 𝑘⦌𝐸)
83	81, 82	xpeq12d 5716	. . . . . . 7 ⊢ (𝑘 = 𝑥 → ({𝑘} × 𝐸) = ({𝑥} × ⦋𝑥 / 𝑘⦌𝐸))
84	77, 80, 83	cbviun 5036	. . . . . 6 ⊢ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) = ∪ 𝑥 ∈ 𝐷 ({𝑥} × ⦋𝑥 / 𝑘⦌𝐸)
85	84	mpteq1i 5238	. . . . 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}))
86		nfcv 2905	. . . . . 6 ⊢ Ⅎ𝑥𝐸
87		nfcv 2905	. . . . . 6 ⊢ Ⅎ𝑥(𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘)
88		nfcv 2905	. . . . . 6 ⊢ Ⅎ𝑦(𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘)
89		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑘𝑦
90		nfmpo2 7514	. . . . . . 7 ⊢ Ⅎ𝑘(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
91		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑘𝑥
92	89, 90, 91	nfov 7461	. . . . . 6 ⊢ Ⅎ𝑘(𝑦(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥)
93		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑗𝑦
94		nfmpo1 7513	. . . . . . 7 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
95		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑗𝑥
96	93, 94, 95	nfov 7461	. . . . . 6 ⊢ Ⅎ𝑗(𝑦(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥)
97		oveq2 7439	. . . . . . 7 ⊢ (𝑘 = 𝑥 → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥))
98		oveq1 7438	. . . . . . 7 ⊢ (𝑗 = 𝑦 → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥) = (𝑦(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥))
99	97, 98	sylan9eq 2797	. . . . . 6 ⊢ ((𝑘 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = (𝑦(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑥))
100	86, 79, 87, 88, 92, 96, 82, 99	cbvmpox 7526	. . . . 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))
101	76, 85, 100	3eqtr4i 2775	. . . 4 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘∪ ◡{𝑧})) = (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘))
102		f1of 6848	. . . . . . 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))
103	66, 102	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}):∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)⟶∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
104		eqid 2737	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}) = (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧})
105	104	fmpt 7130	. . . . . 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))
106	103, 105	sylibr 234	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸)∪ ◡{𝑧} ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
107		eqidd 2738	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}) = (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}))
108	16	feqmptd 6977	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↦ ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑥)))
109		fveq2 6906	. . . . 5 ⊢ (𝑥 = ∪ ◡{𝑧} → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑥) = ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘∪ ◡{𝑧}))
110	106, 107, 108, 109	fmptcof 7150	. . . 4 ⊢ (𝜑 → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∘ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧})) = (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘∪ ◡{𝑧})))
111	12	ex 412	. . . . . . . . 9 ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) → 𝑋 ∈ 𝐵))
112	14	ovmpt4g 7580	. . . . . . . . . 10 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋)
113	112	3expia 1122	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) → (𝑋 ∈ 𝐵 → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋))
114	111, 113	sylcom 30	. . . . . . . 8 ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋))
115	54, 114	sylbird 260	. . . . . . 7 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ∧ 𝑗 ∈ 𝐸) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋))
116	115	3impib 1117	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷 ∧ 𝑗 ∈ 𝐸) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋)
117	116	eqcomd 2743	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷 ∧ 𝑗 ∈ 𝐸) → 𝑋 = (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘))
118	117	mpoeq3dva 7510	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ 𝑋) = (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘)))
119	101, 110, 118	3eqtr4a 2803	. . 3 ⊢ (𝜑 → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∘ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧})) = (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ 𝑋))
120	119	oveq2d 7447	. 2 ⊢ (𝜑 → (𝐺 Σg ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∘ (𝑧 ∈ ∪ 𝑘 ∈ 𝐷 ({𝑘} × 𝐸) ↦ ∪ ◡{𝑧}))) = (𝐺 Σg (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ 𝑋)))
121	67, 120	eqtrd 2777	1 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  Vcvv 3480  ⦋csb 3899  {csn 4626  ⟨cop 4632  ∪ cuni 4907  ∪ ciun 4991   class class class wbr 5143   ↦ cmpt 5225   × cxp 5683  ^◡ccnv 5684   ∘ ccom 5689  Rel wrel 5690  ⟶wf 6557  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  Fincfn 8985  Basecbs 17247  0_gc0g 17484   Σ_g cgsu 17485  CMndccmn 19798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-0g 17486  df-gsum 17487  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-cntz 19335  df-cmn 19800

This theorem is referenced by:  gsumcom  19995  gsumbagdiag  21951