Theorem fsumcom2 15747

Description: Interchange order of summation. Note that 𝐵(𝑗) and 𝐷(𝑘) are not necessarily constant expressions. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.) (Proof shortened by JJ, 2-Aug-2021.)

Hypotheses

Ref	Expression
fsumcom2.1	⊢ (𝜑 → 𝐴 ∈ Fin)
fsumcom2.2	⊢ (𝜑 → 𝐶 ∈ Fin)
fsumcom2.3	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
fsumcom2.4	⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
fsumcom2.5	⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐸 ∈ ℂ)

Assertion

Ref	Expression
fsumcom2	⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑘 ∈ 𝐶 Σ𝑗 ∈ 𝐷 𝐸)

Distinct variable groups:   𝑗,𝑘,𝐴   𝐶,𝑗,𝑘   𝜑,𝑗,𝑘   𝐵,𝑘   𝐷,𝑗

Allowed substitution hints:   𝐵(𝑗)   𝐷(𝑘)   𝐸(𝑗,𝑘)

Proof of Theorem fsumcom2

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

Step	Hyp	Ref	Expression
1		relxp 5659	. . . . . . . . 9 ⊢ Rel ({𝑗} × 𝐵)
2	1	rgenw 3049	. . . . . . . 8 ⊢ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐵)
3		reliun 5782	. . . . . . . 8 ⊢ (Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐵))
4	2, 3	mpbir 231	. . . . . . 7 ⊢ Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
5		relcnv 6078	. . . . . . 7 ⊢ Rel ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)
6		ancom 460	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑗 ∧ 𝑦 = 𝑘) ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗))
7		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
9	7, 8	opth 5439	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ (𝑥 = 𝑗 ∧ 𝑦 = 𝑘))
10	8, 7	opth 5439	. . . . . . . . . . . 12 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗))
11	6, 9, 10	3bitr4i 303	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩)
12	11	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩))
13		fsumcom2.4	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
14	12, 13	anbi12d 632	. . . . . . . . 9 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))))
15	14	2exbidv 1924	. . . . . . . 8 ⊢ (𝜑 → (∃𝑗∃𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))))
16		eliunxp 5804	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗∃𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)))
17	7, 8	opelcnv 5848	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
18		eliunxp 5804	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘∃𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
19		excom 2163	. . . . . . . . 9 ⊢ (∃𝑘∃𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
20	17, 18, 19	3bitri 297	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
21	15, 16, 20	3bitr4g 314	. . . . . . 7 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)))
22	4, 5, 21	eqrelrdv 5758	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
23		nfcv 2892	. . . . . . 7 ⊢ Ⅎ𝑚({𝑗} × 𝐵)
24		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑗{𝑚}
25		nfcsb1v 3889	. . . . . . . 8 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐵
26	24, 25	nfxp 5674	. . . . . . 7 ⊢ Ⅎ𝑗({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)
27		sneq 4602	. . . . . . . 8 ⊢ (𝑗 = 𝑚 → {𝑗} = {𝑚})
28		csbeq1a 3879	. . . . . . . 8 ⊢ (𝑗 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑗⦌𝐵)
29	27, 28	xpeq12d 5672	. . . . . . 7 ⊢ (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵))
30	23, 26, 29	cbviun 5003	. . . . . 6 ⊢ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ∪ 𝑚 ∈ 𝐴 ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)
31		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑛({𝑘} × 𝐷)
32		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎ𝑘{𝑛}
33		nfcsb1v 3889	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐷
34	32, 33	nfxp 5674	. . . . . . . 8 ⊢ Ⅎ𝑘({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)
35		sneq 4602	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → {𝑘} = {𝑛})
36		csbeq1a 3879	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → 𝐷 = ⦋𝑛 / 𝑘⦌𝐷)
37	35, 36	xpeq12d 5672	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ({𝑘} × 𝐷) = ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
38	31, 34, 37	cbviun 5003	. . . . . . 7 ⊢ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)
39	38	cnveqi 5841	. . . . . 6 ⊢ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ◡∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)
40	22, 30, 39	3eqtr3g 2788	. . . . 5 ⊢ (𝜑 → ∪ 𝑚 ∈ 𝐴 ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ◡∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
41	40	sumeq1d 15673	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ ∪ 𝑚 ∈ 𝐴 ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = Σ𝑧 ∈ ◡ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸)
42		vex 3454	. . . . . . . 8 ⊢ 𝑛 ∈ V
43		vex 3454	. . . . . . . 8 ⊢ 𝑚 ∈ V
44	42, 43	op1std 7981	. . . . . . 7 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → (1st ‘𝑤) = 𝑛)
45	44	csbeq1d 3869	. . . . . 6 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
46	42, 43	op2ndd 7982	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → (2nd ‘𝑤) = 𝑚)
47	46	csbeq1d 3869	. . . . . . 7 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → ⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑚 / 𝑗⦌𝐸)
48	47	csbeq2dv 3872	. . . . . 6 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → ⦋𝑛 / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
49	45, 48	eqtrd 2765	. . . . 5 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
50	43, 42	op2ndd 7982	. . . . . . 7 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd ‘𝑧) = 𝑛)
51	50	csbeq1d 3869	. . . . . 6 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸)
52	43, 42	op1std 7981	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → (1st ‘𝑧) = 𝑚)
53	52	csbeq1d 3869	. . . . . . 7 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → ⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑚 / 𝑗⦌𝐸)
54	53	csbeq2dv 3872	. . . . . 6 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → ⦋𝑛 / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
55	51, 54	eqtrd 2765	. . . . 5 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
56		fsumcom2.2	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Fin)
57		snfi 9017	. . . . . . . 8 ⊢ {𝑛} ∈ Fin
58		fsumcom2.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ Fin)
59	58	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐴 ∈ Fin)
60	43, 42	opelcnv 5848	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑚, 𝑛⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑛, 𝑚⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
61	33, 36	opeliunxp2f 8192	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑛, 𝑚⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷))
62	60, 61	sylbbr 236	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷) → ⟨𝑚, 𝑛⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
63	62	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ⟨𝑚, 𝑛⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
64	22	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
65	63, 64	eleqtrrd 2832	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ⟨𝑚, 𝑛⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
66		eliun 4962	. . . . . . . . . . . . 13 ⊢ (⟨𝑚, 𝑛⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
67	65, 66	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ∃𝑗 ∈ 𝐴 ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
68		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
69		opelxp 5677	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) ↔ (𝑚 ∈ {𝑗} ∧ 𝑛 ∈ 𝐵))
70	68, 69	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → (𝑚 ∈ {𝑗} ∧ 𝑛 ∈ 𝐵))
71	70	simpld 494	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 ∈ {𝑗})
72		elsni 4609	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
73	71, 72	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 = 𝑗)
74		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑗 ∈ 𝐴)
75	73, 74	eqeltrd 2829	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 ∈ 𝐴)
76	75	rexlimiva 3127	. . . . . . . . . . . 12 ⊢ (∃𝑗 ∈ 𝐴 ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑚 ∈ 𝐴)
77	67, 76	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → 𝑚 ∈ 𝐴)
78	77	expr 456	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷 → 𝑚 ∈ 𝐴))
79	78	ssrdv 3955	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ⦋𝑛 / 𝑘⦌𝐷 ⊆ 𝐴)
80	59, 79	ssfid 9219	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ⦋𝑛 / 𝑘⦌𝐷 ∈ Fin)
81		xpfi 9276	. . . . . . . 8 ⊢ (({𝑛} ∈ Fin ∧ ⦋𝑛 / 𝑘⦌𝐷 ∈ Fin) → ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin)
82	57, 80, 81	sylancr 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin)
83	82	ralrimiva 3126	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin)
84		iunfi 9301	. . . . . 6 ⊢ ((𝐶 ∈ Fin ∧ ∀𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin) → ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin)
85	56, 83, 84	syl2anc 584	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin)
86		reliun 5782	. . . . . . 7 ⊢ (Rel ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ↔ ∀𝑛 ∈ 𝐶 Rel ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
87		relxp 5659	. . . . . . . 8 ⊢ Rel ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)
88	87	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ 𝐶 → Rel ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
89	86, 88	mprgbir 3052	. . . . . 6 ⊢ Rel ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)
90	89	a1i 11	. . . . 5 ⊢ (𝜑 → Rel ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
91		csbeq1 3868	. . . . . . . 8 ⊢ (𝑚 = (2nd ‘𝑤) → ⦋𝑚 / 𝑗⦌𝐸 = ⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
92	91	csbeq2dv 3872	. . . . . . 7 ⊢ (𝑚 = (2nd ‘𝑤) → ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
93	92	eleq1d 2814	. . . . . 6 ⊢ (𝑚 = (2nd ‘𝑤) → (⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ))
94		csbeq1 3868	. . . . . . . 8 ⊢ (𝑛 = (1st ‘𝑤) → ⦋𝑛 / 𝑘⦌𝐷 = ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
95		csbeq1 3868	. . . . . . . . 9 ⊢ (𝑛 = (1st ‘𝑤) → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
96	95	eleq1d 2814	. . . . . . . 8 ⊢ (𝑛 = (1st ‘𝑤) → (⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
97	94, 96	raleqbidv 3321	. . . . . . 7 ⊢ (𝑛 = (1st ‘𝑤) → (∀𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ ↔ ∀𝑚 ∈ ⦋ (1st ‘𝑤) / 𝑘⦌𝐷⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
98		simpl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → 𝜑)
99	25	nfcri 2884	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵
100	72	equcomd 2019	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ {𝑗} → 𝑗 = 𝑚)
101	100, 28	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ {𝑗} → 𝐵 = ⦋𝑚 / 𝑗⦌𝐵)
102	101	eleq2d 2815	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ {𝑗} → (𝑛 ∈ 𝐵 ↔ 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵))
103	102	biimpa 476	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ {𝑗} ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)
104	69, 103	sylbi 217	. . . . . . . . . . . . 13 ⊢ (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)
105	104	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝐴 → (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵))
106	99, 105	rexlimi 3238	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ 𝐴 ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)
107	67, 106	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)
108		fsumcom2.5	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐸 ∈ ℂ)
109	108	ralrimivva 3181	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ)
110		nfcsb1v 3889	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐸
111	110	nfel1 2909	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ
112	25, 111	nfralw 3287	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ
113		csbeq1a 3879	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑚 → 𝐸 = ⦋𝑚 / 𝑗⦌𝐸)
114	113	eleq1d 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑚 → (𝐸 ∈ ℂ ↔ ⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
115	28, 114	raleqbidv 3321	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑚 → (∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ ↔ ∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
116	112, 115	rspc 3579	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ → ∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
117	109, 116	mpan9 506	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
118		nfcsb1v 3889	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
119	118	nfel1 2909	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ
120		csbeq1a 3879	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ⦋𝑚 / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
121	120	eleq1d 2814	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
122	119, 121	rspc 3579	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵 → (∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
123	117, 122	syl5com 31	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵 → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
124	123	impr 454	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)) → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
125	98, 77, 107, 124	syl12anc 836	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
126	125	ralrimivva 3181	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ 𝐶 ∀𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
127	126	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ∀𝑛 ∈ 𝐶 ∀𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
128		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
129		eliun 4962	. . . . . . . . 9 ⊢ (𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ↔ ∃𝑛 ∈ 𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
130	128, 129	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ∃𝑛 ∈ 𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
131		xp1st 8003	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (1st ‘𝑤) ∈ {𝑛})
132	131	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ {𝑛})
133		elsni 4609	. . . . . . . . . . 11 ⊢ ((1st ‘𝑤) ∈ {𝑛} → (1st ‘𝑤) = 𝑛)
134	132, 133	syl 17	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) = 𝑛)
135		simpl 482	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → 𝑛 ∈ 𝐶)
136	134, 135	eqeltrd 2829	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ 𝐶)
137	136	rexlimiva 3127	. . . . . . . 8 ⊢ (∃𝑛 ∈ 𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (1st ‘𝑤) ∈ 𝐶)
138	130, 137	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ 𝐶)
139	97, 127, 138	rspcdva 3592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ∀𝑚 ∈ ⦋ (1st ‘𝑤) / 𝑘⦌𝐷⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
140		xp2nd 8004	. . . . . . . . . 10 ⊢ (𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (2nd ‘𝑤) ∈ ⦋𝑛 / 𝑘⦌𝐷)
141	140	adantl 481	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋𝑛 / 𝑘⦌𝐷)
142	134	csbeq1d 3869	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ⦋(1st ‘𝑤) / 𝑘⦌𝐷 = ⦋𝑛 / 𝑘⦌𝐷)
143	141, 142	eleqtrrd 2832	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
144	143	rexlimiva 3127	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
145	130, 144	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
146	93, 139, 145	rspcdva 3592	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ)
147	49, 55, 85, 90, 146	fsumcnv 15746	. . . 4 ⊢ (𝜑 → Σ𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = Σ𝑧 ∈ ◡ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸)
148	41, 147	eqtr4d 2768	. . 3 ⊢ (𝜑 → Σ𝑧 ∈ ∪ 𝑚 ∈ 𝐴 ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = Σ𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
149		fsumcom2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
150	149	ralrimiva 3126	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ Fin)
151	25	nfel1 2909	. . . . . 6 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐵 ∈ Fin
152	28	eleq1d 2814	. . . . . 6 ⊢ (𝑗 = 𝑚 → (𝐵 ∈ Fin ↔ ⦋𝑚 / 𝑗⦌𝐵 ∈ Fin))
153	151, 152	rspc 3579	. . . . 5 ⊢ (𝑚 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 𝐵 ∈ Fin → ⦋𝑚 / 𝑗⦌𝐵 ∈ Fin))
154	150, 153	mpan9 506	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ⦋𝑚 / 𝑗⦌𝐵 ∈ Fin)
155	55, 58, 154, 124	fsum2d 15744	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = Σ𝑧 ∈ ∪ 𝑚 ∈ 𝐴 ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸)
156	49, 56, 80, 125	fsum2d 15744	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ 𝐶 Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = Σ𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
157	148, 155, 156	3eqtr4d 2775	. 2 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = Σ𝑛 ∈ 𝐶 Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
158		csbeq1a 3879	. . . . 5 ⊢ (𝑘 = 𝑛 → 𝐸 = ⦋𝑛 / 𝑘⦌𝐸)
159		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑛𝐸
160		nfcsb1v 3889	. . . . 5 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐸
161	158, 159, 160	cbvsum 15668	. . . 4 ⊢ Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑛 ∈ 𝐵 ⦋𝑛 / 𝑘⦌𝐸
162	113	csbeq2dv 3872	. . . . . 6 ⊢ (𝑗 = 𝑚 → ⦋𝑛 / 𝑘⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
163	162	adantr 480	. . . . 5 ⊢ ((𝑗 = 𝑚 ∧ 𝑛 ∈ 𝐵) → ⦋𝑛 / 𝑘⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
164	28, 163	sumeq12dv 15679	. . . 4 ⊢ (𝑗 = 𝑚 → Σ𝑛 ∈ 𝐵 ⦋𝑛 / 𝑘⦌𝐸 = Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
165	161, 164	eqtrid 2777	. . 3 ⊢ (𝑗 = 𝑚 → Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
166		nfcv 2892	. . 3 ⊢ Ⅎ𝑚Σ𝑘 ∈ 𝐵 𝐸
167		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑗𝑛
168	167, 110	nfcsbw 3891	. . . 4 ⊢ Ⅎ𝑗⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
169	25, 168	nfsum 15664	. . 3 ⊢ Ⅎ𝑗Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
170	165, 166, 169	cbvsum 15668	. 2 ⊢ Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑚 ∈ 𝐴 Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
171		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑚𝐸
172	113, 171, 110	cbvsum 15668	. . . 4 ⊢ Σ𝑗 ∈ 𝐷 𝐸 = Σ𝑚 ∈ 𝐷 ⦋𝑚 / 𝑗⦌𝐸
173	120	adantr 480	. . . . 5 ⊢ ((𝑘 = 𝑛 ∧ 𝑚 ∈ 𝐷) → ⦋𝑚 / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
174	36, 173	sumeq12dv 15679	. . . 4 ⊢ (𝑘 = 𝑛 → Σ𝑚 ∈ 𝐷 ⦋𝑚 / 𝑗⦌𝐸 = Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
175	172, 174	eqtrid 2777	. . 3 ⊢ (𝑘 = 𝑛 → Σ𝑗 ∈ 𝐷 𝐸 = Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
176		nfcv 2892	. . 3 ⊢ Ⅎ𝑛Σ𝑗 ∈ 𝐷 𝐸
177	33, 118	nfsum 15664	. . 3 ⊢ Ⅎ𝑘Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
178	175, 176, 177	cbvsum 15668	. 2 ⊢ Σ𝑘 ∈ 𝐶 Σ𝑗 ∈ 𝐷 𝐸 = Σ𝑛 ∈ 𝐶 Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
179	157, 170, 178	3eqtr4g 2790	1 ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑘 ∈ 𝐶 Σ𝑗 ∈ 𝐷 𝐸)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  ⦋csb 3865  {csn 4592  ⟨cop 4598  ∪ ciun 4958   × cxp 5639  ^◡ccnv 5640  Rel wrel 5646  ‘cfv 6514  1^st c1st 7969  2^nd c2nd 7970  Fincfn 8921  ℂcc 11073  Σcsu 15659

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-rp 12959  df-fz 13476  df-fzo 13623  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-sum 15660

This theorem is referenced by:  fsumcom  15748  fsum0diag  15750  fsumdvdsdiag  27101  dvdsflsumcom  27105  fsumfldivdiag  27107  logfac2  27135  chpchtsum  27137  logfaclbnd  27140  dchrisum0lem1  27434