Theorem fsumcom2 14505

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 5227	. . . . . . . . 9 ⊢ Rel ({𝑗} × 𝐵)
2	1	rgenw 2924	. . . . . . . 8 ⊢ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐵)
3		reliun 5239	. . . . . . . 8 ⊢ (Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐵))
4	2, 3	mpbir 221	. . . . . . 7 ⊢ Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
5		relcnv 5503	. . . . . . 7 ⊢ Rel ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)
6		ancom 466	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑗 ∧ 𝑦 = 𝑘) ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗))
7		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
9	7, 8	opth 4945	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ (𝑥 = 𝑗 ∧ 𝑦 = 𝑘))
10	8, 7	opth 4945	. . . . . . . . . . . 12 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗))
11	6, 9, 10	3bitr4i 292	. . . . . . . . . . 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 747	. . . . . . . . 9 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))))
15	14	2exbidv 1852	. . . . . . . 8 ⊢ (𝜑 → (∃𝑗∃𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))))
16		eliunxp 5259	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗∃𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)))
17	7, 8	opelcnv 5304	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
18		eliunxp 5259	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘∃𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
19		excom 2042	. . . . . . . . 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 286	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
21	15, 16, 20	3bitr4g 303	. . . . . . 7 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)))
22	4, 5, 21	eqrelrdv 5216	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
23		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑚({𝑗} × 𝐵)
24		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑗{𝑚}
25		nfcsb1v 3549	. . . . . . . 8 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐵
26	24, 25	nfxp 5142	. . . . . . 7 ⊢ Ⅎ𝑗({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)
27		sneq 4187	. . . . . . . 8 ⊢ (𝑗 = 𝑚 → {𝑗} = {𝑚})
28		csbeq1a 3542	. . . . . . . 8 ⊢ (𝑗 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑗⦌𝐵)
29	27, 28	xpeq12d 5140	. . . . . . 7 ⊢ (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵))
30	23, 26, 29	cbviun 4557	. . . . . 6 ⊢ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ∪ 𝑚 ∈ 𝐴 ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)
31		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑛({𝑘} × 𝐷)
32		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑘{𝑛}
33		nfcsb1v 3549	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐷
34	32, 33	nfxp 5142	. . . . . . . 8 ⊢ Ⅎ𝑘({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)
35		sneq 4187	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → {𝑘} = {𝑛})
36		csbeq1a 3542	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → 𝐷 = ⦋𝑛 / 𝑘⦌𝐷)
37	35, 36	xpeq12d 5140	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ({𝑘} × 𝐷) = ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
38	31, 34, 37	cbviun 4557	. . . . . . 7 ⊢ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)
39	38	cnveqi 5297	. . . . . 6 ⊢ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ◡∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)
40	22, 30, 39	3eqtr3g 2679	. . . . 5 ⊢ (𝜑 → ∪ 𝑚 ∈ 𝐴 ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ◡∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
41	40	sumeq1d 14431	. . . 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 3203	. . . . . . . 8 ⊢ 𝑛 ∈ V
43		vex 3203	. . . . . . . 8 ⊢ 𝑚 ∈ V
44	42, 43	op1std 7178	. . . . . . 7 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → (1st ‘𝑤) = 𝑛)
45	44	csbeq1d 3540	. . . . . 6 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
46	42, 43	op2ndd 7179	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → (2nd ‘𝑤) = 𝑚)
47	46	csbeq1d 3540	. . . . . . 7 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → ⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑚 / 𝑗⦌𝐸)
48	47	csbeq2dv 3992	. . . . . 6 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → ⦋𝑛 / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
49	45, 48	eqtrd 2656	. . . . 5 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
50	43, 42	op2ndd 7179	. . . . . . 7 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd ‘𝑧) = 𝑛)
51	50	csbeq1d 3540	. . . . . 6 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸)
52	43, 42	op1std 7178	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → (1st ‘𝑧) = 𝑚)
53	52	csbeq1d 3540	. . . . . . 7 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → ⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑚 / 𝑗⦌𝐸)
54	53	csbeq2dv 3992	. . . . . 6 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → ⦋𝑛 / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
55	51, 54	eqtrd 2656	. . . . 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 8038	. . . . . . . 8 ⊢ {𝑛} ∈ Fin
58		fsumcom2.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ Fin)
59	58	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐴 ∈ Fin)
60	43, 42	opelcnv 5304	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑚, 𝑛⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑛, 𝑚⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
61	33, 36	opeliunxp2f 7336	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑛, 𝑚⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷))
62	60, 61	sylbbr 226	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷) → ⟨𝑚, 𝑛⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
63	62	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ⟨𝑚, 𝑛⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
64	22	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
65	63, 64	eleqtrrd 2704	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ⟨𝑚, 𝑛⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
66		eliun 4524	. . . . . . . . . . . . 13 ⊢ (⟨𝑚, 𝑛⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
67	65, 66	sylib 208	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ∃𝑗 ∈ 𝐴 ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
68		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
69		opelxp 5146	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) ↔ (𝑚 ∈ {𝑗} ∧ 𝑛 ∈ 𝐵))
70	68, 69	sylib 208	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → (𝑚 ∈ {𝑗} ∧ 𝑛 ∈ 𝐵))
71	70	simpld 475	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 ∈ {𝑗})
72		elsni 4194	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
73	71, 72	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 = 𝑗)
74		simpl 473	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑗 ∈ 𝐴)
75	73, 74	eqeltrd 2701	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 ∈ 𝐴)
76	75	rexlimiva 3028	. . . . . . . . . . . 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 643	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷 → 𝑚 ∈ 𝐴))
79	78	ssrdv 3609	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ⦋𝑛 / 𝑘⦌𝐷 ⊆ 𝐴)
80	59, 79	ssfid 8183	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ⦋𝑛 / 𝑘⦌𝐷 ∈ Fin)
81		xpfi 8231	. . . . . . . 8 ⊢ (({𝑛} ∈ Fin ∧ ⦋𝑛 / 𝑘⦌𝐷 ∈ Fin) → ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin)
82	57, 80, 81	sylancr 695	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin)
83	82	ralrimiva 2966	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin)
84		iunfi 8254	. . . . . 6 ⊢ ((𝐶 ∈ Fin ∧ ∀𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin) → ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin)
85	56, 83, 84	syl2anc 693	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin)
86		reliun 5239	. . . . . . 7 ⊢ (Rel ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ↔ ∀𝑛 ∈ 𝐶 Rel ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
87		relxp 5227	. . . . . . . 8 ⊢ Rel ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)
88	87	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ 𝐶 → Rel ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
89	86, 88	mprgbir 2927	. . . . . 6 ⊢ Rel ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)
90	89	a1i 11	. . . . 5 ⊢ (𝜑 → Rel ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
91		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
92		eliun 4524	. . . . . . . 8 ⊢ (𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ↔ ∃𝑛 ∈ 𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
93	91, 92	sylib 208	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ∃𝑛 ∈ 𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
94		xp2nd 7199	. . . . . . . . . 10 ⊢ (𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (2nd ‘𝑤) ∈ ⦋𝑛 / 𝑘⦌𝐷)
95	94	adantl 482	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋𝑛 / 𝑘⦌𝐷)
96		xp1st 7198	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (1st ‘𝑤) ∈ {𝑛})
97	96	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ {𝑛})
98		elsni 4194	. . . . . . . . . . 11 ⊢ ((1st ‘𝑤) ∈ {𝑛} → (1st ‘𝑤) = 𝑛)
99	97, 98	syl 17	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) = 𝑛)
100	99	csbeq1d 3540	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ⦋(1st ‘𝑤) / 𝑘⦌𝐷 = ⦋𝑛 / 𝑘⦌𝐷)
101	95, 100	eleqtrrd 2704	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
102	101	rexlimiva 3028	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
103	93, 102	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
104		simpl 473	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → 𝑛 ∈ 𝐶)
105	99, 104	eqeltrd 2701	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ 𝐶)
106	105	rexlimiva 3028	. . . . . . . 8 ⊢ (∃𝑛 ∈ 𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (1st ‘𝑤) ∈ 𝐶)
107	93, 106	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ 𝐶)
108		simpl 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → 𝜑)
109	25	nfcri 2758	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵
110	72	equcomd 1946	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ {𝑗} → 𝑗 = 𝑚)
111	110, 28	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ {𝑗} → 𝐵 = ⦋𝑚 / 𝑗⦌𝐵)
112	111	eleq2d 2687	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ {𝑗} → (𝑛 ∈ 𝐵 ↔ 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵))
113	112	biimpa 501	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ {𝑗} ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)
114	69, 113	sylbi 207	. . . . . . . . . . . . 13 ⊢ (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)
115	114	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝐴 → (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵))
116	109, 115	rexlimi 3024	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ 𝐴 ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)
117	67, 116	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)
118		fsumcom2.5	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐸 ∈ ℂ)
119	118	ralrimivva 2971	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ)
120		nfcsb1v 3549	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐸
121	120	nfel1 2779	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ
122	25, 121	nfral 2945	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ
123		csbeq1a 3542	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑚 → 𝐸 = ⦋𝑚 / 𝑗⦌𝐸)
124	123	eleq1d 2686	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑚 → (𝐸 ∈ ℂ ↔ ⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
125	28, 124	raleqbidv 3152	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑚 → (∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ ↔ ∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
126	122, 125	rspc 3303	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ → ∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
127	119, 126	mpan9 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
128		nfcsb1v 3549	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
129	128	nfel1 2779	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ
130		csbeq1a 3542	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ⦋𝑚 / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
131	130	eleq1d 2686	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
132	129, 131	rspc 3303	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵 → (∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
133	127, 132	syl5com 31	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵 → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
134	133	impr 649	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)) → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
135	108, 77, 117, 134	syl12anc 1324	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
136	135	ralrimivva 2971	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ 𝐶 ∀𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
137	136	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ∀𝑛 ∈ 𝐶 ∀𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
138		csbeq1 3536	. . . . . . . . 9 ⊢ (𝑛 = (1st ‘𝑤) → ⦋𝑛 / 𝑘⦌𝐷 = ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
139		csbeq1 3536	. . . . . . . . . 10 ⊢ (𝑛 = (1st ‘𝑤) → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
140	139	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑛 = (1st ‘𝑤) → (⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
141	138, 140	raleqbidv 3152	. . . . . . . 8 ⊢ (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))
142	141	rspcv 3305	. . . . . . 7 ⊢ ((1st ‘𝑤) ∈ 𝐶 → (∀𝑛 ∈ 𝐶 ∀𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ → ∀𝑚 ∈ ⦋ (1st ‘𝑤) / 𝑘⦌𝐷⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
143	107, 137, 142	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ∀𝑚 ∈ ⦋ (1st ‘𝑤) / 𝑘⦌𝐷⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
144		csbeq1 3536	. . . . . . . . 9 ⊢ (𝑚 = (2nd ‘𝑤) → ⦋𝑚 / 𝑗⦌𝐸 = ⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
145	144	csbeq2dv 3992	. . . . . . . 8 ⊢ (𝑚 = (2nd ‘𝑤) → ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
146	145	eleq1d 2686	. . . . . . 7 ⊢ (𝑚 = (2nd ‘𝑤) → (⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ))
147	146	rspcv 3305	. . . . . 6 ⊢ ((2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷 → (∀𝑚 ∈ ⦋ (1st ‘𝑤) / 𝑘⦌𝐷⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ))
148	103, 143, 147	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ)
149	49, 55, 85, 90, 148	fsumcnv 14504	. . . 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)
150	41, 149	eqtr4d 2659	. . 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)
151		fsumcom2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
152	151	ralrimiva 2966	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ Fin)
153	25	nfel1 2779	. . . . . 6 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐵 ∈ Fin
154	28	eleq1d 2686	. . . . . 6 ⊢ (𝑗 = 𝑚 → (𝐵 ∈ Fin ↔ ⦋𝑚 / 𝑗⦌𝐵 ∈ Fin))
155	153, 154	rspc 3303	. . . . 5 ⊢ (𝑚 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 𝐵 ∈ Fin → ⦋𝑚 / 𝑗⦌𝐵 ∈ Fin))
156	152, 155	mpan9 486	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ⦋𝑚 / 𝑗⦌𝐵 ∈ Fin)
157	55, 58, 156, 134	fsum2d 14502	. . 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)
158	49, 56, 80, 135	fsum2d 14502	. . 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)
159	150, 157, 158	3eqtr4d 2666	. 2 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = Σ𝑛 ∈ 𝐶 Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
160		nfcv 2764	. . 3 ⊢ Ⅎ𝑚Σ𝑘 ∈ 𝐵 𝐸
161		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑗𝑛
162	161, 120	nfcsb 3551	. . . 4 ⊢ Ⅎ𝑗⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
163	25, 162	nfsum 14421	. . 3 ⊢ Ⅎ𝑗Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
164		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑛𝐸
165		nfcsb1v 3549	. . . . 5 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐸
166		csbeq1a 3542	. . . . 5 ⊢ (𝑘 = 𝑛 → 𝐸 = ⦋𝑛 / 𝑘⦌𝐸)
167	164, 165, 166	cbvsumi 14427	. . . 4 ⊢ Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑛 ∈ 𝐵 ⦋𝑛 / 𝑘⦌𝐸
168	123	csbeq2dv 3992	. . . . . 6 ⊢ (𝑗 = 𝑚 → ⦋𝑛 / 𝑘⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
169	168	adantr 481	. . . . 5 ⊢ ((𝑗 = 𝑚 ∧ 𝑛 ∈ 𝐵) → ⦋𝑛 / 𝑘⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
170	28, 169	sumeq12dv 14437	. . . 4 ⊢ (𝑗 = 𝑚 → Σ𝑛 ∈ 𝐵 ⦋𝑛 / 𝑘⦌𝐸 = Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
171	167, 170	syl5eq 2668	. . 3 ⊢ (𝑗 = 𝑚 → Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
172	160, 163, 171	cbvsumi 14427	. 2 ⊢ Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑚 ∈ 𝐴 Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
173		nfcv 2764	. . 3 ⊢ Ⅎ𝑛Σ𝑗 ∈ 𝐷 𝐸
174	33, 128	nfsum 14421	. . 3 ⊢ Ⅎ𝑘Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
175		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑚𝐸
176	175, 120, 123	cbvsumi 14427	. . . 4 ⊢ Σ𝑗 ∈ 𝐷 𝐸 = Σ𝑚 ∈ 𝐷 ⦋𝑚 / 𝑗⦌𝐸
177	130	adantr 481	. . . . 5 ⊢ ((𝑘 = 𝑛 ∧ 𝑚 ∈ 𝐷) → ⦋𝑚 / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
178	36, 177	sumeq12dv 14437	. . . 4 ⊢ (𝑘 = 𝑛 → Σ𝑚 ∈ 𝐷 ⦋𝑚 / 𝑗⦌𝐸 = Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
179	176, 178	syl5eq 2668	. . 3 ⊢ (𝑘 = 𝑛 → Σ𝑗 ∈ 𝐷 𝐸 = Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
180	173, 174, 179	cbvsumi 14427	. 2 ⊢ Σ𝑘 ∈ 𝐶 Σ𝑗 ∈ 𝐷 𝐸 = Σ𝑛 ∈ 𝐶 Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
181	159, 172, 180	3eqtr4g 2681	1 ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑘 ∈ 𝐶 Σ𝑗 ∈ 𝐷 𝐸)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  ⦋csb 3533  {csn 4177  ⟨cop 4183  ∪ ciun 4520   × cxp 5112  ^◡ccnv 5113  Rel wrel 5119  ‘cfv 5888  1^st c1st 7166  2^nd c2nd 7167  Fincfn 7955  ℂcc 9934  Σcsu 14416

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-inf2 8538  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  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-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-sup 8348  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  fsumcom  14507  fsum0diag  14509  fsumdvdsdiag  24910  dvdsflsumcom  24914  fsumfldivdiag  24916  logfac2  24942  chpchtsum  24944  logfaclbnd  24947  dchrisum0lem1  25205