Theorem fsumcom2 15851

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 5684	. . . . . . . . 9 ⊢ Rel ({𝑗} × 𝐵)
2	1	rgenw 3086	. . . . . . . 8 ⊢ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐵)
3		reliun 5808	. . . . . . . 8 ⊢ (Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐵))
4	2, 3	mpbir 234	. . . . . . 7 ⊢ Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
5		relcnv 6111	. . . . . . 7 ⊢ Rel ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)
6		ancom 466	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑗 ∧ 𝑦 = 𝑘) ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗))
7		vex 3462	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8		vex 3462	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
9	7, 8	opth 5463	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ (𝑥 = 𝑗 ∧ 𝑦 = 𝑘))
10	8, 7	opth 5463	. . . . . . . . . . . 12 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗))
11	6, 9, 10	3bitr4i 306	. . . . . . . . . . 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 644	. . . . . . . . 9 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))))
15	14	2exbidv 1957	. . . . . . . 8 ⊢ (𝜑 → (∃𝑗∃𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))))
16		eliunxp 5828	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗∃𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)))
17	7, 8	opelcnv 5872	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
18		eliunxp 5828	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑥⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘∃𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
19		excom 2200	. . . . . . . . 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 300	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑗∃𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))
21	15, 16, 20	3bitr4g 317	. . . . . . 7 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)))
22	4, 5, 21	eqrelrdv 5783	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
23		nfcv 2928	. . . . . . 7 ⊢ Ⅎ𝑚({𝑗} × 𝐵)
24		nfcv 2928	. . . . . . . 8 ⊢ Ⅎ𝑗{𝑚}
25		nfcsb1v 3880	. . . . . . . 8 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐵
26	24, 25	nfxp 5699	. . . . . . 7 ⊢ Ⅎ𝑗({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)
27		sneq 4604	. . . . . . . 8 ⊢ (𝑗 = 𝑚 → {𝑗} = {𝑚})
28		csbeq1a 3870	. . . . . . . 8 ⊢ (𝑗 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑗⦌𝐵)
29	27, 28	xpeq12d 5697	. . . . . . 7 ⊢ (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵))
30	23, 26, 29	cbviun 5004	. . . . . 6 ⊢ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ∪ 𝑚 ∈ 𝐴 ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)
31		nfcv 2928	. . . . . . . 8 ⊢ Ⅎ𝑛({𝑘} × 𝐷)
32		nfcv 2928	. . . . . . . . 9 ⊢ Ⅎ𝑘{𝑛}
33		nfcsb1v 3880	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐷
34	32, 33	nfxp 5699	. . . . . . . 8 ⊢ Ⅎ𝑘({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)
35		sneq 4604	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → {𝑘} = {𝑛})
36		csbeq1a 3870	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → 𝐷 = ⦋𝑛 / 𝑘⦌𝐷)
37	35, 36	xpeq12d 5697	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ({𝑘} × 𝐷) = ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
38	31, 34, 37	cbviun 5004	. . . . . . 7 ⊢ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)
39	38	cnveqi 5865	. . . . . 6 ⊢ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ◡∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)
40	22, 30, 39	3eqtr3g 2824	. . . . 5 ⊢ (𝜑 → ∪ 𝑚 ∈ 𝐴 ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ◡∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
41	40	sumeq1d 15777	. . . 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 3462	. . . . . . . 8 ⊢ 𝑛 ∈ V
43		vex 3462	. . . . . . . 8 ⊢ 𝑚 ∈ V
44	42, 43	op1std 8005	. . . . . . 7 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → (1st ‘𝑤) = 𝑛)
45	44	csbeq1d 3860	. . . . . 6 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
46	42, 43	op2ndd 8006	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → (2nd ‘𝑤) = 𝑚)
47	46	csbeq1d 3860	. . . . . . 7 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → ⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑚 / 𝑗⦌𝐸)
48	47	csbeq2dv 3863	. . . . . 6 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → ⦋𝑛 / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
49	45, 48	eqtrd 2801	. . . . 5 ⊢ (𝑤 = ⟨𝑛, 𝑚⟩ → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
50	43, 42	op2ndd 8006	. . . . . . 7 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd ‘𝑧) = 𝑛)
51	50	csbeq1d 3860	. . . . . 6 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸)
52	43, 42	op1std 8005	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → (1st ‘𝑧) = 𝑚)
53	52	csbeq1d 3860	. . . . . . 7 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → ⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑚 / 𝑗⦌𝐸)
54	53	csbeq2dv 3863	. . . . . 6 ⊢ (𝑧 = ⟨𝑚, 𝑛⟩ → ⦋𝑛 / 𝑘⦌⦋(1st ‘𝑧) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
55	51, 54	eqtrd 2801	. . . . 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 9050	. . . . . . . 8 ⊢ {𝑛} ∈ Fin
58		fsumcom2.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ Fin)
59	58	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐴 ∈ Fin)
60	43, 42	opelcnv 5872	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑚, 𝑛⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑛, 𝑚⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
61	33, 36	opeliunxp2f 8215	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑛, 𝑚⟩ ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷))
62	60, 61	sylbbr 239	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷) → ⟨𝑚, 𝑛⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
63	62	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ⟨𝑚, 𝑛⟩ ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
64	22	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))
65	63, 64	eleqtrrd 2869	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ⟨𝑚, 𝑛⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
66		eliun 4965	. . . . . . . . . . . . 13 ⊢ (⟨𝑚, 𝑛⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
67	65, 66	sylib 221	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ∃𝑗 ∈ 𝐴 ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
68		opelxp 5702	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) ↔ (𝑚 ∈ {𝑗} ∧ 𝑛 ∈ 𝐵))
69	68	bilani 510	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → (𝑚 ∈ {𝑗} ∧ 𝑛 ∈ 𝐵))
70	69	simpld 500	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 ∈ {𝑗})
71		elsni 4611	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
72	70, 71	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 = 𝑗)
73		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑗 ∈ 𝐴)
74	72, 73	eqeltrd 2866	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 ∈ 𝐴)
75	74	rexlimiva 3161	. . . . . . . . . . . 12 ⊢ (∃𝑗 ∈ 𝐴 ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑚 ∈ 𝐴)
76	67, 75	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → 𝑚 ∈ 𝐴)
77	76	expr 462	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷 → 𝑚 ∈ 𝐴))
78	77	ssrdv 3946	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ⦋𝑛 / 𝑘⦌𝐷 ⊆ 𝐴)
79	59, 78	ssfid 9239	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ⦋𝑛 / 𝑘⦌𝐷 ∈ Fin)
80		xpfi 9289	. . . . . . . 8 ⊢ (({𝑛} ∈ Fin ∧ ⦋𝑛 / 𝑘⦌𝐷 ∈ Fin) → ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin)
81	57, 79, 80	sylancr 599	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin)
82	81	ralrimiva 3160	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin)
83		iunfi 9310	. . . . . 6 ⊢ ((𝐶 ∈ Fin ∧ ∀𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin) → ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin)
84	56, 82, 83	syl2anc 596	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin)
85		reliun 5808	. . . . . . 7 ⊢ (Rel ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ↔ ∀𝑛 ∈ 𝐶 Rel ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
86		relxp 5684	. . . . . . . 8 ⊢ Rel ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)
87	86	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ 𝐶 → Rel ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
88	85, 87	mprgbir 3089	. . . . . 6 ⊢ Rel ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)
89	88	a1i 11	. . . . 5 ⊢ (𝜑 → Rel ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
90		csbeq1 3859	. . . . . . . 8 ⊢ (𝑚 = (2nd ‘𝑤) → ⦋𝑚 / 𝑗⦌𝐸 = ⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
91	90	csbeq2dv 3863	. . . . . . 7 ⊢ (𝑚 = (2nd ‘𝑤) → ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸)
92	91	eleq1d 2851	. . . . . 6 ⊢ (𝑚 = (2nd ‘𝑤) → (⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ))
93		csbeq1 3859	. . . . . . . 8 ⊢ (𝑛 = (1st ‘𝑤) → ⦋𝑛 / 𝑘⦌𝐷 = ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
94		csbeq1 3859	. . . . . . . . 9 ⊢ (𝑛 = (1st ‘𝑤) → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
95	94	eleq1d 2851	. . . . . . . 8 ⊢ (𝑛 = (1st ‘𝑤) → (⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
96	93, 95	raleqbidv 3341	. . . . . . 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))
97		simpl 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → 𝜑)
98	25	nfcri 2920	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵
99	71	equcomd 2052	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ {𝑗} → 𝑗 = 𝑚)
100	99, 28	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ {𝑗} → 𝐵 = ⦋𝑚 / 𝑗⦌𝐵)
101	100	eleq2d 2852	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ {𝑗} → (𝑛 ∈ 𝐵 ↔ 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵))
102	101	biimpa 482	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ {𝑗} ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)
103	68, 102	sylbi 220	. . . . . . . . . . . . 13 ⊢ (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)
104	103	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝐴 → (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵))
105	98, 104	rexlimi 3268	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ 𝐴 ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)
106	67, 105	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)
107		fsumcom2.5	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐸 ∈ ℂ)
108	107	ralrimivva 3211	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ)
109		nfcsb1v 3880	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐸
110	109	nfel1 2944	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ
111	25, 110	nfralw 3315	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ
112		csbeq1a 3870	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑚 → 𝐸 = ⦋𝑚 / 𝑗⦌𝐸)
113	112	eleq1d 2851	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑚 → (𝐸 ∈ ℂ ↔ ⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
114	28, 113	raleqbidv 3341	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑚 → (∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ ↔ ∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
115	111, 114	rspc 3572	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ → ∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
116	108, 115	mpan9 516	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
117		nfcsb1v 3880	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
118	117	nfel1 2944	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ
119		csbeq1a 3870	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ⦋𝑚 / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
120	119	eleq1d 2851	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
121	118, 120	rspc 3572	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵 → (∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
122	116, 121	syl5com 32	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵 → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ))
123	122	impr 460	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)) → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
124	97, 76, 106, 123	syl12anc 850	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
125	124	ralrimivva 3211	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ 𝐶 ∀𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
126	125	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ∀𝑛 ∈ 𝐶 ∀𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
127		eliun 4965	. . . . . . . . 9 ⊢ (𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ↔ ∃𝑛 ∈ 𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
128	127	bilani 510	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ∃𝑛 ∈ 𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷))
129		xp1st 8027	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (1st ‘𝑤) ∈ {𝑛})
130	129	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ {𝑛})
131		elsni 4611	. . . . . . . . . . 11 ⊢ ((1st ‘𝑤) ∈ {𝑛} → (1st ‘𝑤) = 𝑛)
132	130, 131	syl 18	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) = 𝑛)
133		simpl 488	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → 𝑛 ∈ 𝐶)
134	132, 133	eqeltrd 2866	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ 𝐶)
135	134	rexlimiva 3161	. . . . . . . 8 ⊢ (∃𝑛 ∈ 𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (1st ‘𝑤) ∈ 𝐶)
136	128, 135	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ 𝐶)
137	96, 126, 136	rspcdva 3585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ∀𝑚 ∈ ⦋ (1st ‘𝑤) / 𝑘⦌𝐷⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)
138		xp2nd 8028	. . . . . . . . . 10 ⊢ (𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (2nd ‘𝑤) ∈ ⦋𝑛 / 𝑘⦌𝐷)
139	138	adantl 487	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋𝑛 / 𝑘⦌𝐷)
140	132	csbeq1d 3860	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ⦋(1st ‘𝑤) / 𝑘⦌𝐷 = ⦋𝑛 / 𝑘⦌𝐷)
141	139, 140	eleqtrrd 2869	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
142	141	rexlimiva 3161	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
143	128, 142	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋(1st ‘𝑤) / 𝑘⦌𝐷)
144	92, 137, 143	rspcdva 3585	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd ‘𝑤) / 𝑗⦌𝐸 ∈ ℂ)
145	49, 55, 84, 89, 144	fsumcnv 15850	. . . 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)
146	41, 145	eqtr4d 2804	. . 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)
147		fsumcom2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
148	147	ralrimiva 3160	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ Fin)
149	25	nfel1 2944	. . . . . 6 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐵 ∈ Fin
150	28	eleq1d 2851	. . . . . 6 ⊢ (𝑗 = 𝑚 → (𝐵 ∈ Fin ↔ ⦋𝑚 / 𝑗⦌𝐵 ∈ Fin))
151	149, 150	rspc 3572	. . . . 5 ⊢ (𝑚 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 𝐵 ∈ Fin → ⦋𝑚 / 𝑗⦌𝐵 ∈ Fin))
152	148, 151	mpan9 516	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ⦋𝑚 / 𝑗⦌𝐵 ∈ Fin)
153	55, 58, 152, 123	fsum2d 15848	. . 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)
154	49, 56, 79, 124	fsum2d 15848	. . 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)
155	146, 153, 154	3eqtr4d 2811	. 2 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = Σ𝑛 ∈ 𝐶 Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
156		csbeq1a 3870	. . . . 5 ⊢ (𝑘 = 𝑛 → 𝐸 = ⦋𝑛 / 𝑘⦌𝐸)
157		nfcv 2928	. . . . 5 ⊢ Ⅎ𝑛𝐸
158		nfcsb1v 3880	. . . . 5 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐸
159	156, 157, 158	cbvsum 15772	. . . 4 ⊢ Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑛 ∈ 𝐵 ⦋𝑛 / 𝑘⦌𝐸
160	112	csbeq2dv 3863	. . . . . 6 ⊢ (𝑗 = 𝑚 → ⦋𝑛 / 𝑘⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
161	160	adantr 486	. . . . 5 ⊢ ((𝑗 = 𝑚 ∧ 𝑛 ∈ 𝐵) → ⦋𝑛 / 𝑘⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
162	28, 161	sumeq12dv 15783	. . . 4 ⊢ (𝑗 = 𝑚 → Σ𝑛 ∈ 𝐵 ⦋𝑛 / 𝑘⦌𝐸 = Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
163	159, 162	eqtrid 2813	. . 3 ⊢ (𝑗 = 𝑚 → Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
164		nfcv 2928	. . 3 ⊢ Ⅎ𝑚Σ𝑘 ∈ 𝐵 𝐸
165		nfcv 2928	. . . . 5 ⊢ Ⅎ𝑗𝑛
166	165, 109	nfcsbw 3882	. . . 4 ⊢ Ⅎ𝑗⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
167	25, 166	nfsum 15768	. . 3 ⊢ Ⅎ𝑗Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
168	163, 164, 167	cbvsum 15772	. 2 ⊢ Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑚 ∈ 𝐴 Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
169		nfcv 2928	. . . . 5 ⊢ Ⅎ𝑚𝐸
170	112, 169, 109	cbvsum 15772	. . . 4 ⊢ Σ𝑗 ∈ 𝐷 𝐸 = Σ𝑚 ∈ 𝐷 ⦋𝑚 / 𝑗⦌𝐸
171	119	adantr 486	. . . . 5 ⊢ ((𝑘 = 𝑛 ∧ 𝑚 ∈ 𝐷) → ⦋𝑚 / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
172	36, 171	sumeq12dv 15783	. . . 4 ⊢ (𝑘 = 𝑛 → Σ𝑚 ∈ 𝐷 ⦋𝑚 / 𝑗⦌𝐸 = Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
173	170, 172	eqtrid 2813	. . 3 ⊢ (𝑘 = 𝑛 → Σ𝑗 ∈ 𝐷 𝐸 = Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸)
174		nfcv 2928	. . 3 ⊢ Ⅎ𝑛Σ𝑗 ∈ 𝐷 𝐸
175	33, 117	nfsum 15768	. . 3 ⊢ Ⅎ𝑘Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
176	173, 174, 175	cbvsum 15772	. 2 ⊢ Σ𝑘 ∈ 𝐶 Σ𝑗 ∈ 𝐷 𝐸 = Σ𝑛 ∈ 𝐶 Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸
177	155, 168, 176	3eqtr4g 2826	1 ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑘 ∈ 𝐶 Σ𝑗 ∈ 𝐷 𝐸)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092  ⦋csb 3856  {csn 4594  ⟨cop 4600  ∪ ciun 4961   × cxp 5664  ^◡ccnv 5665  Rel wrel 5671  ‘cfv 6543  1^st c1st 7993  2^nd c2nd 7994  Fincfn 8952  ℂcc 11116  Σcsu 15763

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-inf2 9620  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195  ax-pre-sup 11196

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-oi 9482  df-card 9944  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-div 11890  df-nn 12252  df-2 12321  df-3 12322  df-n0 12523  df-z 12610  df-uz 12881  df-rp 13035  df-fz 13554  df-fzo 13702  df-seq 14058  df-exp 14118  df-hash 14387  df-cj 15176  df-re 15177  df-im 15178  df-sqrt 15312  df-abs 15313  df-clim 15565  df-sum 15764

This theorem is used by:  fsumcom  15852  fsum0diag  15854  fsumdvdsdiag  27385  dvdsflsumcom  27389  fsumfldivdiag  27391  logfac2  27418  chpchtsum  27420  logfaclbnd  27423  dchrisum0lem1  27717