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Theorem fisumcom2 11446
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 (𝜑𝐴 ∈ Fin)
fsumcom2.2 (𝜑𝐶 ∈ Fin)
fsumcom2.3 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
fisumcom2.fi ((𝜑𝑘𝐶) → 𝐷 ∈ Fin)
fsumcom2.4 (𝜑 → ((𝑗𝐴𝑘𝐵) ↔ (𝑘𝐶𝑗𝐷)))
fsumcom2.5 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐸 ∈ ℂ)
Assertion
Ref Expression
fisumcom2 (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐸 = Σ𝑘𝐶 Σ𝑗𝐷 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 fisumcom2
Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 4736 . . . . . . . . 9 Rel ({𝑗} × 𝐵)
21rgenw 2532 . . . . . . . 8 𝑗𝐴 Rel ({𝑗} × 𝐵)
3 reliun 4748 . . . . . . . 8 (Rel 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∀𝑗𝐴 Rel ({𝑗} × 𝐵))
42, 3mpbir 146 . . . . . . 7 Rel 𝑗𝐴 ({𝑗} × 𝐵)
5 relcnv 5007 . . . . . . 7 Rel 𝑘𝐶 ({𝑘} × 𝐷)
6 ancom 266 . . . . . . . . . . . 12 ((𝑥 = 𝑗𝑦 = 𝑘) ↔ (𝑦 = 𝑘𝑥 = 𝑗))
7 vex 2741 . . . . . . . . . . . . 13 𝑥 ∈ V
8 vex 2741 . . . . . . . . . . . . 13 𝑦 ∈ V
97, 8opth 4238 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ (𝑥 = 𝑗𝑦 = 𝑘))
108, 7opth 4238 . . . . . . . . . . . 12 (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ↔ (𝑦 = 𝑘𝑥 = 𝑗))
116, 9, 103bitr4i 212 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩)
1211a1i 9 . . . . . . . . . 10 (𝜑 → (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩))
13 fsumcom2.4 . . . . . . . . . 10 (𝜑 → ((𝑗𝐴𝑘𝐵) ↔ (𝑘𝐶𝑗𝐷)))
1412, 13anbi12d 473 . . . . . . . . 9 (𝜑 → ((⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷))))
15142exbidv 1868 . . . . . . . 8 (𝜑 → (∃𝑗𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷))))
16 eliunxp 4767 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)))
177, 8opelcnv 4810 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
18 eliunxp 4767 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
19 excom 1664 . . . . . . . . 9 (∃𝑘𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
2017, 18, 193bitri 206 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
2115, 16, 203bitr4g 223 . . . . . . 7 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷)))
224, 5, 21eqrelrdv 4723 . . . . . 6 (𝜑 𝑗𝐴 ({𝑗} × 𝐵) = 𝑘𝐶 ({𝑘} × 𝐷))
23 nfcv 2319 . . . . . . 7 𝑚({𝑗} × 𝐵)
24 nfcv 2319 . . . . . . . 8 𝑗{𝑚}
25 nfcsb1v 3091 . . . . . . . 8 𝑗𝑚 / 𝑗𝐵
2624, 25nfxp 4654 . . . . . . 7 𝑗({𝑚} × 𝑚 / 𝑗𝐵)
27 sneq 3604 . . . . . . . 8 (𝑗 = 𝑚 → {𝑗} = {𝑚})
28 csbeq1a 3067 . . . . . . . 8 (𝑗 = 𝑚𝐵 = 𝑚 / 𝑗𝐵)
2927, 28xpeq12d 4652 . . . . . . 7 (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × 𝑚 / 𝑗𝐵))
3023, 26, 29cbviun 3924 . . . . . 6 𝑗𝐴 ({𝑗} × 𝐵) = 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)
31 nfcv 2319 . . . . . . . 8 𝑛({𝑘} × 𝐷)
32 nfcv 2319 . . . . . . . . 9 𝑘{𝑛}
33 nfcsb1v 3091 . . . . . . . . 9 𝑘𝑛 / 𝑘𝐷
3432, 33nfxp 4654 . . . . . . . 8 𝑘({𝑛} × 𝑛 / 𝑘𝐷)
35 sneq 3604 . . . . . . . . 9 (𝑘 = 𝑛 → {𝑘} = {𝑛})
36 csbeq1a 3067 . . . . . . . . 9 (𝑘 = 𝑛𝐷 = 𝑛 / 𝑘𝐷)
3735, 36xpeq12d 4652 . . . . . . . 8 (𝑘 = 𝑛 → ({𝑘} × 𝐷) = ({𝑛} × 𝑛 / 𝑘𝐷))
3831, 34, 37cbviun 3924 . . . . . . 7 𝑘𝐶 ({𝑘} × 𝐷) = 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
3938cnveqi 4803 . . . . . 6 𝑘𝐶 ({𝑘} × 𝐷) = 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
4022, 30, 393eqtr3g 2233 . . . . 5 (𝜑 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵) = 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
4140sumeq1d 11374 . . . 4 (𝜑 → Σ𝑧 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = Σ𝑧 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
42 vex 2741 . . . . . . . 8 𝑛 ∈ V
43 vex 2741 . . . . . . . 8 𝑚 ∈ V
4442, 43op1std 6149 . . . . . . 7 (𝑤 = ⟨𝑛, 𝑚⟩ → (1st𝑤) = 𝑛)
4544csbeq1d 3065 . . . . . 6 (𝑤 = ⟨𝑛, 𝑚⟩ → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑛 / 𝑘(2nd𝑤) / 𝑗𝐸)
4642, 43op2ndd 6150 . . . . . . . 8 (𝑤 = ⟨𝑛, 𝑚⟩ → (2nd𝑤) = 𝑚)
4746csbeq1d 3065 . . . . . . 7 (𝑤 = ⟨𝑛, 𝑚⟩ → (2nd𝑤) / 𝑗𝐸 = 𝑚 / 𝑗𝐸)
4847csbeq2dv 3084 . . . . . 6 (𝑤 = ⟨𝑛, 𝑚⟩ → 𝑛 / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
4945, 48eqtrd 2210 . . . . 5 (𝑤 = ⟨𝑛, 𝑚⟩ → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
5043, 42op2ndd 6150 . . . . . . 7 (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd𝑧) = 𝑛)
5150csbeq1d 3065 . . . . . 6 (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑛 / 𝑘(1st𝑧) / 𝑗𝐸)
5243, 42op1std 6149 . . . . . . . 8 (𝑧 = ⟨𝑚, 𝑛⟩ → (1st𝑧) = 𝑚)
5352csbeq1d 3065 . . . . . . 7 (𝑧 = ⟨𝑚, 𝑛⟩ → (1st𝑧) / 𝑗𝐸 = 𝑚 / 𝑗𝐸)
5453csbeq2dv 3084 . . . . . 6 (𝑧 = ⟨𝑚, 𝑛⟩ → 𝑛 / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
5551, 54eqtrd 2210 . . . . 5 (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
56 fsumcom2.2 . . . . . 6 (𝜑𝐶 ∈ Fin)
57 snfig 6814 . . . . . . . . 9 (𝑛 ∈ V → {𝑛} ∈ Fin)
5857elv 2742 . . . . . . . 8 {𝑛} ∈ Fin
59 fisumcom2.fi . . . . . . . . . 10 ((𝜑𝑘𝐶) → 𝐷 ∈ Fin)
6059ralrimiva 2550 . . . . . . . . 9 (𝜑 → ∀𝑘𝐶 𝐷 ∈ Fin)
6133nfel1 2330 . . . . . . . . . 10 𝑘𝑛 / 𝑘𝐷 ∈ Fin
6236eleq1d 2246 . . . . . . . . . 10 (𝑘 = 𝑛 → (𝐷 ∈ Fin ↔ 𝑛 / 𝑘𝐷 ∈ Fin))
6361, 62rspc 2836 . . . . . . . . 9 (𝑛𝐶 → (∀𝑘𝐶 𝐷 ∈ Fin → 𝑛 / 𝑘𝐷 ∈ Fin))
6460, 63mpan9 281 . . . . . . . 8 ((𝜑𝑛𝐶) → 𝑛 / 𝑘𝐷 ∈ Fin)
65 xpfi 6929 . . . . . . . 8 (({𝑛} ∈ Fin ∧ 𝑛 / 𝑘𝐷 ∈ Fin) → ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
6658, 64, 65sylancr 414 . . . . . . 7 ((𝜑𝑛𝐶) → ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
6766ralrimiva 2550 . . . . . 6 (𝜑 → ∀𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
68 disjsnxp 6238 . . . . . . 7 Disj 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
6968a1i 9 . . . . . 6 (𝜑Disj 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
70 iunfidisj 6945 . . . . . 6 ((𝐶 ∈ Fin ∧ ∀𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin ∧ Disj 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
7156, 67, 69, 70syl3anc 1238 . . . . 5 (𝜑 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
72 reliun 4748 . . . . . . 7 (Rel 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ↔ ∀𝑛𝐶 Rel ({𝑛} × 𝑛 / 𝑘𝐷))
73 relxp 4736 . . . . . . . 8 Rel ({𝑛} × 𝑛 / 𝑘𝐷)
7473a1i 9 . . . . . . 7 (𝑛𝐶 → Rel ({𝑛} × 𝑛 / 𝑘𝐷))
7572, 74mprgbir 2535 . . . . . 6 Rel 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
7675a1i 9 . . . . 5 (𝜑 → Rel 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
77 csbeq1 3061 . . . . . . . 8 (𝑚 = (2nd𝑤) → 𝑚 / 𝑗𝐸 = (2nd𝑤) / 𝑗𝐸)
7877csbeq2dv 3084 . . . . . . 7 (𝑚 = (2nd𝑤) → (1st𝑤) / 𝑘𝑚 / 𝑗𝐸 = (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
7978eleq1d 2246 . . . . . 6 (𝑚 = (2nd𝑤) → ((1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ ↔ (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 ∈ ℂ))
80 csbeq1 3061 . . . . . . . 8 (𝑛 = (1st𝑤) → 𝑛 / 𝑘𝐷 = (1st𝑤) / 𝑘𝐷)
81 csbeq1 3061 . . . . . . . . 9 (𝑛 = (1st𝑤) → 𝑛 / 𝑘𝑚 / 𝑗𝐸 = (1st𝑤) / 𝑘𝑚 / 𝑗𝐸)
8281eleq1d 2246 . . . . . . . 8 (𝑛 = (1st𝑤) → (𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ ↔ (1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
8380, 82raleqbidv 2685 . . . . . . 7 (𝑛 = (1st𝑤) → (∀𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ ↔ ∀𝑚 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
84 simpl 109 . . . . . . . . . 10 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝜑)
8543, 42opelcnv 4810 . . . . . . . . . . . . . . 15 (⟨𝑚, 𝑛⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑛, 𝑚⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
8633, 36opeliunxp2f 6239 . . . . . . . . . . . . . . 15 (⟨𝑛, 𝑚⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ (𝑛𝐶𝑚𝑛 / 𝑘𝐷))
8785, 86sylbbr 136 . . . . . . . . . . . . . 14 ((𝑛𝐶𝑚𝑛 / 𝑘𝐷) → ⟨𝑚, 𝑛⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
8887adantl 277 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → ⟨𝑚, 𝑛⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
8922adantr 276 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑗𝐴 ({𝑗} × 𝐵) = 𝑘𝐶 ({𝑘} × 𝐷))
9088, 89eleqtrrd 2257 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → ⟨𝑚, 𝑛⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵))
91 eliun 3891 . . . . . . . . . . . 12 (⟨𝑚, 𝑛⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
9290, 91sylib 122 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → ∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
93 simpr 110 . . . . . . . . . . . . . . . 16 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
94 opelxp 4657 . . . . . . . . . . . . . . . 16 (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) ↔ (𝑚 ∈ {𝑗} ∧ 𝑛𝐵))
9593, 94sylib 122 . . . . . . . . . . . . . . 15 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → (𝑚 ∈ {𝑗} ∧ 𝑛𝐵))
9695simpld 112 . . . . . . . . . . . . . 14 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 ∈ {𝑗})
97 elsni 3611 . . . . . . . . . . . . . 14 (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
9896, 97syl 14 . . . . . . . . . . . . 13 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 = 𝑗)
99 simpl 109 . . . . . . . . . . . . 13 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑗𝐴)
10098, 99eqeltrd 2254 . . . . . . . . . . . 12 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚𝐴)
101100rexlimiva 2589 . . . . . . . . . . 11 (∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑚𝐴)
10292, 101syl 14 . . . . . . . . . 10 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑚𝐴)
10325nfcri 2313 . . . . . . . . . . . 12 𝑗 𝑛𝑚 / 𝑗𝐵
10497equcomd 1707 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ {𝑗} → 𝑗 = 𝑚)
105104, 28syl 14 . . . . . . . . . . . . . . . 16 (𝑚 ∈ {𝑗} → 𝐵 = 𝑚 / 𝑗𝐵)
106105eleq2d 2247 . . . . . . . . . . . . . . 15 (𝑚 ∈ {𝑗} → (𝑛𝐵𝑛𝑚 / 𝑗𝐵))
107106biimpa 296 . . . . . . . . . . . . . 14 ((𝑚 ∈ {𝑗} ∧ 𝑛𝐵) → 𝑛𝑚 / 𝑗𝐵)
10894, 107sylbi 121 . . . . . . . . . . . . 13 (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛𝑚 / 𝑗𝐵)
109108a1i 9 . . . . . . . . . . . 12 (𝑗𝐴 → (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛𝑚 / 𝑗𝐵))
110103, 109rexlimi 2587 . . . . . . . . . . 11 (∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛𝑚 / 𝑗𝐵)
11192, 110syl 14 . . . . . . . . . 10 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑛𝑚 / 𝑗𝐵)
112 fsumcom2.5 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐸 ∈ ℂ)
113112ralrimivva 2559 . . . . . . . . . . . . 13 (𝜑 → ∀𝑗𝐴𝑘𝐵 𝐸 ∈ ℂ)
114 nfcsb1v 3091 . . . . . . . . . . . . . . . 16 𝑗𝑚 / 𝑗𝐸
115114nfel1 2330 . . . . . . . . . . . . . . 15 𝑗𝑚 / 𝑗𝐸 ∈ ℂ
11625, 115nfralxy 2515 . . . . . . . . . . . . . 14 𝑗𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ
117 csbeq1a 3067 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑚𝐸 = 𝑚 / 𝑗𝐸)
118117eleq1d 2246 . . . . . . . . . . . . . . 15 (𝑗 = 𝑚 → (𝐸 ∈ ℂ ↔ 𝑚 / 𝑗𝐸 ∈ ℂ))
11928, 118raleqbidv 2685 . . . . . . . . . . . . . 14 (𝑗 = 𝑚 → (∀𝑘𝐵 𝐸 ∈ ℂ ↔ ∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ))
120116, 119rspc 2836 . . . . . . . . . . . . 13 (𝑚𝐴 → (∀𝑗𝐴𝑘𝐵 𝐸 ∈ ℂ → ∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ))
121113, 120mpan9 281 . . . . . . . . . . . 12 ((𝜑𝑚𝐴) → ∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ)
122 nfcsb1v 3091 . . . . . . . . . . . . . 14 𝑘𝑛 / 𝑘𝑚 / 𝑗𝐸
123122nfel1 2330 . . . . . . . . . . . . 13 𝑘𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ
124 csbeq1a 3067 . . . . . . . . . . . . . 14 (𝑘 = 𝑛𝑚 / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
125124eleq1d 2246 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑚 / 𝑗𝐸 ∈ ℂ ↔ 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
126123, 125rspc 2836 . . . . . . . . . . . 12 (𝑛𝑚 / 𝑗𝐵 → (∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ → 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
127121, 126syl5com 29 . . . . . . . . . . 11 ((𝜑𝑚𝐴) → (𝑛𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
128127impr 379 . . . . . . . . . 10 ((𝜑 ∧ (𝑚𝐴𝑛𝑚 / 𝑗𝐵)) → 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
12984, 102, 111, 128syl12anc 1236 . . . . . . . . 9 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
130129ralrimivva 2559 . . . . . . . 8 (𝜑 → ∀𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
131130adantr 276 . . . . . . 7 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → ∀𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
132 simpr 110 . . . . . . . . 9 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → 𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
133 eliun 3891 . . . . . . . . 9 (𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ↔ ∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷))
134132, 133sylib 122 . . . . . . . 8 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → ∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷))
135 xp1st 6166 . . . . . . . . . . . 12 (𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (1st𝑤) ∈ {𝑛})
136135adantl 277 . . . . . . . . . . 11 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) ∈ {𝑛})
137 elsni 3611 . . . . . . . . . . 11 ((1st𝑤) ∈ {𝑛} → (1st𝑤) = 𝑛)
138136, 137syl 14 . . . . . . . . . 10 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) = 𝑛)
139 simpl 109 . . . . . . . . . 10 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → 𝑛𝐶)
140138, 139eqeltrd 2254 . . . . . . . . 9 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) ∈ 𝐶)
141140rexlimiva 2589 . . . . . . . 8 (∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (1st𝑤) ∈ 𝐶)
142134, 141syl 14 . . . . . . 7 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) ∈ 𝐶)
14383, 131, 142rspcdva 2847 . . . . . 6 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → ∀𝑚 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
144 xp2nd 6167 . . . . . . . . . 10 (𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (2nd𝑤) ∈ 𝑛 / 𝑘𝐷)
145144adantl 277 . . . . . . . . 9 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (2nd𝑤) ∈ 𝑛 / 𝑘𝐷)
146138csbeq1d 3065 . . . . . . . . 9 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) / 𝑘𝐷 = 𝑛 / 𝑘𝐷)
147145, 146eleqtrrd 2257 . . . . . . . 8 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
148147rexlimiva 2589 . . . . . . 7 (∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
149134, 148syl 14 . . . . . 6 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
15079, 143, 149rspcdva 2847 . . . . 5 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 ∈ ℂ)
15149, 55, 71, 76, 150fsumcnv 11445 . . . 4 (𝜑 → Σ𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = Σ𝑧 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
15241, 151eqtr4d 2213 . . 3 (𝜑 → Σ𝑧 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = Σ𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
153 fsumcom2.1 . . . 4 (𝜑𝐴 ∈ Fin)
154 fsumcom2.3 . . . . . 6 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
155154ralrimiva 2550 . . . . 5 (𝜑 → ∀𝑗𝐴 𝐵 ∈ Fin)
15625nfel1 2330 . . . . . 6 𝑗𝑚 / 𝑗𝐵 ∈ Fin
15728eleq1d 2246 . . . . . 6 (𝑗 = 𝑚 → (𝐵 ∈ Fin ↔ 𝑚 / 𝑗𝐵 ∈ Fin))
158156, 157rspc 2836 . . . . 5 (𝑚𝐴 → (∀𝑗𝐴 𝐵 ∈ Fin → 𝑚 / 𝑗𝐵 ∈ Fin))
159155, 158mpan9 281 . . . 4 ((𝜑𝑚𝐴) → 𝑚 / 𝑗𝐵 ∈ Fin)
16055, 153, 159, 128fsum2d 11443 . . 3 (𝜑 → Σ𝑚𝐴 Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸 = Σ𝑧 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
16149, 56, 64, 129fsum2d 11443 . . 3 (𝜑 → Σ𝑛𝐶 Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 = Σ𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
162152, 160, 1613eqtr4d 2220 . 2 (𝜑 → Σ𝑚𝐴 Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸 = Σ𝑛𝐶 Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸)
163 nfcv 2319 . . 3 𝑚Σ𝑘𝐵 𝐸
164 nfcv 2319 . . . . 5 𝑗𝑛
165164, 114nfcsb 3095 . . . 4 𝑗𝑛 / 𝑘𝑚 / 𝑗𝐸
16625, 165nfsum 11365 . . 3 𝑗Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸
167 nfcv 2319 . . . . 5 𝑛𝐸
168 nfcsb1v 3091 . . . . 5 𝑘𝑛 / 𝑘𝐸
169 csbeq1a 3067 . . . . 5 (𝑘 = 𝑛𝐸 = 𝑛 / 𝑘𝐸)
170167, 168, 169cbvsumi 11370 . . . 4 Σ𝑘𝐵 𝐸 = Σ𝑛𝐵 𝑛 / 𝑘𝐸
171117csbeq2dv 3084 . . . . . 6 (𝑗 = 𝑚𝑛 / 𝑘𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
172171adantr 276 . . . . 5 ((𝑗 = 𝑚𝑛𝐵) → 𝑛 / 𝑘𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
17328, 172sumeq12dv 11380 . . . 4 (𝑗 = 𝑚 → Σ𝑛𝐵 𝑛 / 𝑘𝐸 = Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸)
174170, 173eqtrid 2222 . . 3 (𝑗 = 𝑚 → Σ𝑘𝐵 𝐸 = Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸)
175163, 166, 174cbvsumi 11370 . 2 Σ𝑗𝐴 Σ𝑘𝐵 𝐸 = Σ𝑚𝐴 Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸
176 nfcv 2319 . . 3 𝑛Σ𝑗𝐷 𝐸
17733, 122nfsum 11365 . . 3 𝑘Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸
178 nfcv 2319 . . . . 5 𝑚𝐸
179178, 114, 117cbvsumi 11370 . . . 4 Σ𝑗𝐷 𝐸 = Σ𝑚𝐷 𝑚 / 𝑗𝐸
180124adantr 276 . . . . 5 ((𝑘 = 𝑛𝑚𝐷) → 𝑚 / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
18136, 180sumeq12dv 11380 . . . 4 (𝑘 = 𝑛 → Σ𝑚𝐷 𝑚 / 𝑗𝐸 = Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸)
182179, 181eqtrid 2222 . . 3 (𝑘 = 𝑛 → Σ𝑗𝐷 𝐸 = Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸)
183176, 177, 182cbvsumi 11370 . 2 Σ𝑘𝐶 Σ𝑗𝐷 𝐸 = Σ𝑛𝐶 Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸
184162, 175, 1833eqtr4g 2235 1 (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐸 = Σ𝑘𝐶 Σ𝑗𝐷 𝐸)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456  Vcvv 2738  csb 3058  {csn 3593  cop 3596   ciun 3887  Disj wdisj 3981   × cxp 4625  ccnv 4626  Rel wrel 4632  cfv 5217  1st c1st 6139  2nd c2nd 6140  Fincfn 6740  cc 7809  Σcsu 11361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-disj 3982  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-isom 5226  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-frec 6392  df-1o 6417  df-oadd 6421  df-er 6535  df-en 6741  df-dom 6742  df-fin 6743  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-fz 10009  df-fzo 10143  df-seqfrec 10446  df-exp 10520  df-ihash 10756  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-clim 11287  df-sumdc 11362
This theorem is referenced by:  fsumcom  11447  fisum0diag  11449
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