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Theorem fisumcom2 10832
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 4547 . . . . . . . . 9 Rel ({𝑗} × 𝐵)
21rgenw 2430 . . . . . . . 8 𝑗𝐴 Rel ({𝑗} × 𝐵)
3 reliun 4558 . . . . . . . 8 (Rel 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∀𝑗𝐴 Rel ({𝑗} × 𝐵))
42, 3mpbir 144 . . . . . . 7 Rel 𝑗𝐴 ({𝑗} × 𝐵)
5 relcnv 4810 . . . . . . 7 Rel 𝑘𝐶 ({𝑘} × 𝐷)
6 ancom 262 . . . . . . . . . . . 12 ((𝑥 = 𝑗𝑦 = 𝑘) ↔ (𝑦 = 𝑘𝑥 = 𝑗))
7 vex 2622 . . . . . . . . . . . . 13 𝑥 ∈ V
8 vex 2622 . . . . . . . . . . . . 13 𝑦 ∈ V
97, 8opth 4064 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ (𝑥 = 𝑗𝑦 = 𝑘))
108, 7opth 4064 . . . . . . . . . . . 12 (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ↔ (𝑦 = 𝑘𝑥 = 𝑗))
116, 9, 103bitr4i 210 . . . . . . . . . . 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 457 . . . . . . . . 9 (𝜑 → ((⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷))))
15142exbidv 1796 . . . . . . . 8 (𝜑 → (∃𝑗𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷))))
16 eliunxp 4575 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)))
177, 8opelcnv 4618 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
18 eliunxp 4575 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
19 excom 1599 . . . . . . . . 9 (∃𝑘𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
2017, 18, 193bitri 204 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
2115, 16, 203bitr4g 221 . . . . . . 7 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷)))
224, 5, 21eqrelrdv 4534 . . . . . 6 (𝜑 𝑗𝐴 ({𝑗} × 𝐵) = 𝑘𝐶 ({𝑘} × 𝐷))
23 nfcv 2228 . . . . . . 7 𝑚({𝑗} × 𝐵)
24 nfcv 2228 . . . . . . . 8 𝑗{𝑚}
25 nfcsb1v 2963 . . . . . . . 8 𝑗𝑚 / 𝑗𝐵
2624, 25nfxp 4464 . . . . . . 7 𝑗({𝑚} × 𝑚 / 𝑗𝐵)
27 sneq 3457 . . . . . . . 8 (𝑗 = 𝑚 → {𝑗} = {𝑚})
28 csbeq1a 2941 . . . . . . . 8 (𝑗 = 𝑚𝐵 = 𝑚 / 𝑗𝐵)
2927, 28xpeq12d 4463 . . . . . . 7 (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × 𝑚 / 𝑗𝐵))
3023, 26, 29cbviun 3767 . . . . . 6 𝑗𝐴 ({𝑗} × 𝐵) = 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)
31 nfcv 2228 . . . . . . . 8 𝑛({𝑘} × 𝐷)
32 nfcv 2228 . . . . . . . . 9 𝑘{𝑛}
33 nfcsb1v 2963 . . . . . . . . 9 𝑘𝑛 / 𝑘𝐷
3432, 33nfxp 4464 . . . . . . . 8 𝑘({𝑛} × 𝑛 / 𝑘𝐷)
35 sneq 3457 . . . . . . . . 9 (𝑘 = 𝑛 → {𝑘} = {𝑛})
36 csbeq1a 2941 . . . . . . . . 9 (𝑘 = 𝑛𝐷 = 𝑛 / 𝑘𝐷)
3735, 36xpeq12d 4463 . . . . . . . 8 (𝑘 = 𝑛 → ({𝑘} × 𝐷) = ({𝑛} × 𝑛 / 𝑘𝐷))
3831, 34, 37cbviun 3767 . . . . . . 7 𝑘𝐶 ({𝑘} × 𝐷) = 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
3938cnveqi 4611 . . . . . 6 𝑘𝐶 ({𝑘} × 𝐷) = 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
4022, 30, 393eqtr3g 2143 . . . . 5 (𝜑 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵) = 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
4140sumeq1d 10755 . . . 4 (𝜑 → Σ𝑧 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = Σ𝑧 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
42 vex 2622 . . . . . . . 8 𝑛 ∈ V
43 vex 2622 . . . . . . . 8 𝑚 ∈ V
4442, 43op1std 5919 . . . . . . 7 (𝑤 = ⟨𝑛, 𝑚⟩ → (1st𝑤) = 𝑛)
4544csbeq1d 2939 . . . . . 6 (𝑤 = ⟨𝑛, 𝑚⟩ → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑛 / 𝑘(2nd𝑤) / 𝑗𝐸)
4642, 43op2ndd 5920 . . . . . . . 8 (𝑤 = ⟨𝑛, 𝑚⟩ → (2nd𝑤) = 𝑚)
4746csbeq1d 2939 . . . . . . 7 (𝑤 = ⟨𝑛, 𝑚⟩ → (2nd𝑤) / 𝑗𝐸 = 𝑚 / 𝑗𝐸)
4847csbeq2dv 2956 . . . . . 6 (𝑤 = ⟨𝑛, 𝑚⟩ → 𝑛 / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
4945, 48eqtrd 2120 . . . . 5 (𝑤 = ⟨𝑛, 𝑚⟩ → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
5043, 42op2ndd 5920 . . . . . . 7 (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd𝑧) = 𝑛)
5150csbeq1d 2939 . . . . . 6 (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑛 / 𝑘(1st𝑧) / 𝑗𝐸)
5243, 42op1std 5919 . . . . . . . 8 (𝑧 = ⟨𝑚, 𝑛⟩ → (1st𝑧) = 𝑚)
5352csbeq1d 2939 . . . . . . 7 (𝑧 = ⟨𝑚, 𝑛⟩ → (1st𝑧) / 𝑗𝐸 = 𝑚 / 𝑗𝐸)
5453csbeq2dv 2956 . . . . . 6 (𝑧 = ⟨𝑚, 𝑛⟩ → 𝑛 / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
5551, 54eqtrd 2120 . . . . 5 (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
56 fsumcom2.2 . . . . . 6 (𝜑𝐶 ∈ Fin)
57 snfig 6531 . . . . . . . . 9 (𝑛 ∈ V → {𝑛} ∈ Fin)
5857elv 2623 . . . . . . . 8 {𝑛} ∈ Fin
59 fisumcom2.fi . . . . . . . . . 10 ((𝜑𝑘𝐶) → 𝐷 ∈ Fin)
6059ralrimiva 2446 . . . . . . . . 9 (𝜑 → ∀𝑘𝐶 𝐷 ∈ Fin)
6133nfel1 2239 . . . . . . . . . 10 𝑘𝑛 / 𝑘𝐷 ∈ Fin
6236eleq1d 2156 . . . . . . . . . 10 (𝑘 = 𝑛 → (𝐷 ∈ Fin ↔ 𝑛 / 𝑘𝐷 ∈ Fin))
6361, 62rspc 2716 . . . . . . . . 9 (𝑛𝐶 → (∀𝑘𝐶 𝐷 ∈ Fin → 𝑛 / 𝑘𝐷 ∈ Fin))
6460, 63mpan9 275 . . . . . . . 8 ((𝜑𝑛𝐶) → 𝑛 / 𝑘𝐷 ∈ Fin)
65 xpfi 6640 . . . . . . . 8 (({𝑛} ∈ Fin ∧ 𝑛 / 𝑘𝐷 ∈ Fin) → ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
6658, 64, 65sylancr 405 . . . . . . 7 ((𝜑𝑛𝐶) → ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
6766ralrimiva 2446 . . . . . 6 (𝜑 → ∀𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
68 disjsnxp 6002 . . . . . . 7 Disj 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
6968a1i 9 . . . . . 6 (𝜑Disj 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
70 iunfidisj 6655 . . . . . 6 ((𝐶 ∈ Fin ∧ ∀𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin ∧ Disj 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
7156, 67, 69, 70syl3anc 1174 . . . . 5 (𝜑 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
72 reliun 4558 . . . . . . 7 (Rel 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ↔ ∀𝑛𝐶 Rel ({𝑛} × 𝑛 / 𝑘𝐷))
73 relxp 4547 . . . . . . . 8 Rel ({𝑛} × 𝑛 / 𝑘𝐷)
7473a1i 9 . . . . . . 7 (𝑛𝐶 → Rel ({𝑛} × 𝑛 / 𝑘𝐷))
7572, 74mprgbir 2433 . . . . . 6 Rel 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
7675a1i 9 . . . . 5 (𝜑 → Rel 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
77 csbeq1 2936 . . . . . . . 8 (𝑚 = (2nd𝑤) → 𝑚 / 𝑗𝐸 = (2nd𝑤) / 𝑗𝐸)
7877csbeq2dv 2956 . . . . . . 7 (𝑚 = (2nd𝑤) → (1st𝑤) / 𝑘𝑚 / 𝑗𝐸 = (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
7978eleq1d 2156 . . . . . 6 (𝑚 = (2nd𝑤) → ((1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ ↔ (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 ∈ ℂ))
80 csbeq1 2936 . . . . . . . 8 (𝑛 = (1st𝑤) → 𝑛 / 𝑘𝐷 = (1st𝑤) / 𝑘𝐷)
81 csbeq1 2936 . . . . . . . . 9 (𝑛 = (1st𝑤) → 𝑛 / 𝑘𝑚 / 𝑗𝐸 = (1st𝑤) / 𝑘𝑚 / 𝑗𝐸)
8281eleq1d 2156 . . . . . . . 8 (𝑛 = (1st𝑤) → (𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ ↔ (1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
8380, 82raleqbidv 2574 . . . . . . 7 (𝑛 = (1st𝑤) → (∀𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ ↔ ∀𝑚 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
84 simpl 107 . . . . . . . . . 10 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝜑)
8543, 42opelcnv 4618 . . . . . . . . . . . . . . 15 (⟨𝑚, 𝑛⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑛, 𝑚⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
8633, 36opeliunxp2f 6003 . . . . . . . . . . . . . . 15 (⟨𝑛, 𝑚⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ (𝑛𝐶𝑚𝑛 / 𝑘𝐷))
8785, 86sylbbr 134 . . . . . . . . . . . . . 14 ((𝑛𝐶𝑚𝑛 / 𝑘𝐷) → ⟨𝑚, 𝑛⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
8887adantl 271 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → ⟨𝑚, 𝑛⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
8922adantr 270 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑗𝐴 ({𝑗} × 𝐵) = 𝑘𝐶 ({𝑘} × 𝐷))
9088, 89eleqtrrd 2167 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → ⟨𝑚, 𝑛⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵))
91 eliun 3734 . . . . . . . . . . . 12 (⟨𝑚, 𝑛⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
9290, 91sylib 120 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → ∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
93 simpr 108 . . . . . . . . . . . . . . . 16 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
94 opelxp 4467 . . . . . . . . . . . . . . . 16 (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) ↔ (𝑚 ∈ {𝑗} ∧ 𝑛𝐵))
9593, 94sylib 120 . . . . . . . . . . . . . . 15 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → (𝑚 ∈ {𝑗} ∧ 𝑛𝐵))
9695simpld 110 . . . . . . . . . . . . . 14 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 ∈ {𝑗})
97 elsni 3464 . . . . . . . . . . . . . 14 (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
9896, 97syl 14 . . . . . . . . . . . . 13 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 = 𝑗)
99 simpl 107 . . . . . . . . . . . . 13 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑗𝐴)
10098, 99eqeltrd 2164 . . . . . . . . . . . 12 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚𝐴)
101100rexlimiva 2484 . . . . . . . . . . 11 (∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑚𝐴)
10292, 101syl 14 . . . . . . . . . 10 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑚𝐴)
10325nfcri 2222 . . . . . . . . . . . 12 𝑗 𝑛𝑚 / 𝑗𝐵
10497equcomd 1640 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ {𝑗} → 𝑗 = 𝑚)
105104, 28syl 14 . . . . . . . . . . . . . . . 16 (𝑚 ∈ {𝑗} → 𝐵 = 𝑚 / 𝑗𝐵)
106105eleq2d 2157 . . . . . . . . . . . . . . 15 (𝑚 ∈ {𝑗} → (𝑛𝐵𝑛𝑚 / 𝑗𝐵))
107106biimpa 290 . . . . . . . . . . . . . 14 ((𝑚 ∈ {𝑗} ∧ 𝑛𝐵) → 𝑛𝑚 / 𝑗𝐵)
10894, 107sylbi 119 . . . . . . . . . . . . 13 (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛𝑚 / 𝑗𝐵)
109108a1i 9 . . . . . . . . . . . 12 (𝑗𝐴 → (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛𝑚 / 𝑗𝐵))
110103, 109rexlimi 2482 . . . . . . . . . . 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 2455 . . . . . . . . . . . . 13 (𝜑 → ∀𝑗𝐴𝑘𝐵 𝐸 ∈ ℂ)
114 nfcsb1v 2963 . . . . . . . . . . . . . . . 16 𝑗𝑚 / 𝑗𝐸
115114nfel1 2239 . . . . . . . . . . . . . . 15 𝑗𝑚 / 𝑗𝐸 ∈ ℂ
11625, 115nfralxy 2414 . . . . . . . . . . . . . 14 𝑗𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ
117 csbeq1a 2941 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑚𝐸 = 𝑚 / 𝑗𝐸)
118117eleq1d 2156 . . . . . . . . . . . . . . 15 (𝑗 = 𝑚 → (𝐸 ∈ ℂ ↔ 𝑚 / 𝑗𝐸 ∈ ℂ))
11928, 118raleqbidv 2574 . . . . . . . . . . . . . 14 (𝑗 = 𝑚 → (∀𝑘𝐵 𝐸 ∈ ℂ ↔ ∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ))
120116, 119rspc 2716 . . . . . . . . . . . . 13 (𝑚𝐴 → (∀𝑗𝐴𝑘𝐵 𝐸 ∈ ℂ → ∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ))
121113, 120mpan9 275 . . . . . . . . . . . 12 ((𝜑𝑚𝐴) → ∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ)
122 nfcsb1v 2963 . . . . . . . . . . . . . 14 𝑘𝑛 / 𝑘𝑚 / 𝑗𝐸
123122nfel1 2239 . . . . . . . . . . . . 13 𝑘𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ
124 csbeq1a 2941 . . . . . . . . . . . . . 14 (𝑘 = 𝑛𝑚 / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
125124eleq1d 2156 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑚 / 𝑗𝐸 ∈ ℂ ↔ 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
126123, 125rspc 2716 . . . . . . . . . . . 12 (𝑛𝑚 / 𝑗𝐵 → (∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ → 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
127121, 126syl5com 29 . . . . . . . . . . 11 ((𝜑𝑚𝐴) → (𝑛𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
128127impr 371 . . . . . . . . . 10 ((𝜑 ∧ (𝑚𝐴𝑛𝑚 / 𝑗𝐵)) → 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
12984, 102, 111, 128syl12anc 1172 . . . . . . . . 9 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
130129ralrimivva 2455 . . . . . . . 8 (𝜑 → ∀𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
131130adantr 270 . . . . . . 7 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → ∀𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
132 simpr 108 . . . . . . . . 9 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → 𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
133 eliun 3734 . . . . . . . . 9 (𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ↔ ∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷))
134132, 133sylib 120 . . . . . . . 8 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → ∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷))
135 xp1st 5936 . . . . . . . . . . . 12 (𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (1st𝑤) ∈ {𝑛})
136135adantl 271 . . . . . . . . . . 11 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) ∈ {𝑛})
137 elsni 3464 . . . . . . . . . . 11 ((1st𝑤) ∈ {𝑛} → (1st𝑤) = 𝑛)
138136, 137syl 14 . . . . . . . . . 10 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) = 𝑛)
139 simpl 107 . . . . . . . . . 10 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → 𝑛𝐶)
140138, 139eqeltrd 2164 . . . . . . . . 9 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) ∈ 𝐶)
141140rexlimiva 2484 . . . . . . . 8 (∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (1st𝑤) ∈ 𝐶)
142134, 141syl 14 . . . . . . 7 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) ∈ 𝐶)
14383, 131, 142rspcdva 2727 . . . . . 6 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → ∀𝑚 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
144 xp2nd 5937 . . . . . . . . . 10 (𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (2nd𝑤) ∈ 𝑛 / 𝑘𝐷)
145144adantl 271 . . . . . . . . 9 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (2nd𝑤) ∈ 𝑛 / 𝑘𝐷)
146138csbeq1d 2939 . . . . . . . . 9 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) / 𝑘𝐷 = 𝑛 / 𝑘𝐷)
147145, 146eleqtrrd 2167 . . . . . . . 8 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
148147rexlimiva 2484 . . . . . . 7 (∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
149134, 148syl 14 . . . . . 6 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
15079, 143, 149rspcdva 2727 . . . . 5 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 ∈ ℂ)
15149, 55, 71, 76, 150fsumcnv 10831 . . . 4 (𝜑 → Σ𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = Σ𝑧 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
15241, 151eqtr4d 2123 . . 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 2446 . . . . 5 (𝜑 → ∀𝑗𝐴 𝐵 ∈ Fin)
15625nfel1 2239 . . . . . 6 𝑗𝑚 / 𝑗𝐵 ∈ Fin
15728eleq1d 2156 . . . . . 6 (𝑗 = 𝑚 → (𝐵 ∈ Fin ↔ 𝑚 / 𝑗𝐵 ∈ Fin))
158156, 157rspc 2716 . . . . 5 (𝑚𝐴 → (∀𝑗𝐴 𝐵 ∈ Fin → 𝑚 / 𝑗𝐵 ∈ Fin))
159155, 158mpan9 275 . . . 4 ((𝜑𝑚𝐴) → 𝑚 / 𝑗𝐵 ∈ Fin)
16055, 153, 159, 128fsum2d 10829 . . 3 (𝜑 → Σ𝑚𝐴 Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸 = Σ𝑧 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
16149, 56, 64, 129fsum2d 10829 . . 3 (𝜑 → Σ𝑛𝐶 Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 = Σ𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
162152, 160, 1613eqtr4d 2130 . 2 (𝜑 → Σ𝑚𝐴 Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸 = Σ𝑛𝐶 Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸)
163 nfcv 2228 . . 3 𝑚Σ𝑘𝐵 𝐸
164 nfcv 2228 . . . . 5 𝑗𝑛
165164, 114nfcsb 2965 . . . 4 𝑗𝑛 / 𝑘𝑚 / 𝑗𝐸
16625, 165nfsum 10746 . . 3 𝑗Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸
167 nfcv 2228 . . . . 5 𝑛𝐸
168 nfcsb1v 2963 . . . . 5 𝑘𝑛 / 𝑘𝐸
169 csbeq1a 2941 . . . . 5 (𝑘 = 𝑛𝐸 = 𝑛 / 𝑘𝐸)
170167, 168, 169cbvsumi 10751 . . . 4 Σ𝑘𝐵 𝐸 = Σ𝑛𝐵 𝑛 / 𝑘𝐸
171117csbeq2dv 2956 . . . . . 6 (𝑗 = 𝑚𝑛 / 𝑘𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
172171adantr 270 . . . . 5 ((𝑗 = 𝑚𝑛𝐵) → 𝑛 / 𝑘𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
17328, 172sumeq12dv 10761 . . . 4 (𝑗 = 𝑚 → Σ𝑛𝐵 𝑛 / 𝑘𝐸 = Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸)
174170, 173syl5eq 2132 . . 3 (𝑗 = 𝑚 → Σ𝑘𝐵 𝐸 = Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸)
175163, 166, 174cbvsumi 10751 . 2 Σ𝑗𝐴 Σ𝑘𝐵 𝐸 = Σ𝑚𝐴 Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸
176 nfcv 2228 . . 3 𝑛Σ𝑗𝐷 𝐸
17733, 122nfsum 10746 . . 3 𝑘Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸
178 nfcv 2228 . . . . 5 𝑚𝐸
179178, 114, 117cbvsumi 10751 . . . 4 Σ𝑗𝐷 𝐸 = Σ𝑚𝐷 𝑚 / 𝑗𝐸
180124adantr 270 . . . . 5 ((𝑘 = 𝑛𝑚𝐷) → 𝑚 / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
18136, 180sumeq12dv 10761 . . . 4 (𝑘 = 𝑛 → Σ𝑚𝐷 𝑚 / 𝑗𝐸 = Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸)
182179, 181syl5eq 2132 . . 3 (𝑘 = 𝑛 → Σ𝑗𝐷 𝐸 = Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸)
183176, 177, 182cbvsumi 10751 . 2 Σ𝑘𝐶 Σ𝑗𝐷 𝐸 = Σ𝑛𝐶 Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸
184162, 175, 1833eqtr4g 2145 1 (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐸 = Σ𝑘𝐶 Σ𝑗𝐷 𝐸)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438  wral 2359  wrex 2360  Vcvv 2619  csb 2933  {csn 3446  cop 3449   ciun 3730  Disj wdisj 3822   × cxp 4436  ccnv 4437  Rel wrel 4443  cfv 5015  1st c1st 5909  2nd c2nd 5910  Fincfn 6457  cc 7348  Σcsu 10742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-disj 3823  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6292  df-en 6458  df-dom 6459  df-fin 6460  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-q 9105  df-rp 9135  df-fz 9425  df-fzo 9554  df-iseq 9853  df-seq3 9854  df-exp 9955  df-ihash 10184  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-clim 10667  df-isum 10743
This theorem is referenced by:  fsumcom  10833  fisum0diag  10835
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