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Theorem fsumcom2 15467
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)
fsumcom2.4 (𝜑 → ((𝑗𝐴𝑘𝐵) ↔ (𝑘𝐶𝑗𝐷)))
fsumcom2.5 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐸 ∈ ℂ)
Assertion
Ref Expression
fsumcom2 (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 fsumcom2
Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 5606 . . . . . . . . 9 Rel ({𝑗} × 𝐵)
21rgenw 3077 . . . . . . . 8 𝑗𝐴 Rel ({𝑗} × 𝐵)
3 reliun 5723 . . . . . . . 8 (Rel 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∀𝑗𝐴 Rel ({𝑗} × 𝐵))
42, 3mpbir 230 . . . . . . 7 Rel 𝑗𝐴 ({𝑗} × 𝐵)
5 relcnv 6009 . . . . . . 7 Rel 𝑘𝐶 ({𝑘} × 𝐷)
6 ancom 460 . . . . . . . . . . . 12 ((𝑥 = 𝑗𝑦 = 𝑘) ↔ (𝑦 = 𝑘𝑥 = 𝑗))
7 vex 3434 . . . . . . . . . . . . 13 𝑥 ∈ V
8 vex 3434 . . . . . . . . . . . . 13 𝑦 ∈ V
97, 8opth 5393 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ (𝑥 = 𝑗𝑦 = 𝑘))
108, 7opth 5393 . . . . . . . . . . . 12 (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ↔ (𝑦 = 𝑘𝑥 = 𝑗))
116, 9, 103bitr4i 302 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩)
1211a1i 11 . . . . . . . . . 10 (𝜑 → (⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩))
13 fsumcom2.4 . . . . . . . . . 10 (𝜑 → ((𝑗𝐴𝑘𝐵) ↔ (𝑘𝐶𝑗𝐷)))
1412, 13anbi12d 630 . . . . . . . . 9 (𝜑 → ((⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷))))
15142exbidv 1930 . . . . . . . 8 (𝜑 → (∃𝑗𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷))))
16 eliunxp 5743 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝑘(⟨𝑥, 𝑦⟩ = ⟨𝑗, 𝑘⟩ ∧ (𝑗𝐴𝑘𝐵)))
177, 8opelcnv 5787 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
18 eliunxp 5743 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
19 excom 2165 . . . . . . . . 9 (∃𝑘𝑗(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
2017, 18, 193bitri 296 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ∃𝑗𝑘(⟨𝑦, 𝑥⟩ = ⟨𝑘, 𝑗⟩ ∧ (𝑘𝐶𝑗𝐷)))
2115, 16, 203bitr4g 313 . . . . . . 7 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷)))
224, 5, 21eqrelrdv 5699 . . . . . 6 (𝜑 𝑗𝐴 ({𝑗} × 𝐵) = 𝑘𝐶 ({𝑘} × 𝐷))
23 nfcv 2908 . . . . . . 7 𝑚({𝑗} × 𝐵)
24 nfcv 2908 . . . . . . . 8 𝑗{𝑚}
25 nfcsb1v 3861 . . . . . . . 8 𝑗𝑚 / 𝑗𝐵
2624, 25nfxp 5621 . . . . . . 7 𝑗({𝑚} × 𝑚 / 𝑗𝐵)
27 sneq 4576 . . . . . . . 8 (𝑗 = 𝑚 → {𝑗} = {𝑚})
28 csbeq1a 3850 . . . . . . . 8 (𝑗 = 𝑚𝐵 = 𝑚 / 𝑗𝐵)
2927, 28xpeq12d 5619 . . . . . . 7 (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × 𝑚 / 𝑗𝐵))
3023, 26, 29cbviun 4970 . . . . . 6 𝑗𝐴 ({𝑗} × 𝐵) = 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)
31 nfcv 2908 . . . . . . . 8 𝑛({𝑘} × 𝐷)
32 nfcv 2908 . . . . . . . . 9 𝑘{𝑛}
33 nfcsb1v 3861 . . . . . . . . 9 𝑘𝑛 / 𝑘𝐷
3432, 33nfxp 5621 . . . . . . . 8 𝑘({𝑛} × 𝑛 / 𝑘𝐷)
35 sneq 4576 . . . . . . . . 9 (𝑘 = 𝑛 → {𝑘} = {𝑛})
36 csbeq1a 3850 . . . . . . . . 9 (𝑘 = 𝑛𝐷 = 𝑛 / 𝑘𝐷)
3735, 36xpeq12d 5619 . . . . . . . 8 (𝑘 = 𝑛 → ({𝑘} × 𝐷) = ({𝑛} × 𝑛 / 𝑘𝐷))
3831, 34, 37cbviun 4970 . . . . . . 7 𝑘𝐶 ({𝑘} × 𝐷) = 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
3938cnveqi 5780 . . . . . 6 𝑘𝐶 ({𝑘} × 𝐷) = 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
4022, 30, 393eqtr3g 2802 . . . . 5 (𝜑 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵) = 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
4140sumeq1d 15394 . . . 4 (𝜑 → Σ𝑧 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = Σ𝑧 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
42 vex 3434 . . . . . . . 8 𝑛 ∈ V
43 vex 3434 . . . . . . . 8 𝑚 ∈ V
4442, 43op1std 7827 . . . . . . 7 (𝑤 = ⟨𝑛, 𝑚⟩ → (1st𝑤) = 𝑛)
4544csbeq1d 3840 . . . . . 6 (𝑤 = ⟨𝑛, 𝑚⟩ → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑛 / 𝑘(2nd𝑤) / 𝑗𝐸)
4642, 43op2ndd 7828 . . . . . . . 8 (𝑤 = ⟨𝑛, 𝑚⟩ → (2nd𝑤) = 𝑚)
4746csbeq1d 3840 . . . . . . 7 (𝑤 = ⟨𝑛, 𝑚⟩ → (2nd𝑤) / 𝑗𝐸 = 𝑚 / 𝑗𝐸)
4847csbeq2dv 3843 . . . . . 6 (𝑤 = ⟨𝑛, 𝑚⟩ → 𝑛 / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
4945, 48eqtrd 2779 . . . . 5 (𝑤 = ⟨𝑛, 𝑚⟩ → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
5043, 42op2ndd 7828 . . . . . . 7 (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd𝑧) = 𝑛)
5150csbeq1d 3840 . . . . . 6 (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑛 / 𝑘(1st𝑧) / 𝑗𝐸)
5243, 42op1std 7827 . . . . . . . 8 (𝑧 = ⟨𝑚, 𝑛⟩ → (1st𝑧) = 𝑚)
5352csbeq1d 3840 . . . . . . 7 (𝑧 = ⟨𝑚, 𝑛⟩ → (1st𝑧) / 𝑗𝐸 = 𝑚 / 𝑗𝐸)
5453csbeq2dv 3843 . . . . . 6 (𝑧 = ⟨𝑚, 𝑛⟩ → 𝑛 / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
5551, 54eqtrd 2779 . . . . 5 (𝑧 = ⟨𝑚, 𝑛⟩ → (2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
56 fsumcom2.2 . . . . . 6 (𝜑𝐶 ∈ Fin)
57 snfi 8804 . . . . . . . 8 {𝑛} ∈ Fin
58 fsumcom2.1 . . . . . . . . . 10 (𝜑𝐴 ∈ Fin)
5958adantr 480 . . . . . . . . 9 ((𝜑𝑛𝐶) → 𝐴 ∈ Fin)
6043, 42opelcnv 5787 . . . . . . . . . . . . . . . 16 (⟨𝑚, 𝑛⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ ⟨𝑛, 𝑚⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
6133, 36opeliunxp2f 8010 . . . . . . . . . . . . . . . 16 (⟨𝑛, 𝑚⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷) ↔ (𝑛𝐶𝑚𝑛 / 𝑘𝐷))
6260, 61sylbbr 235 . . . . . . . . . . . . . . 15 ((𝑛𝐶𝑚𝑛 / 𝑘𝐷) → ⟨𝑚, 𝑛⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
6362adantl 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → ⟨𝑚, 𝑛⟩ ∈ 𝑘𝐶 ({𝑘} × 𝐷))
6422adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑗𝐴 ({𝑗} × 𝐵) = 𝑘𝐶 ({𝑘} × 𝐷))
6563, 64eleqtrrd 2843 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → ⟨𝑚, 𝑛⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵))
66 eliun 4933 . . . . . . . . . . . . 13 (⟨𝑚, 𝑛⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
6765, 66sylib 217 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → ∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
68 simpr 484 . . . . . . . . . . . . . . . . 17 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵))
69 opelxp 5624 . . . . . . . . . . . . . . . . 17 (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) ↔ (𝑚 ∈ {𝑗} ∧ 𝑛𝐵))
7068, 69sylib 217 . . . . . . . . . . . . . . . 16 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → (𝑚 ∈ {𝑗} ∧ 𝑛𝐵))
7170simpld 494 . . . . . . . . . . . . . . 15 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 ∈ {𝑗})
72 elsni 4583 . . . . . . . . . . . . . . 15 (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
7371, 72syl 17 . . . . . . . . . . . . . 14 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚 = 𝑗)
74 simpl 482 . . . . . . . . . . . . . 14 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑗𝐴)
7573, 74eqeltrd 2840 . . . . . . . . . . . . 13 ((𝑗𝐴 ∧ ⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵)) → 𝑚𝐴)
7675rexlimiva 3211 . . . . . . . . . . . 12 (∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑚𝐴)
7767, 76syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑚𝐴)
7877expr 456 . . . . . . . . . 10 ((𝜑𝑛𝐶) → (𝑚𝑛 / 𝑘𝐷𝑚𝐴))
7978ssrdv 3931 . . . . . . . . 9 ((𝜑𝑛𝐶) → 𝑛 / 𝑘𝐷𝐴)
8059, 79ssfid 9003 . . . . . . . 8 ((𝜑𝑛𝐶) → 𝑛 / 𝑘𝐷 ∈ Fin)
81 xpfi 9046 . . . . . . . 8 (({𝑛} ∈ Fin ∧ 𝑛 / 𝑘𝐷 ∈ Fin) → ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
8257, 80, 81sylancr 586 . . . . . . 7 ((𝜑𝑛𝐶) → ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
8382ralrimiva 3109 . . . . . 6 (𝜑 → ∀𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
84 iunfi 9068 . . . . . 6 ((𝐶 ∈ Fin ∧ ∀𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin) → 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
8556, 83, 84syl2anc 583 . . . . 5 (𝜑 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ∈ Fin)
86 reliun 5723 . . . . . . 7 (Rel 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ↔ ∀𝑛𝐶 Rel ({𝑛} × 𝑛 / 𝑘𝐷))
87 relxp 5606 . . . . . . . 8 Rel ({𝑛} × 𝑛 / 𝑘𝐷)
8887a1i 11 . . . . . . 7 (𝑛𝐶 → Rel ({𝑛} × 𝑛 / 𝑘𝐷))
8986, 88mprgbir 3080 . . . . . 6 Rel 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)
9089a1i 11 . . . . 5 (𝜑 → Rel 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
91 csbeq1 3839 . . . . . . . 8 (𝑚 = (2nd𝑤) → 𝑚 / 𝑗𝐸 = (2nd𝑤) / 𝑗𝐸)
9291csbeq2dv 3843 . . . . . . 7 (𝑚 = (2nd𝑤) → (1st𝑤) / 𝑘𝑚 / 𝑗𝐸 = (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
9392eleq1d 2824 . . . . . 6 (𝑚 = (2nd𝑤) → ((1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ ↔ (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 ∈ ℂ))
94 csbeq1 3839 . . . . . . . 8 (𝑛 = (1st𝑤) → 𝑛 / 𝑘𝐷 = (1st𝑤) / 𝑘𝐷)
95 csbeq1 3839 . . . . . . . . 9 (𝑛 = (1st𝑤) → 𝑛 / 𝑘𝑚 / 𝑗𝐸 = (1st𝑤) / 𝑘𝑚 / 𝑗𝐸)
9695eleq1d 2824 . . . . . . . 8 (𝑛 = (1st𝑤) → (𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ ↔ (1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
9794, 96raleqbidv 3334 . . . . . . 7 (𝑛 = (1st𝑤) → (∀𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ ↔ ∀𝑚 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
98 simpl 482 . . . . . . . . . 10 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝜑)
9925nfcri 2895 . . . . . . . . . . . 12 𝑗 𝑛𝑚 / 𝑗𝐵
10072equcomd 2025 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ {𝑗} → 𝑗 = 𝑚)
101100, 28syl 17 . . . . . . . . . . . . . . . 16 (𝑚 ∈ {𝑗} → 𝐵 = 𝑚 / 𝑗𝐵)
102101eleq2d 2825 . . . . . . . . . . . . . . 15 (𝑚 ∈ {𝑗} → (𝑛𝐵𝑛𝑚 / 𝑗𝐵))
103102biimpa 476 . . . . . . . . . . . . . 14 ((𝑚 ∈ {𝑗} ∧ 𝑛𝐵) → 𝑛𝑚 / 𝑗𝐵)
10469, 103sylbi 216 . . . . . . . . . . . . 13 (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛𝑚 / 𝑗𝐵)
105104a1i 11 . . . . . . . . . . . 12 (𝑗𝐴 → (⟨𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛𝑚 / 𝑗𝐵))
10699, 105rexlimi 3245 . . . . . . . . . . 11 (∃𝑗𝐴𝑚, 𝑛⟩ ∈ ({𝑗} × 𝐵) → 𝑛𝑚 / 𝑗𝐵)
10767, 106syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑛𝑚 / 𝑗𝐵)
108 fsumcom2.5 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐸 ∈ ℂ)
109108ralrimivva 3116 . . . . . . . . . . . . 13 (𝜑 → ∀𝑗𝐴𝑘𝐵 𝐸 ∈ ℂ)
110 nfcsb1v 3861 . . . . . . . . . . . . . . . 16 𝑗𝑚 / 𝑗𝐸
111110nfel1 2924 . . . . . . . . . . . . . . 15 𝑗𝑚 / 𝑗𝐸 ∈ ℂ
11225, 111nfralw 3151 . . . . . . . . . . . . . 14 𝑗𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ
113 csbeq1a 3850 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑚𝐸 = 𝑚 / 𝑗𝐸)
114113eleq1d 2824 . . . . . . . . . . . . . . 15 (𝑗 = 𝑚 → (𝐸 ∈ ℂ ↔ 𝑚 / 𝑗𝐸 ∈ ℂ))
11528, 114raleqbidv 3334 . . . . . . . . . . . . . 14 (𝑗 = 𝑚 → (∀𝑘𝐵 𝐸 ∈ ℂ ↔ ∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ))
116112, 115rspc 3547 . . . . . . . . . . . . 13 (𝑚𝐴 → (∀𝑗𝐴𝑘𝐵 𝐸 ∈ ℂ → ∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ))
117109, 116mpan9 506 . . . . . . . . . . . 12 ((𝜑𝑚𝐴) → ∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ)
118 nfcsb1v 3861 . . . . . . . . . . . . . 14 𝑘𝑛 / 𝑘𝑚 / 𝑗𝐸
119118nfel1 2924 . . . . . . . . . . . . 13 𝑘𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ
120 csbeq1a 3850 . . . . . . . . . . . . . 14 (𝑘 = 𝑛𝑚 / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
121120eleq1d 2824 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝑚 / 𝑗𝐸 ∈ ℂ ↔ 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
122119, 121rspc 3547 . . . . . . . . . . . 12 (𝑛𝑚 / 𝑗𝐵 → (∀𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐸 ∈ ℂ → 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
123117, 122syl5com 31 . . . . . . . . . . 11 ((𝜑𝑚𝐴) → (𝑛𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ))
124123impr 454 . . . . . . . . . 10 ((𝜑 ∧ (𝑚𝐴𝑛𝑚 / 𝑗𝐵)) → 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
12598, 77, 107, 124syl12anc 833 . . . . . . . . 9 ((𝜑 ∧ (𝑛𝐶𝑚𝑛 / 𝑘𝐷)) → 𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
126125ralrimivva 3116 . . . . . . . 8 (𝜑 → ∀𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
127126adantr 480 . . . . . . 7 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → ∀𝑛𝐶𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
128 simpr 484 . . . . . . . . 9 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → 𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷))
129 eliun 4933 . . . . . . . . 9 (𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷) ↔ ∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷))
130128, 129sylib 217 . . . . . . . 8 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → ∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷))
131 xp1st 7849 . . . . . . . . . . . 12 (𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (1st𝑤) ∈ {𝑛})
132131adantl 481 . . . . . . . . . . 11 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) ∈ {𝑛})
133 elsni 4583 . . . . . . . . . . 11 ((1st𝑤) ∈ {𝑛} → (1st𝑤) = 𝑛)
134132, 133syl 17 . . . . . . . . . 10 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) = 𝑛)
135 simpl 482 . . . . . . . . . 10 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → 𝑛𝐶)
136134, 135eqeltrd 2840 . . . . . . . . 9 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) ∈ 𝐶)
137136rexlimiva 3211 . . . . . . . 8 (∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (1st𝑤) ∈ 𝐶)
138130, 137syl 17 . . . . . . 7 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) ∈ 𝐶)
13997, 127, 138rspcdva 3562 . . . . . 6 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → ∀𝑚 (1st𝑤) / 𝑘𝐷(1st𝑤) / 𝑘𝑚 / 𝑗𝐸 ∈ ℂ)
140 xp2nd 7850 . . . . . . . . . 10 (𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (2nd𝑤) ∈ 𝑛 / 𝑘𝐷)
141140adantl 481 . . . . . . . . 9 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (2nd𝑤) ∈ 𝑛 / 𝑘𝐷)
142134csbeq1d 3840 . . . . . . . . 9 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) / 𝑘𝐷 = 𝑛 / 𝑘𝐷)
143141, 142eleqtrrd 2843 . . . . . . . 8 ((𝑛𝐶𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷)) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
144143rexlimiva 3211 . . . . . . 7 (∃𝑛𝐶 𝑤 ∈ ({𝑛} × 𝑛 / 𝑘𝐷) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
145130, 144syl 17 . . . . . 6 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → (2nd𝑤) ∈ (1st𝑤) / 𝑘𝐷)
14693, 139, 145rspcdva 3562 . . . . 5 ((𝜑𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)) → (1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 ∈ ℂ)
14749, 55, 85, 90, 146fsumcnv 15466 . . . 4 (𝜑 → Σ𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸 = Σ𝑧 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
14841, 147eqtr4d 2782 . . 3 (𝜑 → Σ𝑧 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸 = Σ𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
149 fsumcom2.3 . . . . . 6 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
150149ralrimiva 3109 . . . . 5 (𝜑 → ∀𝑗𝐴 𝐵 ∈ Fin)
15125nfel1 2924 . . . . . 6 𝑗𝑚 / 𝑗𝐵 ∈ Fin
15228eleq1d 2824 . . . . . 6 (𝑗 = 𝑚 → (𝐵 ∈ Fin ↔ 𝑚 / 𝑗𝐵 ∈ Fin))
153151, 152rspc 3547 . . . . 5 (𝑚𝐴 → (∀𝑗𝐴 𝐵 ∈ Fin → 𝑚 / 𝑗𝐵 ∈ Fin))
154150, 153mpan9 506 . . . 4 ((𝜑𝑚𝐴) → 𝑚 / 𝑗𝐵 ∈ Fin)
15555, 58, 154, 124fsum2d 15464 . . 3 (𝜑 → Σ𝑚𝐴 Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸 = Σ𝑧 𝑚𝐴 ({𝑚} × 𝑚 / 𝑗𝐵)(2nd𝑧) / 𝑘(1st𝑧) / 𝑗𝐸)
15649, 56, 80, 125fsum2d 15464 . . 3 (𝜑 → Σ𝑛𝐶 Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸 = Σ𝑤 𝑛𝐶 ({𝑛} × 𝑛 / 𝑘𝐷)(1st𝑤) / 𝑘(2nd𝑤) / 𝑗𝐸)
157148, 155, 1563eqtr4d 2789 . 2 (𝜑 → Σ𝑚𝐴 Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸 = Σ𝑛𝐶 Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸)
158 nfcv 2908 . . 3 𝑚Σ𝑘𝐵 𝐸
159 nfcv 2908 . . . . 5 𝑗𝑛
160159, 110nfcsbw 3863 . . . 4 𝑗𝑛 / 𝑘𝑚 / 𝑗𝐸
16125, 160nfsum 15383 . . 3 𝑗Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸
162 nfcv 2908 . . . . 5 𝑛𝐸
163 nfcsb1v 3861 . . . . 5 𝑘𝑛 / 𝑘𝐸
164 csbeq1a 3850 . . . . 5 (𝑘 = 𝑛𝐸 = 𝑛 / 𝑘𝐸)
165162, 163, 164cbvsumi 15390 . . . 4 Σ𝑘𝐵 𝐸 = Σ𝑛𝐵 𝑛 / 𝑘𝐸
166113csbeq2dv 3843 . . . . . 6 (𝑗 = 𝑚𝑛 / 𝑘𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
167166adantr 480 . . . . 5 ((𝑗 = 𝑚𝑛𝐵) → 𝑛 / 𝑘𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
16828, 167sumeq12dv 15399 . . . 4 (𝑗 = 𝑚 → Σ𝑛𝐵 𝑛 / 𝑘𝐸 = Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸)
169165, 168eqtrid 2791 . . 3 (𝑗 = 𝑚 → Σ𝑘𝐵 𝐸 = Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸)
170158, 161, 169cbvsumi 15390 . 2 Σ𝑗𝐴 Σ𝑘𝐵 𝐸 = Σ𝑚𝐴 Σ𝑛 𝑚 / 𝑗𝐵𝑛 / 𝑘𝑚 / 𝑗𝐸
171 nfcv 2908 . . 3 𝑛Σ𝑗𝐷 𝐸
17233, 118nfsum 15383 . . 3 𝑘Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸
173 nfcv 2908 . . . . 5 𝑚𝐸
174173, 110, 113cbvsumi 15390 . . . 4 Σ𝑗𝐷 𝐸 = Σ𝑚𝐷 𝑚 / 𝑗𝐸
175120adantr 480 . . . . 5 ((𝑘 = 𝑛𝑚𝐷) → 𝑚 / 𝑗𝐸 = 𝑛 / 𝑘𝑚 / 𝑗𝐸)
17636, 175sumeq12dv 15399 . . . 4 (𝑘 = 𝑛 → Σ𝑚𝐷 𝑚 / 𝑗𝐸 = Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸)
177174, 176eqtrid 2791 . . 3 (𝑘 = 𝑛 → Σ𝑗𝐷 𝐸 = Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸)
178171, 172, 177cbvsumi 15390 . 2 Σ𝑘𝐶 Σ𝑗𝐷 𝐸 = Σ𝑛𝐶 Σ𝑚 𝑛 / 𝑘𝐷𝑛 / 𝑘𝑚 / 𝑗𝐸
179157, 170, 1783eqtr4g 2804 1 (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐸 = Σ𝑘𝐶 Σ𝑗𝐷 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1541  wex 1785  wcel 2109  wral 3065  wrex 3066  csb 3836  {csn 4566  cop 4572   ciun 4929   × cxp 5586  ccnv 5587  Rel wrel 5593  cfv 6430  1st c1st 7815  2nd c2nd 7816  Fincfn 8707  cc 10853  Σcsu 15378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-inf2 9360  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932  ax-pre-sup 10933
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-int 4885  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-se 5544  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-isom 6439  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-1st 7817  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-1o 8281  df-er 8472  df-en 8708  df-dom 8709  df-sdom 8710  df-fin 8711  df-sup 9162  df-oi 9230  df-card 9681  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-div 11616  df-nn 11957  df-2 12019  df-3 12020  df-n0 12217  df-z 12303  df-uz 12565  df-rp 12713  df-fz 13222  df-fzo 13365  df-seq 13703  df-exp 13764  df-hash 14026  df-cj 14791  df-re 14792  df-im 14793  df-sqrt 14927  df-abs 14928  df-clim 15178  df-sum 15379
This theorem is referenced by:  fsumcom  15468  fsum0diag  15470  fsumdvdsdiag  26314  dvdsflsumcom  26318  fsumfldivdiag  26320  logfac2  26346  chpchtsum  26348  logfaclbnd  26351  dchrisum0lem1  26645
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