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Theorem fsum2dlemstep 11196
 Description: Lemma for fsum2d 11197- induction step. (Contributed by Mario Carneiro, 23-Apr-2014.) (Revised by Jim Kingdon, 8-Oct-2022.)
Hypotheses
Ref Expression
fsum2d.1 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
fsum2d.2 (𝜑𝐴 ∈ Fin)
fsum2d.3 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
fsum2d.4 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
fsum2d.5 (𝜑 → ¬ 𝑦𝑥)
fsum2d.6 (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
fsum2dlemstep.x (𝜑𝑥 ∈ Fin)
fsum2d.7 (𝜓 ↔ Σ𝑗𝑥 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
Assertion
Ref Expression
fsum2dlemstep ((𝜑𝜓) → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
Distinct variable groups:   𝑗,𝑘,𝑥,𝑦,𝑧,𝐴   𝐵,𝑘,𝑥,𝑦,𝑧   𝐷,𝑗,𝑘,𝑥,𝑦   𝑥,𝐶,𝑦,𝑧   𝜑,𝑗,𝑘,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑧,𝑗,𝑘)   𝐵(𝑗)   𝐶(𝑗,𝑘)   𝐷(𝑧)

Proof of Theorem fsum2dlemstep
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . 4 ((𝜑𝜓) → 𝜓)
2 fsum2d.7 . . . 4 (𝜓 ↔ Σ𝑗𝑥 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
31, 2sylib 121 . . 3 ((𝜑𝜓) → Σ𝑗𝑥 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
4 nfcv 2279 . . . . . 6 𝑚Σ𝑘𝐵 𝐶
5 nfcsb1v 3030 . . . . . . 7 𝑗𝑚 / 𝑗𝐵
6 nfcsb1v 3030 . . . . . . 7 𝑗𝑚 / 𝑗𝐶
75, 6nfsum 11119 . . . . . 6 𝑗Σ𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶
8 csbeq1a 3007 . . . . . . 7 (𝑗 = 𝑚𝐵 = 𝑚 / 𝑗𝐵)
9 csbeq1a 3007 . . . . . . . 8 (𝑗 = 𝑚𝐶 = 𝑚 / 𝑗𝐶)
109adantr 274 . . . . . . 7 ((𝑗 = 𝑚𝑘𝐵) → 𝐶 = 𝑚 / 𝑗𝐶)
118, 10sumeq12dv 11134 . . . . . 6 (𝑗 = 𝑚 → Σ𝑘𝐵 𝐶 = Σ𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶)
124, 7, 11cbvsumi 11124 . . . . 5 Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶 = Σ𝑚 ∈ {𝑦𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶
13 fsum2d.6 . . . . . . . . 9 (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
1413unssbd 3249 . . . . . . . 8 (𝜑 → {𝑦} ⊆ 𝐴)
15 vex 2684 . . . . . . . . 9 𝑦 ∈ V
1615snss 3644 . . . . . . . 8 (𝑦𝐴 ↔ {𝑦} ⊆ 𝐴)
1714, 16sylibr 133 . . . . . . 7 (𝜑𝑦𝐴)
18 fsum2d.3 . . . . . . . . . 10 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
1918ralrimiva 2503 . . . . . . . . 9 (𝜑 → ∀𝑗𝐴 𝐵 ∈ Fin)
20 nfcsb1v 3030 . . . . . . . . . . 11 𝑗𝑦 / 𝑗𝐵
2120nfel1 2290 . . . . . . . . . 10 𝑗𝑦 / 𝑗𝐵 ∈ Fin
22 csbeq1a 3007 . . . . . . . . . . 11 (𝑗 = 𝑦𝐵 = 𝑦 / 𝑗𝐵)
2322eleq1d 2206 . . . . . . . . . 10 (𝑗 = 𝑦 → (𝐵 ∈ Fin ↔ 𝑦 / 𝑗𝐵 ∈ Fin))
2421, 23rspc 2778 . . . . . . . . 9 (𝑦𝐴 → (∀𝑗𝐴 𝐵 ∈ Fin → 𝑦 / 𝑗𝐵 ∈ Fin))
2517, 19, 24sylc 62 . . . . . . . 8 (𝜑𝑦 / 𝑗𝐵 ∈ Fin)
26 fsum2d.4 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
2726ralrimivva 2512 . . . . . . . . . 10 (𝜑 → ∀𝑗𝐴𝑘𝐵 𝐶 ∈ ℂ)
28 nfcsb1v 3030 . . . . . . . . . . . . 13 𝑗𝑦 / 𝑗𝐶
2928nfel1 2290 . . . . . . . . . . . 12 𝑗𝑦 / 𝑗𝐶 ∈ ℂ
3020, 29nfralxy 2469 . . . . . . . . . . 11 𝑗𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ
31 csbeq1a 3007 . . . . . . . . . . . . 13 (𝑗 = 𝑦𝐶 = 𝑦 / 𝑗𝐶)
3231eleq1d 2206 . . . . . . . . . . . 12 (𝑗 = 𝑦 → (𝐶 ∈ ℂ ↔ 𝑦 / 𝑗𝐶 ∈ ℂ))
3322, 32raleqbidv 2636 . . . . . . . . . . 11 (𝑗 = 𝑦 → (∀𝑘𝐵 𝐶 ∈ ℂ ↔ ∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ))
3430, 33rspc 2778 . . . . . . . . . 10 (𝑦𝐴 → (∀𝑗𝐴𝑘𝐵 𝐶 ∈ ℂ → ∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ))
3517, 27, 34sylc 62 . . . . . . . . 9 (𝜑 → ∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ)
3635r19.21bi 2518 . . . . . . . 8 ((𝜑𝑘𝑦 / 𝑗𝐵) → 𝑦 / 𝑗𝐶 ∈ ℂ)
3725, 36fsumcl 11162 . . . . . . 7 (𝜑 → Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ)
38 csbeq1 3001 . . . . . . . . 9 (𝑚 = 𝑦𝑚 / 𝑗𝐵 = 𝑦 / 𝑗𝐵)
39 csbeq1 3001 . . . . . . . . . 10 (𝑚 = 𝑦𝑚 / 𝑗𝐶 = 𝑦 / 𝑗𝐶)
4039adantr 274 . . . . . . . . 9 ((𝑚 = 𝑦𝑘𝑚 / 𝑗𝐵) → 𝑚 / 𝑗𝐶 = 𝑦 / 𝑗𝐶)
4138, 40sumeq12dv 11134 . . . . . . . 8 (𝑚 = 𝑦 → Σ𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶)
4241sumsn 11173 . . . . . . 7 ((𝑦𝐴 ∧ Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ) → Σ𝑚 ∈ {𝑦𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶)
4317, 37, 42syl2anc 408 . . . . . 6 (𝜑 → Σ𝑚 ∈ {𝑦𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶)
44 nfcv 2279 . . . . . . . 8 𝑚𝑦 / 𝑗𝐶
45 nfcsb1v 3030 . . . . . . . 8 𝑘𝑚 / 𝑘𝑦 / 𝑗𝐶
46 csbeq1a 3007 . . . . . . . 8 (𝑘 = 𝑚𝑦 / 𝑗𝐶 = 𝑚 / 𝑘𝑦 / 𝑗𝐶)
4744, 45, 46cbvsumi 11124 . . . . . . 7 Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 = Σ𝑚 𝑦 / 𝑗𝐵𝑚 / 𝑘𝑦 / 𝑗𝐶
48 csbeq1 3001 . . . . . . . . 9 (𝑚 = (2nd𝑧) → 𝑚 / 𝑘𝑦 / 𝑗𝐶 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
49 snfig 6701 . . . . . . . . . . 11 (𝑦 ∈ V → {𝑦} ∈ Fin)
5049elv 2685 . . . . . . . . . 10 {𝑦} ∈ Fin
51 xpfi 6811 . . . . . . . . . 10 (({𝑦} ∈ Fin ∧ 𝑦 / 𝑗𝐵 ∈ Fin) → ({𝑦} × 𝑦 / 𝑗𝐵) ∈ Fin)
5250, 25, 51sylancr 410 . . . . . . . . 9 (𝜑 → ({𝑦} × 𝑦 / 𝑗𝐵) ∈ Fin)
53 2ndconst 6112 . . . . . . . . . 10 (𝑦𝐴 → (2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵)):({𝑦} × 𝑦 / 𝑗𝐵)–1-1-onto𝑦 / 𝑗𝐵)
5417, 53syl 14 . . . . . . . . 9 (𝜑 → (2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵)):({𝑦} × 𝑦 / 𝑗𝐵)–1-1-onto𝑦 / 𝑗𝐵)
55 fvres 5438 . . . . . . . . . 10 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) → ((2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵))‘𝑧) = (2nd𝑧))
5655adantl 275 . . . . . . . . 9 ((𝜑𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)) → ((2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵))‘𝑧) = (2nd𝑧))
5745nfel1 2290 . . . . . . . . . . 11 𝑘𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ
5846eleq1d 2206 . . . . . . . . . . 11 (𝑘 = 𝑚 → (𝑦 / 𝑗𝐶 ∈ ℂ ↔ 𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ))
5957, 58rspc 2778 . . . . . . . . . 10 (𝑚𝑦 / 𝑗𝐵 → (∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ → 𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ))
6035, 59mpan9 279 . . . . . . . . 9 ((𝜑𝑚𝑦 / 𝑗𝐵) → 𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ)
6148, 52, 54, 56, 60fsumf1o 11152 . . . . . . . 8 (𝜑 → Σ𝑚 𝑦 / 𝑗𝐵𝑚 / 𝑘𝑦 / 𝑗𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
62 elxp 4551 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) ↔ ∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)))
63 nfv 1508 . . . . . . . . . . . . . . 15 𝑗 𝑧 = ⟨𝑚, 𝑘
64 nfv 1508 . . . . . . . . . . . . . . . 16 𝑗 𝑚 ∈ {𝑦}
6520nfcri 2273 . . . . . . . . . . . . . . . 16 𝑗 𝑘𝑦 / 𝑗𝐵
6664, 65nfan 1544 . . . . . . . . . . . . . . 15 𝑗(𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)
6763, 66nfan 1544 . . . . . . . . . . . . . 14 𝑗(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵))
6867nfex 1616 . . . . . . . . . . . . 13 𝑗𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵))
69 nfv 1508 . . . . . . . . . . . . 13 𝑚𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵))
70 opeq1 3700 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
7170eqeq2d 2149 . . . . . . . . . . . . . . 15 (𝑚 = 𝑗 → (𝑧 = ⟨𝑚, 𝑘⟩ ↔ 𝑧 = ⟨𝑗, 𝑘⟩))
72 eqtr2 2156 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝑗𝑚 = 𝑦) → 𝑗 = 𝑦)
7372, 22syl 14 . . . . . . . . . . . . . . . . . . 19 ((𝑚 = 𝑗𝑚 = 𝑦) → 𝐵 = 𝑦 / 𝑗𝐵)
7473eleq2d 2207 . . . . . . . . . . . . . . . . . 18 ((𝑚 = 𝑗𝑚 = 𝑦) → (𝑘𝐵𝑘𝑦 / 𝑗𝐵))
7574pm5.32da 447 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑗 → ((𝑚 = 𝑦𝑘𝐵) ↔ (𝑚 = 𝑦𝑘𝑦 / 𝑗𝐵)))
76 velsn 3539 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ {𝑦} ↔ 𝑚 = 𝑦)
7776anbi1i 453 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵) ↔ (𝑚 = 𝑦𝑘𝑦 / 𝑗𝐵))
7875, 77syl6rbbr 198 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵) ↔ (𝑚 = 𝑦𝑘𝐵)))
79 equequ1 1688 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑗 → (𝑚 = 𝑦𝑗 = 𝑦))
8079anbi1d 460 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ((𝑚 = 𝑦𝑘𝐵) ↔ (𝑗 = 𝑦𝑘𝐵)))
8178, 80bitrd 187 . . . . . . . . . . . . . . 15 (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵) ↔ (𝑗 = 𝑦𝑘𝐵)))
8271, 81anbi12d 464 . . . . . . . . . . . . . 14 (𝑚 = 𝑗 → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)) ↔ (𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵))))
8382exbidv 1797 . . . . . . . . . . . . 13 (𝑚 = 𝑗 → (∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)) ↔ ∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵))))
8468, 69, 83cbvex 1729 . . . . . . . . . . . 12 (∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)) ↔ ∃𝑗𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)))
8562, 84bitri 183 . . . . . . . . . . 11 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) ↔ ∃𝑗𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)))
86 nfv 1508 . . . . . . . . . . . 12 𝑗𝜑
87 nfcv 2279 . . . . . . . . . . . . . 14 𝑗(2nd𝑧)
8887, 28nfcsb 3032 . . . . . . . . . . . . 13 𝑗(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
8988nfeq2 2291 . . . . . . . . . . . 12 𝑗 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
90 nfv 1508 . . . . . . . . . . . . 13 𝑘𝜑
91 nfcsb1v 3030 . . . . . . . . . . . . . 14 𝑘(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
9291nfeq2 2291 . . . . . . . . . . . . 13 𝑘 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
93 fsum2d.1 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
9493ad2antlr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = 𝐶)
9531ad2antrl 481 . . . . . . . . . . . . . . 15 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐶 = 𝑦 / 𝑗𝐶)
96 fveq2 5414 . . . . . . . . . . . . . . . . . 18 (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd𝑧) = (2nd ‘⟨𝑗, 𝑘⟩))
97 vex 2684 . . . . . . . . . . . . . . . . . . 19 𝑗 ∈ V
98 vex 2684 . . . . . . . . . . . . . . . . . . 19 𝑘 ∈ V
9997, 98op2nd 6038 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨𝑗, 𝑘⟩) = 𝑘
10096, 99syl6req 2187 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd𝑧))
101100ad2antlr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝑘 = (2nd𝑧))
102 csbeq1a 3007 . . . . . . . . . . . . . . . 16 (𝑘 = (2nd𝑧) → 𝑦 / 𝑗𝐶 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
103101, 102syl 14 . . . . . . . . . . . . . . 15 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝑦 / 𝑗𝐶 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
10494, 95, 1033eqtrd 2174 . . . . . . . . . . . . . 14 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
105104expl 375 . . . . . . . . . . . . 13 (𝜑 → ((𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
10690, 92, 105exlimd 1576 . . . . . . . . . . . 12 (𝜑 → (∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
10786, 89, 106exlimd 1576 . . . . . . . . . . 11 (𝜑 → (∃𝑗𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
10885, 107syl5bi 151 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
109108imp 123 . . . . . . . . 9 ((𝜑𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
110109sumeq2dv 11130 . . . . . . . 8 (𝜑 → Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
11161, 110eqtr4d 2173 . . . . . . 7 (𝜑 → Σ𝑚 𝑦 / 𝑗𝐵𝑚 / 𝑘𝑦 / 𝑗𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
11247, 111syl5eq 2182 . . . . . 6 (𝜑 → Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
11343, 112eqtrd 2170 . . . . 5 (𝜑 → Σ𝑚 ∈ {𝑦𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
11412, 113syl5eq 2182 . . . 4 (𝜑 → Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
115114adantr 274 . . 3 ((𝜑𝜓) → Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
1163, 115oveq12d 5785 . 2 ((𝜑𝜓) → (Σ𝑗𝑥 Σ𝑘𝐵 𝐶 + Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶) = (Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 + Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷))
117 fsum2d.5 . . . . 5 (𝜑 → ¬ 𝑦𝑥)
118 disjsn 3580 . . . . 5 ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝑥)
119117, 118sylibr 133 . . . 4 (𝜑 → (𝑥 ∩ {𝑦}) = ∅)
120 eqidd 2138 . . . 4 (𝜑 → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦}))
121 fsum2dlemstep.x . . . . 5 (𝜑𝑥 ∈ Fin)
12250a1i 9 . . . . 5 (𝜑 → {𝑦} ∈ Fin)
123 unfidisj 6803 . . . . 5 ((𝑥 ∈ Fin ∧ {𝑦} ∈ Fin ∧ (𝑥 ∩ {𝑦}) = ∅) → (𝑥 ∪ {𝑦}) ∈ Fin)
124121, 122, 119, 123syl3anc 1216 . . . 4 (𝜑 → (𝑥 ∪ {𝑦}) ∈ Fin)
12513sselda 3092 . . . . 5 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝑗𝐴)
12626anassrs 397 . . . . . 6 (((𝜑𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
12718, 126fsumcl 11162 . . . . 5 ((𝜑𝑗𝐴) → Σ𝑘𝐵 𝐶 ∈ ℂ)
128125, 127syldan 280 . . . 4 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → Σ𝑘𝐵 𝐶 ∈ ℂ)
129119, 120, 124, 128fsumsplit 11169 . . 3 (𝜑 → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘𝐵 𝐶 = (Σ𝑗𝑥 Σ𝑘𝐵 𝐶 + Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶))
130129adantr 274 . 2 ((𝜑𝜓) → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘𝐵 𝐶 = (Σ𝑗𝑥 Σ𝑘𝐵 𝐶 + Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶))
131 eliun 3812 . . . . . . . . . 10 (𝑧 𝑗𝑥 ({𝑗} × 𝐵) ↔ ∃𝑗𝑥 𝑧 ∈ ({𝑗} × 𝐵))
132 xp1st 6056 . . . . . . . . . . . . . 14 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) ∈ {𝑗})
133 elsni 3540 . . . . . . . . . . . . . 14 ((1st𝑧) ∈ {𝑗} → (1st𝑧) = 𝑗)
134132, 133syl 14 . . . . . . . . . . . . 13 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) = 𝑗)
135134adantl 275 . . . . . . . . . . . 12 ((𝑗𝑥𝑧 ∈ ({𝑗} × 𝐵)) → (1st𝑧) = 𝑗)
136 simpl 108 . . . . . . . . . . . 12 ((𝑗𝑥𝑧 ∈ ({𝑗} × 𝐵)) → 𝑗𝑥)
137135, 136eqeltrd 2214 . . . . . . . . . . 11 ((𝑗𝑥𝑧 ∈ ({𝑗} × 𝐵)) → (1st𝑧) ∈ 𝑥)
138137rexlimiva 2542 . . . . . . . . . 10 (∃𝑗𝑥 𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) ∈ 𝑥)
139131, 138sylbi 120 . . . . . . . . 9 (𝑧 𝑗𝑥 ({𝑗} × 𝐵) → (1st𝑧) ∈ 𝑥)
140 xp1st 6056 . . . . . . . . 9 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) → (1st𝑧) ∈ {𝑦})
141139, 140anim12i 336 . . . . . . . 8 ((𝑧 𝑗𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)) → ((1st𝑧) ∈ 𝑥 ∧ (1st𝑧) ∈ {𝑦}))
142 elin 3254 . . . . . . . 8 (𝑧 ∈ ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) ↔ (𝑧 𝑗𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)))
143 elin 3254 . . . . . . . 8 ((1st𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ ((1st𝑧) ∈ 𝑥 ∧ (1st𝑧) ∈ {𝑦}))
144141, 142, 1433imtr4i 200 . . . . . . 7 (𝑧 ∈ ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) → (1st𝑧) ∈ (𝑥 ∩ {𝑦}))
145119eleq2d 2207 . . . . . . . 8 (𝜑 → ((1st𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ (1st𝑧) ∈ ∅))
146 noel 3362 . . . . . . . . 9 ¬ (1st𝑧) ∈ ∅
147146pm2.21i 635 . . . . . . . 8 ((1st𝑧) ∈ ∅ → 𝑧 ∈ ∅)
148145, 147syl6bi 162 . . . . . . 7 (𝜑 → ((1st𝑧) ∈ (𝑥 ∩ {𝑦}) → 𝑧 ∈ ∅))
149144, 148syl5 32 . . . . . 6 (𝜑 → (𝑧 ∈ ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) → 𝑧 ∈ ∅))
150149ssrdv 3098 . . . . 5 (𝜑 → ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) ⊆ ∅)
151 ss0 3398 . . . . 5 (( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) ⊆ ∅ → ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) = ∅)
152150, 151syl 14 . . . 4 (𝜑 → ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) = ∅)
153 iunxun 3887 . . . . . 6 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵))
154 nfcv 2279 . . . . . . . . 9 𝑚({𝑗} × 𝐵)
155 nfcv 2279 . . . . . . . . . 10 𝑗{𝑚}
156155, 5nfxp 4561 . . . . . . . . 9 𝑗({𝑚} × 𝑚 / 𝑗𝐵)
157 sneq 3533 . . . . . . . . . 10 (𝑗 = 𝑚 → {𝑗} = {𝑚})
158157, 8xpeq12d 4559 . . . . . . . . 9 (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × 𝑚 / 𝑗𝐵))
159154, 156, 158cbviun 3845 . . . . . . . 8 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = 𝑚 ∈ {𝑦} ({𝑚} × 𝑚 / 𝑗𝐵)
160 sneq 3533 . . . . . . . . . 10 (𝑚 = 𝑦 → {𝑚} = {𝑦})
161160, 38xpeq12d 4559 . . . . . . . . 9 (𝑚 = 𝑦 → ({𝑚} × 𝑚 / 𝑗𝐵) = ({𝑦} × 𝑦 / 𝑗𝐵))
16215, 161iunxsn 3884 . . . . . . . 8 𝑚 ∈ {𝑦} ({𝑚} × 𝑚 / 𝑗𝐵) = ({𝑦} × 𝑦 / 𝑗𝐵)
163159, 162eqtri 2158 . . . . . . 7 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ({𝑦} × 𝑦 / 𝑗𝐵)
164163uneq2i 3222 . . . . . 6 ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵)) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × 𝑦 / 𝑗𝐵))
165153, 164eqtri 2158 . . . . 5 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × 𝑦 / 𝑗𝐵))
166165a1i 9 . . . 4 (𝜑 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × 𝑦 / 𝑗𝐵)))
167 snfig 6701 . . . . . . . 8 (𝑗 ∈ V → {𝑗} ∈ Fin)
168167elv 2685 . . . . . . 7 {𝑗} ∈ Fin
169125, 18syldan 280 . . . . . . 7 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ Fin)
170 xpfi 6811 . . . . . . 7 (({𝑗} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑗} × 𝐵) ∈ Fin)
171168, 169, 170sylancr 410 . . . . . 6 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → ({𝑗} × 𝐵) ∈ Fin)
172171ralrimiva 2503 . . . . 5 (𝜑 → ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
173 disjsnxp 6127 . . . . . 6 Disj 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)
174173a1i 9 . . . . 5 (𝜑Disj 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵))
175 iunfidisj 6827 . . . . 5 (((𝑥 ∪ {𝑦}) ∈ Fin ∧ ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin ∧ Disj 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) → 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
176124, 172, 174, 175syl3anc 1216 . . . 4 (𝜑 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
177 eliun 3812 . . . . . 6 (𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ↔ ∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵))
178 elxp 4551 . . . . . . . 8 (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵)))
179 simprl 520 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑧 = ⟨𝑚, 𝑘⟩)
180 simprrl 528 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑚 ∈ {𝑗})
181 elsni 3540 . . . . . . . . . . . . . . 15 (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
182180, 181syl 14 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑚 = 𝑗)
183182opeq1d 3706 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
184179, 183eqtrd 2170 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑧 = ⟨𝑗, 𝑘⟩)
185184, 93syl 14 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝐷 = 𝐶)
186 simpll 518 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝜑)
187125adantr 274 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑗𝐴)
188 simprrr 529 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑘𝐵)
189186, 187, 188, 26syl12anc 1214 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝐶 ∈ ℂ)
190185, 189eqeltrd 2214 . . . . . . . . . 10 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝐷 ∈ ℂ)
191190ex 114 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵)) → 𝐷 ∈ ℂ))
192191exlimdvv 1869 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → (∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵)) → 𝐷 ∈ ℂ))
193178, 192syl5bi 151 . . . . . . 7 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → (𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
194193rexlimdva 2547 . . . . . 6 (𝜑 → (∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
195177, 194syl5bi 151 . . . . 5 (𝜑 → (𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
196195imp 123 . . . 4 ((𝜑𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) → 𝐷 ∈ ℂ)
197152, 166, 176, 196fsumsplit 11169 . . 3 (𝜑 → Σ𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 + Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷))
198197adantr 274 . 2 ((𝜑𝜓) → Σ𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 + Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷))
199116, 130, 1983eqtr4d 2180 1 ((𝜑𝜓) → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2414  ∃wrex 2415  Vcvv 2681  ⦋csb 2998   ∪ cun 3064   ∩ cin 3065   ⊆ wss 3066  ∅c0 3358  {csn 3522  ⟨cop 3525  ∪ ciun 3808  Disj wdisj 3901   × cxp 4532   ↾ cres 4536  –1-1-onto→wf1o 5117  ‘cfv 5118  (class class class)co 5767  1st c1st 6029  2nd c2nd 6030  Fincfn 6627  ℂcc 7611   + caddc 7616  Σcsu 11115 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733 This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-disj 3902  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-frec 6281  df-1o 6306  df-oadd 6310  df-er 6422  df-en 6628  df-dom 6629  df-fin 6630  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-fz 9784  df-fzo 9913  df-seqfrec 10212  df-exp 10286  df-ihash 10515  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-clim 11041  df-sumdc 11116 This theorem is referenced by:  fsum2d  11197
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