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Theorem fsum2dlem 15113
Description: Lemma for fsum2d 15114- induction step. (Contributed by Mario Carneiro, 23-Apr-2014.)
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)
fsum2d.7 (𝜓 ↔ Σ𝑗𝑥 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
Assertion
Ref Expression
fsum2dlem ((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 fsum2dlem
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 simpr 485 . . . 4 ((𝜑𝜓) → 𝜓)
2 fsum2d.7 . . . 4 (𝜓 ↔ Σ𝑗𝑥 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
31, 2sylib 219 . . 3 ((𝜑𝜓) → Σ𝑗𝑥 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
4 nfcv 2974 . . . . . 6 𝑚Σ𝑘𝐵 𝐶
5 nfcsb1v 3904 . . . . . . 7 𝑗𝑚 / 𝑗𝐵
6 nfcsb1v 3904 . . . . . . 7 𝑗𝑚 / 𝑗𝐶
75, 6nfsumw 15035 . . . . . 6 𝑗Σ𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶
8 csbeq1a 3894 . . . . . . 7 (𝑗 = 𝑚𝐵 = 𝑚 / 𝑗𝐵)
9 csbeq1a 3894 . . . . . . . 8 (𝑗 = 𝑚𝐶 = 𝑚 / 𝑗𝐶)
109adantr 481 . . . . . . 7 ((𝑗 = 𝑚𝑘𝐵) → 𝐶 = 𝑚 / 𝑗𝐶)
118, 10sumeq12dv 15051 . . . . . 6 (𝑗 = 𝑚 → Σ𝑘𝐵 𝐶 = Σ𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶)
124, 7, 11cbvsumi 15042 . . . . 5 Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶 = Σ𝑚 ∈ {𝑦𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶
13 fsum2d.6 . . . . . . . . 9 (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
1413unssbd 4161 . . . . . . . 8 (𝜑 → {𝑦} ⊆ 𝐴)
15 vex 3495 . . . . . . . . 9 𝑦 ∈ V
1615snss 4710 . . . . . . . 8 (𝑦𝐴 ↔ {𝑦} ⊆ 𝐴)
1714, 16sylibr 235 . . . . . . 7 (𝜑𝑦𝐴)
18 fsum2d.3 . . . . . . . . . 10 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
1918ralrimiva 3179 . . . . . . . . 9 (𝜑 → ∀𝑗𝐴 𝐵 ∈ Fin)
20 nfcsb1v 3904 . . . . . . . . . . 11 𝑗𝑦 / 𝑗𝐵
2120nfel1 2991 . . . . . . . . . 10 𝑗𝑦 / 𝑗𝐵 ∈ Fin
22 csbeq1a 3894 . . . . . . . . . . 11 (𝑗 = 𝑦𝐵 = 𝑦 / 𝑗𝐵)
2322eleq1d 2894 . . . . . . . . . 10 (𝑗 = 𝑦 → (𝐵 ∈ Fin ↔ 𝑦 / 𝑗𝐵 ∈ Fin))
2421, 23rspc 3608 . . . . . . . . 9 (𝑦𝐴 → (∀𝑗𝐴 𝐵 ∈ Fin → 𝑦 / 𝑗𝐵 ∈ Fin))
2517, 19, 24sylc 65 . . . . . . . 8 (𝜑𝑦 / 𝑗𝐵 ∈ Fin)
26 fsum2d.4 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)
2726ralrimivva 3188 . . . . . . . . . 10 (𝜑 → ∀𝑗𝐴𝑘𝐵 𝐶 ∈ ℂ)
28 nfcsb1v 3904 . . . . . . . . . . . . 13 𝑗𝑦 / 𝑗𝐶
2928nfel1 2991 . . . . . . . . . . . 12 𝑗𝑦 / 𝑗𝐶 ∈ ℂ
3020, 29nfralw 3222 . . . . . . . . . . 11 𝑗𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ
31 csbeq1a 3894 . . . . . . . . . . . . 13 (𝑗 = 𝑦𝐶 = 𝑦 / 𝑗𝐶)
3231eleq1d 2894 . . . . . . . . . . . 12 (𝑗 = 𝑦 → (𝐶 ∈ ℂ ↔ 𝑦 / 𝑗𝐶 ∈ ℂ))
3322, 32raleqbidv 3399 . . . . . . . . . . 11 (𝑗 = 𝑦 → (∀𝑘𝐵 𝐶 ∈ ℂ ↔ ∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ))
3430, 33rspc 3608 . . . . . . . . . 10 (𝑦𝐴 → (∀𝑗𝐴𝑘𝐵 𝐶 ∈ ℂ → ∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ))
3517, 27, 34sylc 65 . . . . . . . . 9 (𝜑 → ∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ)
3635r19.21bi 3205 . . . . . . . 8 ((𝜑𝑘𝑦 / 𝑗𝐵) → 𝑦 / 𝑗𝐶 ∈ ℂ)
3725, 36fsumcl 15078 . . . . . . 7 (𝜑 → Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ)
38 csbeq1 3883 . . . . . . . . 9 (𝑚 = 𝑦𝑚 / 𝑗𝐵 = 𝑦 / 𝑗𝐵)
39 csbeq1 3883 . . . . . . . . . 10 (𝑚 = 𝑦𝑚 / 𝑗𝐶 = 𝑦 / 𝑗𝐶)
4039adantr 481 . . . . . . . . 9 ((𝑚 = 𝑦𝑘𝑚 / 𝑗𝐵) → 𝑚 / 𝑗𝐶 = 𝑦 / 𝑗𝐶)
4138, 40sumeq12dv 15051 . . . . . . . 8 (𝑚 = 𝑦 → Σ𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶)
4241sumsn 15089 . . . . . . 7 ((𝑦𝐴 ∧ Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ) → Σ𝑚 ∈ {𝑦𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶)
4317, 37, 42syl2anc 584 . . . . . 6 (𝜑 → Σ𝑚 ∈ {𝑦𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶)
44 nfcv 2974 . . . . . . . 8 𝑚𝑦 / 𝑗𝐶
45 nfcsb1v 3904 . . . . . . . 8 𝑘𝑚 / 𝑘𝑦 / 𝑗𝐶
46 csbeq1a 3894 . . . . . . . 8 (𝑘 = 𝑚𝑦 / 𝑗𝐶 = 𝑚 / 𝑘𝑦 / 𝑗𝐶)
4744, 45, 46cbvsumi 15042 . . . . . . 7 Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 = Σ𝑚 𝑦 / 𝑗𝐵𝑚 / 𝑘𝑦 / 𝑗𝐶
48 csbeq1 3883 . . . . . . . . 9 (𝑚 = (2nd𝑧) → 𝑚 / 𝑘𝑦 / 𝑗𝐶 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
49 snfi 8582 . . . . . . . . . 10 {𝑦} ∈ Fin
50 xpfi 8777 . . . . . . . . . 10 (({𝑦} ∈ Fin ∧ 𝑦 / 𝑗𝐵 ∈ Fin) → ({𝑦} × 𝑦 / 𝑗𝐵) ∈ Fin)
5149, 25, 50sylancr 587 . . . . . . . . 9 (𝜑 → ({𝑦} × 𝑦 / 𝑗𝐵) ∈ Fin)
52 2ndconst 7785 . . . . . . . . . 10 (𝑦𝐴 → (2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵)):({𝑦} × 𝑦 / 𝑗𝐵)–1-1-onto𝑦 / 𝑗𝐵)
5317, 52syl 17 . . . . . . . . 9 (𝜑 → (2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵)):({𝑦} × 𝑦 / 𝑗𝐵)–1-1-onto𝑦 / 𝑗𝐵)
54 fvres 6682 . . . . . . . . . 10 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) → ((2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵))‘𝑧) = (2nd𝑧))
5554adantl 482 . . . . . . . . 9 ((𝜑𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)) → ((2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵))‘𝑧) = (2nd𝑧))
5645nfel1 2991 . . . . . . . . . . 11 𝑘𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ
5746eleq1d 2894 . . . . . . . . . . 11 (𝑘 = 𝑚 → (𝑦 / 𝑗𝐶 ∈ ℂ ↔ 𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ))
5856, 57rspc 3608 . . . . . . . . . 10 (𝑚𝑦 / 𝑗𝐵 → (∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ → 𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ))
5935, 58mpan9 507 . . . . . . . . 9 ((𝜑𝑚𝑦 / 𝑗𝐵) → 𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ)
6048, 51, 53, 55, 59fsumf1o 15068 . . . . . . . 8 (𝜑 → Σ𝑚 𝑦 / 𝑗𝐵𝑚 / 𝑘𝑦 / 𝑗𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
61 elxp 5571 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) ↔ ∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)))
62 nfv 1906 . . . . . . . . . . . . . . 15 𝑗 𝑧 = ⟨𝑚, 𝑘
63 nfv 1906 . . . . . . . . . . . . . . . 16 𝑗 𝑚 ∈ {𝑦}
6420nfcri 2968 . . . . . . . . . . . . . . . 16 𝑗 𝑘𝑦 / 𝑗𝐵
6563, 64nfan 1891 . . . . . . . . . . . . . . 15 𝑗(𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)
6662, 65nfan 1891 . . . . . . . . . . . . . 14 𝑗(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵))
6766nfex 2334 . . . . . . . . . . . . 13 𝑗𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵))
68 nfv 1906 . . . . . . . . . . . . 13 𝑚𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵))
69 opeq1 4795 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
7069eqeq2d 2829 . . . . . . . . . . . . . . 15 (𝑚 = 𝑗 → (𝑧 = ⟨𝑚, 𝑘⟩ ↔ 𝑧 = ⟨𝑗, 𝑘⟩))
71 eqtr2 2839 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝑗𝑚 = 𝑦) → 𝑗 = 𝑦)
7271, 22syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑚 = 𝑗𝑚 = 𝑦) → 𝐵 = 𝑦 / 𝑗𝐵)
7372eleq2d 2895 . . . . . . . . . . . . . . . . . 18 ((𝑚 = 𝑗𝑚 = 𝑦) → (𝑘𝐵𝑘𝑦 / 𝑗𝐵))
7473pm5.32da 579 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑗 → ((𝑚 = 𝑦𝑘𝐵) ↔ (𝑚 = 𝑦𝑘𝑦 / 𝑗𝐵)))
75 velsn 4573 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ {𝑦} ↔ 𝑚 = 𝑦)
7675anbi1i 623 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵) ↔ (𝑚 = 𝑦𝑘𝑦 / 𝑗𝐵))
7774, 76syl6rbbr 291 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵) ↔ (𝑚 = 𝑦𝑘𝐵)))
78 equequ1 2023 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑗 → (𝑚 = 𝑦𝑗 = 𝑦))
7978anbi1d 629 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ((𝑚 = 𝑦𝑘𝐵) ↔ (𝑗 = 𝑦𝑘𝐵)))
8077, 79bitrd 280 . . . . . . . . . . . . . . 15 (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵) ↔ (𝑗 = 𝑦𝑘𝐵)))
8170, 80anbi12d 630 . . . . . . . . . . . . . 14 (𝑚 = 𝑗 → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)) ↔ (𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵))))
8281exbidv 1913 . . . . . . . . . . . . 13 (𝑚 = 𝑗 → (∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)) ↔ ∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵))))
8367, 68, 82cbvexv1 2353 . . . . . . . . . . . 12 (∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)) ↔ ∃𝑗𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)))
8461, 83bitri 276 . . . . . . . . . . 11 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) ↔ ∃𝑗𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)))
85 nfv 1906 . . . . . . . . . . . 12 𝑗𝜑
86 nfcv 2974 . . . . . . . . . . . . . 14 𝑗(2nd𝑧)
8786, 28nfcsbw 3906 . . . . . . . . . . . . 13 𝑗(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
8887nfeq2 2992 . . . . . . . . . . . 12 𝑗 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
89 nfv 1906 . . . . . . . . . . . . 13 𝑘𝜑
90 nfcsb1v 3904 . . . . . . . . . . . . . 14 𝑘(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
9190nfeq2 2992 . . . . . . . . . . . . 13 𝑘 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
92 fsum2d.1 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
9392ad2antlr 723 . . . . . . . . . . . . . . 15 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = 𝐶)
9431ad2antrl 724 . . . . . . . . . . . . . . 15 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐶 = 𝑦 / 𝑗𝐶)
95 fveq2 6663 . . . . . . . . . . . . . . . . . 18 (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd𝑧) = (2nd ‘⟨𝑗, 𝑘⟩))
96 vex 3495 . . . . . . . . . . . . . . . . . . 19 𝑗 ∈ V
97 vex 3495 . . . . . . . . . . . . . . . . . . 19 𝑘 ∈ V
9896, 97op2nd 7687 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨𝑗, 𝑘⟩) = 𝑘
9995, 98syl6req 2870 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd𝑧))
10099ad2antlr 723 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝑘 = (2nd𝑧))
101 csbeq1a 3894 . . . . . . . . . . . . . . . 16 (𝑘 = (2nd𝑧) → 𝑦 / 𝑗𝐶 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
102100, 101syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝑦 / 𝑗𝐶 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
10393, 94, 1023eqtrd 2857 . . . . . . . . . . . . . 14 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
104103expl 458 . . . . . . . . . . . . 13 (𝜑 → ((𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
10589, 91, 104exlimd 2208 . . . . . . . . . . . 12 (𝜑 → (∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
10685, 88, 105exlimd 2208 . . . . . . . . . . 11 (𝜑 → (∃𝑗𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
10784, 106syl5bi 243 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
108107imp 407 . . . . . . . . 9 ((𝜑𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
109108sumeq2dv 15048 . . . . . . . 8 (𝜑 → Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
11060, 109eqtr4d 2856 . . . . . . 7 (𝜑 → Σ𝑚 𝑦 / 𝑗𝐵𝑚 / 𝑘𝑦 / 𝑗𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
11147, 110syl5eq 2865 . . . . . 6 (𝜑 → Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
11243, 111eqtrd 2853 . . . . 5 (𝜑 → Σ𝑚 ∈ {𝑦𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
11312, 112syl5eq 2865 . . . 4 (𝜑 → Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
114113adantr 481 . . 3 ((𝜑𝜓) → Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
1153, 114oveq12d 7163 . 2 ((𝜑𝜓) → (Σ𝑗𝑥 Σ𝑘𝐵 𝐶 + Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶) = (Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 + Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷))
116 fsum2d.5 . . . . 5 (𝜑 → ¬ 𝑦𝑥)
117 disjsn 4639 . . . . 5 ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝑥)
118116, 117sylibr 235 . . . 4 (𝜑 → (𝑥 ∩ {𝑦}) = ∅)
119 eqidd 2819 . . . 4 (𝜑 → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦}))
120 fsum2d.2 . . . . 5 (𝜑𝐴 ∈ Fin)
121120, 13ssfid 8729 . . . 4 (𝜑 → (𝑥 ∪ {𝑦}) ∈ Fin)
12213sselda 3964 . . . . 5 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝑗𝐴)
12326anassrs 468 . . . . . 6 (((𝜑𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
12418, 123fsumcl 15078 . . . . 5 ((𝜑𝑗𝐴) → Σ𝑘𝐵 𝐶 ∈ ℂ)
125122, 124syldan 591 . . . 4 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → Σ𝑘𝐵 𝐶 ∈ ℂ)
126118, 119, 121, 125fsumsplit 15085 . . 3 (𝜑 → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘𝐵 𝐶 = (Σ𝑗𝑥 Σ𝑘𝐵 𝐶 + Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶))
127126adantr 481 . 2 ((𝜑𝜓) → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘𝐵 𝐶 = (Σ𝑗𝑥 Σ𝑘𝐵 𝐶 + Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶))
128 eliun 4914 . . . . . . . . . 10 (𝑧 𝑗𝑥 ({𝑗} × 𝐵) ↔ ∃𝑗𝑥 𝑧 ∈ ({𝑗} × 𝐵))
129 xp1st 7710 . . . . . . . . . . . . . 14 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) ∈ {𝑗})
130 elsni 4574 . . . . . . . . . . . . . 14 ((1st𝑧) ∈ {𝑗} → (1st𝑧) = 𝑗)
131129, 130syl 17 . . . . . . . . . . . . 13 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) = 𝑗)
132131adantl 482 . . . . . . . . . . . 12 ((𝑗𝑥𝑧 ∈ ({𝑗} × 𝐵)) → (1st𝑧) = 𝑗)
133 simpl 483 . . . . . . . . . . . 12 ((𝑗𝑥𝑧 ∈ ({𝑗} × 𝐵)) → 𝑗𝑥)
134132, 133eqeltrd 2910 . . . . . . . . . . 11 ((𝑗𝑥𝑧 ∈ ({𝑗} × 𝐵)) → (1st𝑧) ∈ 𝑥)
135134rexlimiva 3278 . . . . . . . . . 10 (∃𝑗𝑥 𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) ∈ 𝑥)
136128, 135sylbi 218 . . . . . . . . 9 (𝑧 𝑗𝑥 ({𝑗} × 𝐵) → (1st𝑧) ∈ 𝑥)
137 xp1st 7710 . . . . . . . . 9 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) → (1st𝑧) ∈ {𝑦})
138136, 137anim12i 612 . . . . . . . 8 ((𝑧 𝑗𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)) → ((1st𝑧) ∈ 𝑥 ∧ (1st𝑧) ∈ {𝑦}))
139 elin 4166 . . . . . . . 8 (𝑧 ∈ ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) ↔ (𝑧 𝑗𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)))
140 elin 4166 . . . . . . . 8 ((1st𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ ((1st𝑧) ∈ 𝑥 ∧ (1st𝑧) ∈ {𝑦}))
141138, 139, 1403imtr4i 293 . . . . . . 7 (𝑧 ∈ ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) → (1st𝑧) ∈ (𝑥 ∩ {𝑦}))
142118eleq2d 2895 . . . . . . . 8 (𝜑 → ((1st𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ (1st𝑧) ∈ ∅))
143 noel 4293 . . . . . . . . 9 ¬ (1st𝑧) ∈ ∅
144143pm2.21i 119 . . . . . . . 8 ((1st𝑧) ∈ ∅ → 𝑧 ∈ ∅)
145142, 144syl6bi 254 . . . . . . 7 (𝜑 → ((1st𝑧) ∈ (𝑥 ∩ {𝑦}) → 𝑧 ∈ ∅))
146141, 145syl5 34 . . . . . 6 (𝜑 → (𝑧 ∈ ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) → 𝑧 ∈ ∅))
147146ssrdv 3970 . . . . 5 (𝜑 → ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) ⊆ ∅)
148 ss0 4349 . . . . 5 (( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) ⊆ ∅ → ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) = ∅)
149147, 148syl 17 . . . 4 (𝜑 → ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) = ∅)
150 iunxun 5007 . . . . . 6 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵))
151 nfcv 2974 . . . . . . . . 9 𝑚({𝑗} × 𝐵)
152 nfcv 2974 . . . . . . . . . 10 𝑗{𝑚}
153152, 5nfxp 5581 . . . . . . . . 9 𝑗({𝑚} × 𝑚 / 𝑗𝐵)
154 sneq 4567 . . . . . . . . . 10 (𝑗 = 𝑚 → {𝑗} = {𝑚})
155154, 8xpeq12d 5579 . . . . . . . . 9 (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × 𝑚 / 𝑗𝐵))
156151, 153, 155cbviun 4952 . . . . . . . 8 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = 𝑚 ∈ {𝑦} ({𝑚} × 𝑚 / 𝑗𝐵)
157 sneq 4567 . . . . . . . . . 10 (𝑚 = 𝑦 → {𝑚} = {𝑦})
158157, 38xpeq12d 5579 . . . . . . . . 9 (𝑚 = 𝑦 → ({𝑚} × 𝑚 / 𝑗𝐵) = ({𝑦} × 𝑦 / 𝑗𝐵))
15915, 158iunxsn 5004 . . . . . . . 8 𝑚 ∈ {𝑦} ({𝑚} × 𝑚 / 𝑗𝐵) = ({𝑦} × 𝑦 / 𝑗𝐵)
160156, 159eqtri 2841 . . . . . . 7 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ({𝑦} × 𝑦 / 𝑗𝐵)
161160uneq2i 4133 . . . . . 6 ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵)) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × 𝑦 / 𝑗𝐵))
162150, 161eqtri 2841 . . . . 5 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × 𝑦 / 𝑗𝐵))
163162a1i 11 . . . 4 (𝜑 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × 𝑦 / 𝑗𝐵)))
164 snfi 8582 . . . . . . 7 {𝑗} ∈ Fin
165122, 18syldan 591 . . . . . . 7 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ Fin)
166 xpfi 8777 . . . . . . 7 (({𝑗} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑗} × 𝐵) ∈ Fin)
167164, 165, 166sylancr 587 . . . . . 6 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → ({𝑗} × 𝐵) ∈ Fin)
168167ralrimiva 3179 . . . . 5 (𝜑 → ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
169 iunfi 8800 . . . . 5 (((𝑥 ∪ {𝑦}) ∈ Fin ∧ ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) → 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
170121, 168, 169syl2anc 584 . . . 4 (𝜑 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
171 eliun 4914 . . . . . 6 (𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ↔ ∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵))
172 elxp 5571 . . . . . . . 8 (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵)))
173 simprl 767 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑧 = ⟨𝑚, 𝑘⟩)
174 simprrl 777 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑚 ∈ {𝑗})
175 elsni 4574 . . . . . . . . . . . . . . 15 (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
176174, 175syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑚 = 𝑗)
177176opeq1d 4801 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
178173, 177eqtrd 2853 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑧 = ⟨𝑗, 𝑘⟩)
179178, 92syl 17 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝐷 = 𝐶)
180 simpll 763 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝜑)
181122adantr 481 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑗𝐴)
182 simprrr 778 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑘𝐵)
183180, 181, 182, 26syl12anc 832 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝐶 ∈ ℂ)
184179, 183eqeltrd 2910 . . . . . . . . . 10 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝐷 ∈ ℂ)
185184ex 413 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵)) → 𝐷 ∈ ℂ))
186185exlimdvv 1926 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → (∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵)) → 𝐷 ∈ ℂ))
187172, 186syl5bi 243 . . . . . . 7 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → (𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
188187rexlimdva 3281 . . . . . 6 (𝜑 → (∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
189171, 188syl5bi 243 . . . . 5 (𝜑 → (𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
190189imp 407 . . . 4 ((𝜑𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) → 𝐷 ∈ ℂ)
191149, 163, 170, 190fsumsplit 15085 . . 3 (𝜑 → Σ𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 + Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷))
192191adantr 481 . 2 ((𝜑𝜓) → Σ𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 + Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷))
193115, 127, 1923eqtr4d 2863 1 ((𝜑𝜓) → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  wral 3135  wrex 3136  csb 3880  cun 3931  cin 3932  wss 3933  c0 4288  {csn 4557  cop 4563   ciun 4910   × cxp 5546  cres 5550  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  Fincfn 8497  cc 10523   + caddc 10528  Σcsu 15030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031
This theorem is referenced by:  fsum2d  15114
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