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Theorem fsum2dlem 15660
Description: Lemma for fsum2d 15661- 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 486 . . . 4 ((𝜑𝜓) → 𝜓)
2 fsum2d.7 . . . 4 (𝜓 ↔ Σ𝑗𝑥 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
31, 2sylib 217 . . 3 ((𝜑𝜓) → Σ𝑗𝑥 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)
4 nfcv 2904 . . . . . 6 𝑚Σ𝑘𝐵 𝐶
5 nfcsb1v 3881 . . . . . . 7 𝑗𝑚 / 𝑗𝐵
6 nfcsb1v 3881 . . . . . . 7 𝑗𝑚 / 𝑗𝐶
75, 6nfsum 15581 . . . . . 6 𝑗Σ𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶
8 csbeq1a 3870 . . . . . . 7 (𝑗 = 𝑚𝐵 = 𝑚 / 𝑗𝐵)
9 csbeq1a 3870 . . . . . . . 8 (𝑗 = 𝑚𝐶 = 𝑚 / 𝑗𝐶)
109adantr 482 . . . . . . 7 ((𝑗 = 𝑚𝑘𝐵) → 𝐶 = 𝑚 / 𝑗𝐶)
118, 10sumeq12dv 15596 . . . . . 6 (𝑗 = 𝑚 → Σ𝑘𝐵 𝐶 = Σ𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶)
124, 7, 11cbvsumi 15587 . . . . 5 Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶 = Σ𝑚 ∈ {𝑦𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶
13 fsum2d.6 . . . . . . . . 9 (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
1413unssbd 4149 . . . . . . . 8 (𝜑 → {𝑦} ⊆ 𝐴)
15 vex 3448 . . . . . . . . 9 𝑦 ∈ V
1615snss 4747 . . . . . . . 8 (𝑦𝐴 ↔ {𝑦} ⊆ 𝐴)
1714, 16sylibr 233 . . . . . . 7 (𝜑𝑦𝐴)
18 fsum2d.3 . . . . . . . . . 10 ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)
1918ralrimiva 3140 . . . . . . . . 9 (𝜑 → ∀𝑗𝐴 𝐵 ∈ Fin)
20 nfcsb1v 3881 . . . . . . . . . . 11 𝑗𝑦 / 𝑗𝐵
2120nfel1 2920 . . . . . . . . . 10 𝑗𝑦 / 𝑗𝐵 ∈ Fin
22 csbeq1a 3870 . . . . . . . . . . 11 (𝑗 = 𝑦𝐵 = 𝑦 / 𝑗𝐵)
2322eleq1d 2819 . . . . . . . . . 10 (𝑗 = 𝑦 → (𝐵 ∈ Fin ↔ 𝑦 / 𝑗𝐵 ∈ Fin))
2421, 23rspc 3568 . . . . . . . . 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 3194 . . . . . . . . . 10 (𝜑 → ∀𝑗𝐴𝑘𝐵 𝐶 ∈ ℂ)
28 nfcsb1v 3881 . . . . . . . . . . . . 13 𝑗𝑦 / 𝑗𝐶
2928nfel1 2920 . . . . . . . . . . . 12 𝑗𝑦 / 𝑗𝐶 ∈ ℂ
3020, 29nfralw 3293 . . . . . . . . . . 11 𝑗𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ
31 csbeq1a 3870 . . . . . . . . . . . . 13 (𝑗 = 𝑦𝐶 = 𝑦 / 𝑗𝐶)
3231eleq1d 2819 . . . . . . . . . . . 12 (𝑗 = 𝑦 → (𝐶 ∈ ℂ ↔ 𝑦 / 𝑗𝐶 ∈ ℂ))
3322, 32raleqbidv 3318 . . . . . . . . . . 11 (𝑗 = 𝑦 → (∀𝑘𝐵 𝐶 ∈ ℂ ↔ ∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ))
3430, 33rspc 3568 . . . . . . . . . 10 (𝑦𝐴 → (∀𝑗𝐴𝑘𝐵 𝐶 ∈ ℂ → ∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ))
3517, 27, 34sylc 65 . . . . . . . . 9 (𝜑 → ∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ)
3635r19.21bi 3233 . . . . . . . 8 ((𝜑𝑘𝑦 / 𝑗𝐵) → 𝑦 / 𝑗𝐶 ∈ ℂ)
3725, 36fsumcl 15623 . . . . . . 7 (𝜑 → Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ)
38 csbeq1 3859 . . . . . . . . 9 (𝑚 = 𝑦𝑚 / 𝑗𝐵 = 𝑦 / 𝑗𝐵)
39 csbeq1 3859 . . . . . . . . . 10 (𝑚 = 𝑦𝑚 / 𝑗𝐶 = 𝑦 / 𝑗𝐶)
4039adantr 482 . . . . . . . . 9 ((𝑚 = 𝑦𝑘𝑚 / 𝑗𝐵) → 𝑚 / 𝑗𝐶 = 𝑦 / 𝑗𝐶)
4138, 40sumeq12dv 15596 . . . . . . . 8 (𝑚 = 𝑦 → Σ𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶)
4241sumsn 15636 . . . . . . 7 ((𝑦𝐴 ∧ Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ) → Σ𝑚 ∈ {𝑦𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶)
4317, 37, 42syl2anc 585 . . . . . 6 (𝜑 → Σ𝑚 ∈ {𝑦𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶)
44 nfcv 2904 . . . . . . . 8 𝑚𝑦 / 𝑗𝐶
45 nfcsb1v 3881 . . . . . . . 8 𝑘𝑚 / 𝑘𝑦 / 𝑗𝐶
46 csbeq1a 3870 . . . . . . . 8 (𝑘 = 𝑚𝑦 / 𝑗𝐶 = 𝑚 / 𝑘𝑦 / 𝑗𝐶)
4744, 45, 46cbvsumi 15587 . . . . . . 7 Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 = Σ𝑚 𝑦 / 𝑗𝐵𝑚 / 𝑘𝑦 / 𝑗𝐶
48 csbeq1 3859 . . . . . . . . 9 (𝑚 = (2nd𝑧) → 𝑚 / 𝑘𝑦 / 𝑗𝐶 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
49 snfi 8991 . . . . . . . . . 10 {𝑦} ∈ Fin
50 xpfi 9264 . . . . . . . . . 10 (({𝑦} ∈ Fin ∧ 𝑦 / 𝑗𝐵 ∈ Fin) → ({𝑦} × 𝑦 / 𝑗𝐵) ∈ Fin)
5149, 25, 50sylancr 588 . . . . . . . . 9 (𝜑 → ({𝑦} × 𝑦 / 𝑗𝐵) ∈ Fin)
52 2ndconst 8034 . . . . . . . . . 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 6862 . . . . . . . . . 10 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) → ((2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵))‘𝑧) = (2nd𝑧))
5554adantl 483 . . . . . . . . 9 ((𝜑𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)) → ((2nd ↾ ({𝑦} × 𝑦 / 𝑗𝐵))‘𝑧) = (2nd𝑧))
5645nfel1 2920 . . . . . . . . . . 11 𝑘𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ
5746eleq1d 2819 . . . . . . . . . . 11 (𝑘 = 𝑚 → (𝑦 / 𝑗𝐶 ∈ ℂ ↔ 𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ))
5856, 57rspc 3568 . . . . . . . . . 10 (𝑚𝑦 / 𝑗𝐵 → (∀𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 ∈ ℂ → 𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ))
5935, 58mpan9 508 . . . . . . . . 9 ((𝜑𝑚𝑦 / 𝑗𝐵) → 𝑚 / 𝑘𝑦 / 𝑗𝐶 ∈ ℂ)
6048, 51, 53, 55, 59fsumf1o 15613 . . . . . . . 8 (𝜑 → Σ𝑚 𝑦 / 𝑗𝐵𝑚 / 𝑘𝑦 / 𝑗𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
61 elxp 5657 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) ↔ ∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)))
62 nfv 1918 . . . . . . . . . . . . . . 15 𝑗 𝑧 = ⟨𝑚, 𝑘
63 nfv 1918 . . . . . . . . . . . . . . . 16 𝑗 𝑚 ∈ {𝑦}
6420nfcri 2891 . . . . . . . . . . . . . . . 16 𝑗 𝑘𝑦 / 𝑗𝐵
6563, 64nfan 1903 . . . . . . . . . . . . . . 15 𝑗(𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)
6662, 65nfan 1903 . . . . . . . . . . . . . 14 𝑗(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵))
6766nfex 2318 . . . . . . . . . . . . 13 𝑗𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵))
68 nfv 1918 . . . . . . . . . . . . 13 𝑚𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵))
69 opeq1 4831 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
7069eqeq2d 2744 . . . . . . . . . . . . . . 15 (𝑚 = 𝑗 → (𝑧 = ⟨𝑚, 𝑘⟩ ↔ 𝑧 = ⟨𝑗, 𝑘⟩))
71 velsn 4603 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ {𝑦} ↔ 𝑚 = 𝑦)
7271anbi1i 625 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵) ↔ (𝑚 = 𝑦𝑘𝑦 / 𝑗𝐵))
73 eqtr2 2757 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝑗𝑚 = 𝑦) → 𝑗 = 𝑦)
7473, 22syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑚 = 𝑗𝑚 = 𝑦) → 𝐵 = 𝑦 / 𝑗𝐵)
7574eleq2d 2820 . . . . . . . . . . . . . . . . . 18 ((𝑚 = 𝑗𝑚 = 𝑦) → (𝑘𝐵𝑘𝑦 / 𝑗𝐵))
7675pm5.32da 580 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑗 → ((𝑚 = 𝑦𝑘𝐵) ↔ (𝑚 = 𝑦𝑘𝑦 / 𝑗𝐵)))
7772, 76bitr4id 290 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵) ↔ (𝑚 = 𝑦𝑘𝐵)))
78 equequ1 2029 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑗 → (𝑚 = 𝑦𝑗 = 𝑦))
7978anbi1d 631 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ((𝑚 = 𝑦𝑘𝐵) ↔ (𝑗 = 𝑦𝑘𝐵)))
8077, 79bitrd 279 . . . . . . . . . . . . . . 15 (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵) ↔ (𝑗 = 𝑦𝑘𝐵)))
8170, 80anbi12d 632 . . . . . . . . . . . . . 14 (𝑚 = 𝑗 → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)) ↔ (𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵))))
8281exbidv 1925 . . . . . . . . . . . . 13 (𝑚 = 𝑗 → (∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)) ↔ ∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵))))
8367, 68, 82cbvexv1 2339 . . . . . . . . . . . 12 (∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘𝑦 / 𝑗𝐵)) ↔ ∃𝑗𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)))
8461, 83bitri 275 . . . . . . . . . . 11 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) ↔ ∃𝑗𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)))
85 nfv 1918 . . . . . . . . . . . 12 𝑗𝜑
86 nfcv 2904 . . . . . . . . . . . . . 14 𝑗(2nd𝑧)
8786, 28nfcsbw 3883 . . . . . . . . . . . . 13 𝑗(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
8887nfeq2 2921 . . . . . . . . . . . 12 𝑗 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
89 nfv 1918 . . . . . . . . . . . . 13 𝑘𝜑
90 nfcsb1v 3881 . . . . . . . . . . . . . 14 𝑘(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
9190nfeq2 2921 . . . . . . . . . . . . 13 𝑘 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶
92 fsum2d.1 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
9392ad2antlr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = 𝐶)
9431ad2antrl 727 . . . . . . . . . . . . . . 15 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐶 = 𝑦 / 𝑗𝐶)
95 fveq2 6843 . . . . . . . . . . . . . . . . . 18 (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd𝑧) = (2nd ‘⟨𝑗, 𝑘⟩))
96 vex 3448 . . . . . . . . . . . . . . . . . . 19 𝑗 ∈ V
97 vex 3448 . . . . . . . . . . . . . . . . . . 19 𝑘 ∈ V
9896, 97op2nd 7931 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨𝑗, 𝑘⟩) = 𝑘
9995, 98eqtr2di 2790 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd𝑧))
10099ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝑘 = (2nd𝑧))
101 csbeq1a 3870 . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . 14 (((𝜑𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
104103expl 459 . . . . . . . . . . . . 13 (𝜑 → ((𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
10589, 91, 104exlimd 2212 . . . . . . . . . . . 12 (𝜑 → (∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
10685, 88, 105exlimd 2212 . . . . . . . . . . 11 (𝜑 → (∃𝑗𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦𝑘𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
10784, 106biimtrid 241 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶))
108107imp 408 . . . . . . . . 9 ((𝜑𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)) → 𝐷 = (2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
109108sumeq2dv 15593 . . . . . . . 8 (𝜑 → Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)(2nd𝑧) / 𝑘𝑦 / 𝑗𝐶)
11060, 109eqtr4d 2776 . . . . . . 7 (𝜑 → Σ𝑚 𝑦 / 𝑗𝐵𝑚 / 𝑘𝑦 / 𝑗𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
11147, 110eqtrid 2785 . . . . . 6 (𝜑 → Σ𝑘 𝑦 / 𝑗𝐵𝑦 / 𝑗𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
11243, 111eqtrd 2773 . . . . 5 (𝜑 → Σ𝑚 ∈ {𝑦𝑘 𝑚 / 𝑗𝐵𝑚 / 𝑗𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
11312, 112eqtrid 2785 . . . 4 (𝜑 → Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
114113adantr 482 . . 3 ((𝜑𝜓) → Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶 = Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷)
1153, 114oveq12d 7376 . 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 4673 . . . . 5 ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝑥)
118116, 117sylibr 233 . . . 4 (𝜑 → (𝑥 ∩ {𝑦}) = ∅)
119 eqidd 2734 . . . 4 (𝜑 → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦}))
120 fsum2d.2 . . . . 5 (𝜑𝐴 ∈ Fin)
121120, 13ssfid 9214 . . . 4 (𝜑 → (𝑥 ∪ {𝑦}) ∈ Fin)
12213sselda 3945 . . . . 5 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝑗𝐴)
12326anassrs 469 . . . . . 6 (((𝜑𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
12418, 123fsumcl 15623 . . . . 5 ((𝜑𝑗𝐴) → Σ𝑘𝐵 𝐶 ∈ ℂ)
125122, 124syldan 592 . . . 4 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → Σ𝑘𝐵 𝐶 ∈ ℂ)
126118, 119, 121, 125fsumsplit 15631 . . 3 (𝜑 → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘𝐵 𝐶 = (Σ𝑗𝑥 Σ𝑘𝐵 𝐶 + Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶))
127126adantr 482 . 2 ((𝜑𝜓) → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘𝐵 𝐶 = (Σ𝑗𝑥 Σ𝑘𝐵 𝐶 + Σ𝑗 ∈ {𝑦𝑘𝐵 𝐶))
128 eliun 4959 . . . . . . . . . 10 (𝑧 𝑗𝑥 ({𝑗} × 𝐵) ↔ ∃𝑗𝑥 𝑧 ∈ ({𝑗} × 𝐵))
129 xp1st 7954 . . . . . . . . . . . . . 14 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) ∈ {𝑗})
130 elsni 4604 . . . . . . . . . . . . . 14 ((1st𝑧) ∈ {𝑗} → (1st𝑧) = 𝑗)
131129, 130syl 17 . . . . . . . . . . . . 13 (𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) = 𝑗)
132131adantl 483 . . . . . . . . . . . 12 ((𝑗𝑥𝑧 ∈ ({𝑗} × 𝐵)) → (1st𝑧) = 𝑗)
133 simpl 484 . . . . . . . . . . . 12 ((𝑗𝑥𝑧 ∈ ({𝑗} × 𝐵)) → 𝑗𝑥)
134132, 133eqeltrd 2834 . . . . . . . . . . 11 ((𝑗𝑥𝑧 ∈ ({𝑗} × 𝐵)) → (1st𝑧) ∈ 𝑥)
135134rexlimiva 3141 . . . . . . . . . 10 (∃𝑗𝑥 𝑧 ∈ ({𝑗} × 𝐵) → (1st𝑧) ∈ 𝑥)
136128, 135sylbi 216 . . . . . . . . 9 (𝑧 𝑗𝑥 ({𝑗} × 𝐵) → (1st𝑧) ∈ 𝑥)
137 xp1st 7954 . . . . . . . . 9 (𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵) → (1st𝑧) ∈ {𝑦})
138136, 137anim12i 614 . . . . . . . 8 ((𝑧 𝑗𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)) → ((1st𝑧) ∈ 𝑥 ∧ (1st𝑧) ∈ {𝑦}))
139 elin 3927 . . . . . . . 8 (𝑧 ∈ ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) ↔ (𝑧 𝑗𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)))
140 elin 3927 . . . . . . . 8 ((1st𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ ((1st𝑧) ∈ 𝑥 ∧ (1st𝑧) ∈ {𝑦}))
141138, 139, 1403imtr4i 292 . . . . . . 7 (𝑧 ∈ ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) → (1st𝑧) ∈ (𝑥 ∩ {𝑦}))
142118eleq2d 2820 . . . . . . . 8 (𝜑 → ((1st𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ (1st𝑧) ∈ ∅))
143 noel 4291 . . . . . . . . 9 ¬ (1st𝑧) ∈ ∅
144143pm2.21i 119 . . . . . . . 8 ((1st𝑧) ∈ ∅ → 𝑧 ∈ ∅)
145142, 144syl6bi 253 . . . . . . 7 (𝜑 → ((1st𝑧) ∈ (𝑥 ∩ {𝑦}) → 𝑧 ∈ ∅))
146141, 145syl5 34 . . . . . 6 (𝜑 → (𝑧 ∈ ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) → 𝑧 ∈ ∅))
147146ssrdv 3951 . . . . 5 (𝜑 → ( 𝑗𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × 𝑦 / 𝑗𝐵)) ⊆ ∅)
148 ss0 4359 . . . . 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 5055 . . . . . 6 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵))
151 nfcv 2904 . . . . . . . . 9 𝑚({𝑗} × 𝐵)
152 nfcv 2904 . . . . . . . . . 10 𝑗{𝑚}
153152, 5nfxp 5667 . . . . . . . . 9 𝑗({𝑚} × 𝑚 / 𝑗𝐵)
154 sneq 4597 . . . . . . . . . 10 (𝑗 = 𝑚 → {𝑗} = {𝑚})
155154, 8xpeq12d 5665 . . . . . . . . 9 (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × 𝑚 / 𝑗𝐵))
156151, 153, 155cbviun 4997 . . . . . . . 8 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = 𝑚 ∈ {𝑦} ({𝑚} × 𝑚 / 𝑗𝐵)
157 sneq 4597 . . . . . . . . . 10 (𝑚 = 𝑦 → {𝑚} = {𝑦})
158157, 38xpeq12d 5665 . . . . . . . . 9 (𝑚 = 𝑦 → ({𝑚} × 𝑚 / 𝑗𝐵) = ({𝑦} × 𝑦 / 𝑗𝐵))
15915, 158iunxsn 5052 . . . . . . . 8 𝑚 ∈ {𝑦} ({𝑚} × 𝑚 / 𝑗𝐵) = ({𝑦} × 𝑦 / 𝑗𝐵)
160156, 159eqtri 2761 . . . . . . 7 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ({𝑦} × 𝑦 / 𝑗𝐵)
161160uneq2i 4121 . . . . . 6 ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵)) = ( 𝑗𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × 𝑦 / 𝑗𝐵))
162150, 161eqtri 2761 . . . . 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 8991 . . . . . . 7 {𝑗} ∈ Fin
165122, 18syldan 592 . . . . . . 7 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ Fin)
166 xpfi 9264 . . . . . . 7 (({𝑗} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑗} × 𝐵) ∈ Fin)
167164, 165, 166sylancr 588 . . . . . 6 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → ({𝑗} × 𝐵) ∈ Fin)
168167ralrimiva 3140 . . . . 5 (𝜑 → ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
169 iunfi 9287 . . . . 5 (((𝑥 ∪ {𝑦}) ∈ Fin ∧ ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) → 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
170121, 168, 169syl2anc 585 . . . 4 (𝜑 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
171 eliun 4959 . . . . . 6 (𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ↔ ∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵))
172 elxp 5657 . . . . . . . 8 (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵)))
173 simprl 770 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑧 = ⟨𝑚, 𝑘⟩)
174 simprrl 780 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑚 ∈ {𝑗})
175 elsni 4604 . . . . . . . . . . . . . . 15 (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
176174, 175syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑚 = 𝑗)
177176opeq1d 4837 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
178173, 177eqtrd 2773 . . . . . . . . . . . 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 766 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝜑)
181122adantr 482 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑗𝐴)
182 simprrr 781 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝑘𝐵)
183180, 181, 182, 26syl12anc 836 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝐶 ∈ ℂ)
184179, 183eqeltrd 2834 . . . . . . . . . 10 (((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵))) → 𝐷 ∈ ℂ)
185184ex 414 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵)) → 𝐷 ∈ ℂ))
186185exlimdvv 1938 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → (∃𝑚𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘𝐵)) → 𝐷 ∈ ℂ))
187172, 186biimtrid 241 . . . . . . 7 ((𝜑𝑗 ∈ (𝑥 ∪ {𝑦})) → (𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
188187rexlimdva 3149 . . . . . 6 (𝜑 → (∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
189171, 188biimtrid 241 . . . . 5 (𝜑 → (𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
190189imp 408 . . . 4 ((𝜑𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) → 𝐷 ∈ ℂ)
191149, 163, 170, 190fsumsplit 15631 . . 3 (𝜑 → Σ𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 + Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷))
192191adantr 482 . 2 ((𝜑𝜓) → Σ𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷 + Σ𝑧 ∈ ({𝑦} × 𝑦 / 𝑗𝐵)𝐷))
193115, 127, 1923eqtr4d 2783 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 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wral 3061  wrex 3070  csb 3856  cun 3909  cin 3910  wss 3911  c0 4283  {csn 4587  cop 4593   ciun 4955   × cxp 5632  cres 5636  1-1-ontowf1o 6496  cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  Fincfn 8886  cc 11054   + caddc 11059  Σcsu 15576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-z 12505  df-uz 12769  df-rp 12921  df-fz 13431  df-fzo 13574  df-seq 13913  df-exp 13974  df-hash 14237  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-clim 15376  df-sum 15577
This theorem is referenced by:  fsum2d  15661
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