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Theorem sge0resplit 44637
Description: Σ^ splits into two parts, when it's a real number. This is a special case of sge0split 44640. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0resplit.a (𝜑𝐴𝑉)
sge0resplit.b (𝜑𝐵𝑊)
sge0resplit.u 𝑈 = (𝐴𝐵)
sge0resplit.in0 (𝜑 → (𝐴𝐵) = ∅)
sge0resplit.f (𝜑𝐹:𝑈⟶(0[,]+∞))
sge0resplit.re (𝜑 → (Σ^𝐹) ∈ ℝ)
Assertion
Ref Expression
sge0resplit (𝜑 → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))

Proof of Theorem sge0resplit
Dummy variables 𝑎 𝑏 𝑟 𝑢 𝑣 𝑥 𝑦 𝑡 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sge0resplit.a . . . . . . 7 (𝜑𝐴𝑉)
2 sge0resplit.f . . . . . . . 8 (𝜑𝐹:𝑈⟶(0[,]+∞))
3 ssun1 4132 . . . . . . . . . 10 𝐴 ⊆ (𝐴𝐵)
4 sge0resplit.u . . . . . . . . . . 11 𝑈 = (𝐴𝐵)
54eqcomi 2745 . . . . . . . . . 10 (𝐴𝐵) = 𝑈
63, 5sseqtri 3980 . . . . . . . . 9 𝐴𝑈
76a1i 11 . . . . . . . 8 (𝜑𝐴𝑈)
82, 7fssresd 6709 . . . . . . 7 (𝜑 → (𝐹𝐴):𝐴⟶(0[,]+∞))
94a1i 11 . . . . . . . . 9 (𝜑𝑈 = (𝐴𝐵))
10 sge0resplit.b . . . . . . . . . 10 (𝜑𝐵𝑊)
11 unexg 7683 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
121, 10, 11syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐴𝐵) ∈ V)
139, 12eqeltrd 2838 . . . . . . . 8 (𝜑𝑈 ∈ V)
14 sge0resplit.re . . . . . . . 8 (𝜑 → (Σ^𝐹) ∈ ℝ)
1513, 2, 14sge0ssre 44628 . . . . . . 7 (𝜑 → (Σ^‘(𝐹𝐴)) ∈ ℝ)
161, 8, 15sge0supre 44620 . . . . . 6 (𝜑 → (Σ^‘(𝐹𝐴)) = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ, < ))
1716, 15eqeltrrd 2839 . . . . 5 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ, < ) ∈ ℝ)
18 ssun2 4133 . . . . . . . . . 10 𝐵 ⊆ (𝐴𝐵)
1918, 5sseqtri 3980 . . . . . . . . 9 𝐵𝑈
2019a1i 11 . . . . . . . 8 (𝜑𝐵𝑈)
212, 20fssresd 6709 . . . . . . 7 (𝜑 → (𝐹𝐵):𝐵⟶(0[,]+∞))
2213, 2, 14sge0ssre 44628 . . . . . . 7 (𝜑 → (Σ^‘(𝐹𝐵)) ∈ ℝ)
2310, 21, 22sge0supre 44620 . . . . . 6 (𝜑 → (Σ^‘(𝐹𝐵)) = sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ, < ))
2423, 22eqeltrrd 2839 . . . . 5 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ, < ) ∈ ℝ)
25 rexadd 13151 . . . . 5 ((sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ, < ) ∈ ℝ ∧ sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ, < ) ∈ ℝ) → (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ, < ) +𝑒 sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ, < )) = (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ, < ) + sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ, < )))
2617, 24, 25syl2anc 584 . . . 4 (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ, < ) +𝑒 sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ, < )) = (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ, < ) + sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ, < )))
2713, 2, 14sge0rern 44619 . . . . . . . 8 (𝜑 → ¬ +∞ ∈ ran 𝐹)
28 nelrnres 43396 . . . . . . . 8 (¬ +∞ ∈ ran 𝐹 → ¬ +∞ ∈ ran (𝐹𝐴))
2927, 28syl 17 . . . . . . 7 (𝜑 → ¬ +∞ ∈ ran (𝐹𝐴))
308, 29fge0iccico 44601 . . . . . 6 (𝜑 → (𝐹𝐴):𝐴⟶(0[,)+∞))
3130sge0rnre 44595 . . . . 5 (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ⊆ ℝ)
32 sge0rnn0 44599 . . . . . 6 ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ≠ ∅
3332a1i 11 . . . . 5 (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ≠ ∅)
341, 30sge0reval 44603 . . . . . . . 8 (𝜑 → (Σ^‘(𝐹𝐴)) = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ*, < ))
3534eqcomd 2742 . . . . . . 7 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ*, < ) = (Σ^‘(𝐹𝐴)))
3635, 15eqeltrd 2838 . . . . . 6 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ*, < ) ∈ ℝ)
37 supxrre3 43549 . . . . . . 7 ((ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ⊆ ℝ ∧ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ≠ ∅) → (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ*, < ) ∈ ℝ ↔ ∃𝑤 ∈ ℝ ∀𝑡 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))𝑡𝑤))
3831, 33, 37syl2anc 584 . . . . . 6 (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ*, < ) ∈ ℝ ↔ ∃𝑤 ∈ ℝ ∀𝑡 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))𝑡𝑤))
3936, 38mpbid 231 . . . . 5 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))𝑡𝑤)
40 nelrnres 43396 . . . . . . . 8 (¬ +∞ ∈ ran 𝐹 → ¬ +∞ ∈ ran (𝐹𝐵))
4127, 40syl 17 . . . . . . 7 (𝜑 → ¬ +∞ ∈ ran (𝐹𝐵))
4221, 41fge0iccico 44601 . . . . . 6 (𝜑 → (𝐹𝐵):𝐵⟶(0[,)+∞))
4342sge0rnre 44595 . . . . 5 (𝜑 → ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) ⊆ ℝ)
44 sge0rnn0 44599 . . . . . 6 ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) ≠ ∅
4544a1i 11 . . . . 5 (𝜑 → ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) ≠ ∅)
4610, 42sge0reval 44603 . . . . . . . 8 (𝜑 → (Σ^‘(𝐹𝐵)) = sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ*, < ))
4746eqcomd 2742 . . . . . . 7 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ*, < ) = (Σ^‘(𝐹𝐵)))
4847, 22eqeltrd 2838 . . . . . 6 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ*, < ) ∈ ℝ)
49 supxrre3 43549 . . . . . . 7 ((ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) ⊆ ℝ ∧ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) ≠ ∅) → (sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ*, < ) ∈ ℝ ↔ ∃𝑤 ∈ ℝ ∀𝑡 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑡𝑤))
5043, 45, 49syl2anc 584 . . . . . 6 (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ*, < ) ∈ ℝ ↔ ∃𝑤 ∈ ℝ ∀𝑡 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑡𝑤))
5148, 50mpbid 231 . . . . 5 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑡𝑤)
52 eqid 2736 . . . . 5 {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} = {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)}
5331, 33, 39, 43, 45, 51, 52supadd 12123 . . . 4 (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ, < ) + sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ, < )) = sup({𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)}, ℝ, < ))
54 simpl 483 . . . . . . . . . 10 ((𝜑𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)}) → 𝜑)
55 vex 3449 . . . . . . . . . . . . 13 𝑟 ∈ V
56 eqeq1 2740 . . . . . . . . . . . . . . 15 (𝑧 = 𝑟 → (𝑧 = (𝑣 + 𝑢) ↔ 𝑟 = (𝑣 + 𝑢)))
5756rexbidv 3175 . . . . . . . . . . . . . 14 (𝑧 = 𝑟 → (∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢) ↔ ∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢)))
5857rexbidv 3175 . . . . . . . . . . . . 13 (𝑧 = 𝑟 → (∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢) ↔ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢)))
5955, 58elab 3630 . . . . . . . . . . . 12 (𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} ↔ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))
6059biimpi 215 . . . . . . . . . . 11 (𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))
6160adantl 482 . . . . . . . . . 10 ((𝜑𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)}) → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))
62 simpl 483 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)))) → 𝜑)
63 vex 3449 . . . . . . . . . . . . . . . . . . . 20 𝑣 ∈ V
64 sumeq1 15573 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → Σ𝑦𝑥 ((𝐹𝐴)‘𝑦) = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦))
6564cbvmptv 5218 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) = (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑎 ((𝐹𝐴)‘𝑦))
6665elrnmpt 5911 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ V → (𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ↔ ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦)))
6763, 66ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ↔ ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦))
6867biimpi 215 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦))
6968adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦))
70 vex 3449 . . . . . . . . . . . . . . . . . . . 20 𝑢 ∈ V
71 sumeq1 15573 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑏 → Σ𝑦𝑥 ((𝐹𝐵)‘𝑦) = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))
7271cbvmptv 5218 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) = (𝑏 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))
7372elrnmpt 5911 . . . . . . . . . . . . . . . . . . . 20 (𝑢 ∈ V → (𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) ↔ ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)))
7470, 73ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) ↔ ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))
7574biimpi 215 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) → ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))
7675adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))) → ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))
7769, 76jca 512 . . . . . . . . . . . . . . . 16 ((𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))) → (∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)))
78 reeanv 3217 . . . . . . . . . . . . . . . 16 (∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)) ↔ (∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)))
7977, 78sylibr 233 . . . . . . . . . . . . . . 15 ((𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)))
8079adantl 482 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)))) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)))
81 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
82 elinel1 4155 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → 𝑎 ∈ 𝒫 𝐴)
83 elpwi 4567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 ∈ 𝒫 𝐴𝑎𝐴)
84 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎𝐴𝑎𝐴)
8584, 6sstrdi 3956 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎𝐴𝑎𝑈)
8683, 85syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ∈ 𝒫 𝐴𝑎𝑈)
8782, 86syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → 𝑎𝑈)
8887adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎𝑈)
89 elinel1 4155 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → 𝑏 ∈ 𝒫 𝐵)
90 elpwi 4567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏 ∈ 𝒫 𝐵𝑏𝐵)
91 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏𝐵𝑏𝐵)
9291, 19sstrdi 3956 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏𝐵𝑏𝑈)
9390, 92syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑏 ∈ 𝒫 𝐵𝑏𝑈)
9489, 93syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → 𝑏𝑈)
9594adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑏𝑈)
9688, 95unssd 4146 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑎𝑏) ⊆ 𝑈)
97 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑎 ∈ V
98 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑏 ∈ V
9997, 98unex 7680 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎𝑏) ∈ V
10099elpw 4564 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎𝑏) ∈ 𝒫 𝑈 ↔ (𝑎𝑏) ⊆ 𝑈)
10196, 100sylibr 233 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑎𝑏) ∈ 𝒫 𝑈)
102 elinel2 4156 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → 𝑎 ∈ Fin)
103102adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎 ∈ Fin)
104 elinel2 4156 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → 𝑏 ∈ Fin)
105104adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑏 ∈ Fin)
106 unfi 9116 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ Fin ∧ 𝑏 ∈ Fin) → (𝑎𝑏) ∈ Fin)
107103, 105, 106syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑎𝑏) ∈ Fin)
108101, 107elind 4154 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑎𝑏) ∈ (𝒫 𝑈 ∩ Fin))
109108adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝑎𝑏) ∈ (𝒫 𝑈 ∩ Fin))
110109ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → (𝑎𝑏) ∈ (𝒫 𝑈 ∩ Fin))
111 simpl 483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)) → 𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦))
112 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)) → 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))
113111, 112oveq12d 7375 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)) → (𝑣 + 𝑢) = (Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) + Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)))
114113adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) ∧ (𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))) → (𝑣 + 𝑢) = (Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) + Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)))
11582, 83syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → 𝑎𝐴)
116115sselda 3944 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦𝑎) → 𝑦𝐴)
117 fvres 6861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦𝐴 → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
118116, 117syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦𝑎) → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
119118sumeq2dv 15588 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) = Σ𝑦𝑎 (𝐹𝑦))
120119adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) = Σ𝑦𝑎 (𝐹𝑦))
12189, 90syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → 𝑏𝐵)
122121sselda 3944 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑏 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝑦𝑏) → 𝑦𝐵)
123 fvres 6861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦𝐵 → ((𝐹𝐵)‘𝑦) = (𝐹𝑦))
124122, 123syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑏 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝑦𝑏) → ((𝐹𝐵)‘𝑦) = (𝐹𝑦))
125124sumeq2dv 15588 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → Σ𝑦𝑏 ((𝐹𝐵)‘𝑦) = Σ𝑦𝑏 (𝐹𝑦))
126125adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → Σ𝑦𝑏 ((𝐹𝐵)‘𝑦) = Σ𝑦𝑏 (𝐹𝑦))
127120, 126oveq12d 7375 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) + Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)) = (Σ𝑦𝑎 (𝐹𝑦) + Σ𝑦𝑏 (𝐹𝑦)))
128127adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) ∧ (𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))) → (Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) + Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)) = (Σ𝑦𝑎 (𝐹𝑦) + Σ𝑦𝑏 (𝐹𝑦)))
129114, 128eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) ∧ (𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))) → (𝑣 + 𝑢) = (Σ𝑦𝑎 (𝐹𝑦) + Σ𝑦𝑏 (𝐹𝑦)))
130129ad4ant23 751 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → (𝑣 + 𝑢) = (Σ𝑦𝑎 (𝐹𝑦) + Σ𝑦𝑏 (𝐹𝑦)))
131 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → 𝑟 = (𝑣 + 𝑢))
132 sge0resplit.in0 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐴𝐵) = ∅)
133132adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝐴𝐵) = ∅)
134115ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → 𝑎𝐴)
135121adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑏𝐵)
136135adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → 𝑏𝐵)
137 ssin0 43253 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐴𝐵) = ∅ ∧ 𝑎𝐴𝑏𝐵) → (𝑎𝑏) = ∅)
138133, 134, 136, 137syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝑎𝑏) = ∅)
139 eqidd 2737 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝑎𝑏) = (𝑎𝑏))
140107adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝑎𝑏) ∈ Fin)
141 rge0ssre 13373 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0[,)+∞) ⊆ ℝ
142 ax-resscn 11108 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ℝ ⊆ ℂ
143141, 142sstri 3953 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0[,)+∞) ⊆ ℂ
1442, 27fge0iccico 44601 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐹:𝑈⟶(0[,)+∞))
145144ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎𝑏)) → 𝐹:𝑈⟶(0[,)+∞))
14696sselda 3944 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) ∧ 𝑦 ∈ (𝑎𝑏)) → 𝑦𝑈)
147146adantll 712 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎𝑏)) → 𝑦𝑈)
148145, 147ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎𝑏)) → (𝐹𝑦) ∈ (0[,)+∞))
149143, 148sselid 3942 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎𝑏)) → (𝐹𝑦) ∈ ℂ)
150138, 139, 140, 149fsumsplit 15626 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → Σ𝑦 ∈ (𝑎𝑏)(𝐹𝑦) = (Σ𝑦𝑎 (𝐹𝑦) + Σ𝑦𝑏 (𝐹𝑦)))
151150ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → Σ𝑦 ∈ (𝑎𝑏)(𝐹𝑦) = (Σ𝑦𝑎 (𝐹𝑦) + Σ𝑦𝑏 (𝐹𝑦)))
152130, 131, 1513eqtr4d 2786 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → 𝑟 = Σ𝑦 ∈ (𝑎𝑏)(𝐹𝑦))
153 sumeq1 15573 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (𝑎𝑏) → Σ𝑦𝑥 (𝐹𝑦) = Σ𝑦 ∈ (𝑎𝑏)(𝐹𝑦))
154153rspceeqv 3595 . . . . . . . . . . . . . . . . . . . . 21 (((𝑎𝑏) ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦 ∈ (𝑎𝑏)(𝐹𝑦)) → ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑟 = Σ𝑦𝑥 (𝐹𝑦))
155110, 152, 154syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑟 = Σ𝑦𝑥 (𝐹𝑦))
15655a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → 𝑟 ∈ V)
15781, 155, 156elrnmptd 5916 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))) ∧ 𝑟 = (𝑣 + 𝑢)) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
158157ex 413 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ (𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))) → (𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))))
159158ex 413 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → ((𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)) → (𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))))
160159ex 413 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → ((𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)) → (𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))))))
161160rexlimdvv 3204 . . . . . . . . . . . . . . 15 (𝜑 → (∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)) → (𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))))
162161imp 407 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))) → (𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))))
16362, 80, 162syl2anc 584 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)))) → (𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))))
164163ex 413 . . . . . . . . . . . 12 (𝜑 → ((𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))) → (𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))))
165164rexlimdvv 3204 . . . . . . . . . . 11 (𝜑 → (∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))))
166165imp 407 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢)) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
16754, 61, 166syl2anc 584 . . . . . . . . 9 ((𝜑𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)}) → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
168167ex 413 . . . . . . . 8 (𝜑 → (𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} → 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))))
16981elrnmpt 5911 . . . . . . . . . . . . 13 (𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) → (𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑟 = Σ𝑦𝑥 (𝐹𝑦)))
170169ibi 266 . . . . . . . . . . . 12 (𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) → ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑟 = Σ𝑦𝑥 (𝐹𝑦))
171170adantl 482 . . . . . . . . . . 11 ((𝜑𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑟 = Σ𝑦𝑥 (𝐹𝑦))
172 nfv 1917 . . . . . . . . . . . . 13 𝑥𝜑
173 nfcv 2907 . . . . . . . . . . . . . 14 𝑥𝑟
174 nfmpt1 5213 . . . . . . . . . . . . . . 15 𝑥(𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
175174nfrn 5907 . . . . . . . . . . . . . 14 𝑥ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
176173, 175nfel 2921 . . . . . . . . . . . . 13 𝑥 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
177172, 176nfan 1902 . . . . . . . . . . . 12 𝑥(𝜑𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
178 nfmpt1 5213 . . . . . . . . . . . . . 14 𝑥(𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))
179178nfrn 5907 . . . . . . . . . . . . 13 𝑥ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))
180 nfmpt1 5213 . . . . . . . . . . . . . . 15 𝑥(𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))
181180nfrn 5907 . . . . . . . . . . . . . 14 𝑥ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))
182 nfv 1917 . . . . . . . . . . . . . 14 𝑥 𝑟 = (𝑣 + 𝑢)
183181, 182nfrexw 3296 . . . . . . . . . . . . 13 𝑥𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢)
184179, 183nfrexw 3296 . . . . . . . . . . . 12 𝑥𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢)
185 inss2 4189 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐴) ⊆ 𝐴
186185sseli 3940 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑥𝐴) → 𝑦𝐴)
187186adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦 ∈ (𝑥𝐴)) → 𝑦𝐴)
188117eqcomd 2742 . . . . . . . . . . . . . . . . . . 19 (𝑦𝐴 → (𝐹𝑦) = ((𝐹𝐴)‘𝑦))
189187, 188syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦 ∈ (𝑥𝐴)) → (𝐹𝑦) = ((𝐹𝐴)‘𝑦))
190189sumeq2dv 15588 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) = Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦))
191 sumeq1 15573 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → Σ𝑦𝑥 ((𝐹𝐴)‘𝑦) = Σ𝑦𝑧 ((𝐹𝐴)‘𝑦))
192191cbvmptv 5218 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) = (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑧 ((𝐹𝐴)‘𝑦))
193 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑥 ∈ V
194193inex1 5274 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝐴) ∈ V
195194elpw 4564 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝐴) ∈ 𝒫 𝐴 ↔ (𝑥𝐴) ⊆ 𝐴)
196185, 195mpbir 230 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐴) ∈ 𝒫 𝐴
197196a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐴) ∈ 𝒫 𝐴)
198 elinel2 4156 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ Fin)
199 inss1 4188 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝐴) ⊆ 𝑥
200199a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐴) ⊆ 𝑥)
201 ssfi 9117 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ Fin ∧ (𝑥𝐴) ⊆ 𝑥) → (𝑥𝐴) ∈ Fin)
202198, 200, 201syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐴) ∈ Fin)
203197, 202elind 4154 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐴) ∈ (𝒫 𝐴 ∩ Fin))
204 eqidd 2737 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) = Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦))
205 sumeq1 15573 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑥𝐴) → Σ𝑦𝑧 ((𝐹𝐴)‘𝑦) = Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦))
206205rspceeqv 3595 . . . . . . . . . . . . . . . . . . 19 (((𝑥𝐴) ∈ (𝒫 𝐴 ∩ Fin) ∧ Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) = Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦)) → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) = Σ𝑦𝑧 ((𝐹𝐴)‘𝑦))
207203, 204, 206syl2anc 584 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) = Σ𝑦𝑧 ((𝐹𝐴)‘𝑦))
208 sumex 15572 . . . . . . . . . . . . . . . . . . 19 Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) ∈ V
209208a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) ∈ V)
210192, 207, 209elrnmptd 5916 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)))
211190, 210eqeltrd 2838 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)))
2122113ad2ant2 1134 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)))
213 sumeq1 15573 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → Σ𝑦𝑥 ((𝐹𝐵)‘𝑦) = Σ𝑦𝑧 ((𝐹𝐵)‘𝑦))
214213cbvmptv 5218 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) = (𝑧 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑧 ((𝐹𝐵)‘𝑦))
215 inss2 4189 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐵) ⊆ 𝐵
216193inex1 5274 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝐵) ∈ V
217216elpw 4564 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝐵) ∈ 𝒫 𝐵 ↔ (𝑥𝐵) ⊆ 𝐵)
218215, 217mpbir 230 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝐵) ∈ 𝒫 𝐵
219218a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → (𝑥𝐵) ∈ 𝒫 𝐵)
220 inss1 4188 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝐵) ⊆ 𝑥
221220a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐵) ⊆ 𝑥)
222 ssfi 9117 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ Fin ∧ (𝑥𝐵) ⊆ 𝑥) → (𝑥𝐵) ∈ Fin)
223198, 221, 222syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐵) ∈ Fin)
2242233ad2ant2 1134 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → (𝑥𝐵) ∈ Fin)
225219, 224elind 4154 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → (𝑥𝐵) ∈ (𝒫 𝐵 ∩ Fin))
226215sseli 3940 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (𝑥𝐵) → 𝑦𝐵)
227123eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐵 → (𝐹𝑦) = ((𝐹𝐵)‘𝑦))
228226, 227syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝑥𝐵) → (𝐹𝑦) = ((𝐹𝐵)‘𝑦))
229228sumeq2i 15584 . . . . . . . . . . . . . . . . . . . 20 Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) = Σ𝑦 ∈ (𝑥𝐵)((𝐹𝐵)‘𝑦)
230229a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) = Σ𝑦 ∈ (𝑥𝐵)((𝐹𝐵)‘𝑦))
2312303adant3 1132 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) = Σ𝑦 ∈ (𝑥𝐵)((𝐹𝐵)‘𝑦))
232 sumeq1 15573 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑥𝐵) → Σ𝑦𝑧 ((𝐹𝐵)‘𝑦) = Σ𝑦 ∈ (𝑥𝐵)((𝐹𝐵)‘𝑦))
233232rspceeqv 3595 . . . . . . . . . . . . . . . . . 18 (((𝑥𝐵) ∈ (𝒫 𝐵 ∩ Fin) ∧ Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) = Σ𝑦 ∈ (𝑥𝐵)((𝐹𝐵)‘𝑦)) → ∃𝑧 ∈ (𝒫 𝐵 ∩ Fin)Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) = Σ𝑦𝑧 ((𝐹𝐵)‘𝑦))
234225, 231, 233syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → ∃𝑧 ∈ (𝒫 𝐵 ∩ Fin)Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) = Σ𝑦𝑧 ((𝐹𝐵)‘𝑦))
235 sumex 15572 . . . . . . . . . . . . . . . . . 18 Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ∈ V
236235a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ∈ V)
237214, 234, 236elrnmptd 5916 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)))
238 simp3 1138 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → 𝑟 = Σ𝑦𝑥 (𝐹𝑦))
239185a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑥𝐴) ⊆ 𝐴)
240215a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑥𝐵) ⊆ 𝐵)
241 ssin0 43253 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝐵) = ∅ ∧ (𝑥𝐴) ⊆ 𝐴 ∧ (𝑥𝐵) ⊆ 𝐵) → ((𝑥𝐴) ∩ (𝑥𝐵)) = ∅)
242132, 239, 240, 241syl3anc 1371 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑥𝐴) ∩ (𝑥𝐵)) = ∅)
243242adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ((𝑥𝐴) ∩ (𝑥𝐵)) = ∅)
244 elinel1 4155 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ 𝒫 𝑈)
245 elpwi 4567 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ 𝒫 𝑈𝑥𝑈)
246244, 245syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥𝑈)
2474ineq2i 4169 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝑈) = (𝑥 ∩ (𝐴𝐵))
248247a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑈 → (𝑥𝑈) = (𝑥 ∩ (𝐴𝐵)))
249 dfss 3928 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝑈𝑥 = (𝑥𝑈))
250249biimpi 215 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑈𝑥 = (𝑥𝑈))
251 indi 4233 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∩ (𝐴𝐵)) = ((𝑥𝐴) ∪ (𝑥𝐵))
252251eqcomi 2745 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥𝐴) ∪ (𝑥𝐵)) = (𝑥 ∩ (𝐴𝐵))
253252a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑈 → ((𝑥𝐴) ∪ (𝑥𝐵)) = (𝑥 ∩ (𝐴𝐵)))
254248, 250, 2533eqtr4d 2786 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝑈𝑥 = ((𝑥𝐴) ∪ (𝑥𝐵)))
255246, 254syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 = ((𝑥𝐴) ∪ (𝑥𝐵)))
256255adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑥 = ((𝑥𝐴) ∪ (𝑥𝐵)))
257198adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑥 ∈ Fin)
258144ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑈⟶(0[,)+∞))
259246sselda 3944 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑈)
260259adantll 712 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑈)
261258, 260ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,)+∞))
262143, 261sselid 3942 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℂ)
263243, 256, 257, 262fsumsplit 15626 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
2642633adant3 1132 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → Σ𝑦𝑥 (𝐹𝑦) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
265238, 264eqtrd 2776 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → 𝑟 = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
266 oveq2 7365 . . . . . . . . . . . . . . . . 17 (𝑢 = Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) → (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + 𝑢) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
267266rspceeqv 3595 . . . . . . . . . . . . . . . 16 ((Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) ∧ 𝑟 = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦))) → ∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + 𝑢))
268237, 265, 267syl2anc 584 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → ∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + 𝑢))
269 oveq1 7364 . . . . . . . . . . . . . . . . . 18 (𝑣 = Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) → (𝑣 + 𝑢) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + 𝑢))
270269eqeq2d 2747 . . . . . . . . . . . . . . . . 17 (𝑣 = Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) → (𝑟 = (𝑣 + 𝑢) ↔ 𝑟 = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + 𝑢)))
271270rexbidv 3175 . . . . . . . . . . . . . . . 16 (𝑣 = Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) → (∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢) ↔ ∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + 𝑢)))
272271rspcev 3581 . . . . . . . . . . . . . . 15 ((Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ∧ ∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + 𝑢)) → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))
273212, 268, 272syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))
2742733exp 1119 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑟 = Σ𝑦𝑥 (𝐹𝑦) → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))))
275274adantr 481 . . . . . . . . . . . 12 ((𝜑𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑟 = Σ𝑦𝑥 (𝐹𝑦) → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))))
276177, 184, 275rexlimd 3249 . . . . . . . . . . 11 ((𝜑𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → (∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑟 = Σ𝑦𝑥 (𝐹𝑦) → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢)))
277171, 276mpd 15 . . . . . . . . . 10 ((𝜑𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢))
278277, 59sylibr 233 . . . . . . . . 9 ((𝜑𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → 𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)})
279278ex 413 . . . . . . . 8 (𝜑 → (𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) → 𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)}))
280168, 279impbid 211 . . . . . . 7 (𝜑 → (𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} ↔ 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))))
281280alrimiv 1930 . . . . . 6 (𝜑 → ∀𝑟(𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} ↔ 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))))
282 dfcleq 2729 . . . . . 6 ({𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} = ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ↔ ∀𝑟(𝑟 ∈ {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} ↔ 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))))
283281, 282sylibr 233 . . . . 5 (𝜑 → {𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)} = ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
284283supeq1d 9382 . . . 4 (𝜑 → sup({𝑧 ∣ ∃𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑧 = (𝑣 + 𝑢)}, ℝ, < ) = sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ))
28526, 53, 2843eqtrrd 2781 . . 3 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) = (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ, < ) +𝑒 sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ, < )))
28613, 2, 14sge0supre 44620 . . 3 (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ))
28716, 23oveq12d 7375 . . 3 (𝜑 → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ, < ) +𝑒 sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ, < )))
288285, 286, 2873eqtr4d 2786 . 2 (𝜑 → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
289 rexadd 13151 . . 3 (((Σ^‘(𝐹𝐴)) ∈ ℝ ∧ (Σ^‘(𝐹𝐵)) ∈ ℝ) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))
29015, 22, 289syl2anc 584 . 2 (𝜑 → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))
291288, 290eqtrd 2776 1 (𝜑 → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cun 3908  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560   class class class wbr 5105  cmpt 5188  ran crn 5634  cres 5635  wf 6492  cfv 6496  (class class class)co 7357  Fincfn 8883  supcsup 9376  cc 11049  cr 11050  0cc0 11051   + caddc 11054  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190   +𝑒 cxad 13031  [,)cico 13266  [,]cicc 13267  Σcsu 15570  Σ^csumge0 44593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-xadd 13034  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-sumge0 44594
This theorem is referenced by:  sge0split  44640
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