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Theorem sge0resplit 46421
Description: Σ^ splits into two parts, when it's a real number. This is a special case of sge0split 46424. (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 4178 . . . . . . . . . 10 𝐴 ⊆ (𝐴𝐵)
4 sge0resplit.u . . . . . . . . . . 11 𝑈 = (𝐴𝐵)
54eqcomi 2746 . . . . . . . . . 10 (𝐴𝐵) = 𝑈
63, 5sseqtri 4032 . . . . . . . . 9 𝐴𝑈
76a1i 11 . . . . . . . 8 (𝜑𝐴𝑈)
82, 7fssresd 6775 . . . . . . 7 (𝜑 → (𝐹𝐴):𝐴⟶(0[,]+∞))
94a1i 11 . . . . . . . . 9 (𝜑𝑈 = (𝐴𝐵))
10 sge0resplit.b . . . . . . . . . 10 (𝜑𝐵𝑊)
11 unexg 7763 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
121, 10, 11syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐴𝐵) ∈ V)
139, 12eqeltrd 2841 . . . . . . . 8 (𝜑𝑈 ∈ V)
14 sge0resplit.re . . . . . . . 8 (𝜑 → (Σ^𝐹) ∈ ℝ)
1513, 2, 14sge0ssre 46412 . . . . . . 7 (𝜑 → (Σ^‘(𝐹𝐴)) ∈ ℝ)
161, 8, 15sge0supre 46404 . . . . . 6 (𝜑 → (Σ^‘(𝐹𝐴)) = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ, < ))
1716, 15eqeltrrd 2842 . . . . 5 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ, < ) ∈ ℝ)
18 ssun2 4179 . . . . . . . . . 10 𝐵 ⊆ (𝐴𝐵)
1918, 5sseqtri 4032 . . . . . . . . 9 𝐵𝑈
2019a1i 11 . . . . . . . 8 (𝜑𝐵𝑈)
212, 20fssresd 6775 . . . . . . 7 (𝜑 → (𝐹𝐵):𝐵⟶(0[,]+∞))
2213, 2, 14sge0ssre 46412 . . . . . . 7 (𝜑 → (Σ^‘(𝐹𝐵)) ∈ ℝ)
2310, 21, 22sge0supre 46404 . . . . . 6 (𝜑 → (Σ^‘(𝐹𝐵)) = sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ, < ))
2423, 22eqeltrrd 2842 . . . . 5 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ, < ) ∈ ℝ)
25 rexadd 13274 . . . . 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 46403 . . . . . . . 8 (𝜑 → ¬ +∞ ∈ ran 𝐹)
28 nelrnres 45192 . . . . . . . 8 (¬ +∞ ∈ ran 𝐹 → ¬ +∞ ∈ ran (𝐹𝐴))
2927, 28syl 17 . . . . . . 7 (𝜑 → ¬ +∞ ∈ ran (𝐹𝐴))
308, 29fge0iccico 46385 . . . . . 6 (𝜑 → (𝐹𝐴):𝐴⟶(0[,)+∞))
3130sge0rnre 46379 . . . . 5 (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ⊆ ℝ)
32 sge0rnn0 46383 . . . . . 6 ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ≠ ∅
3332a1i 11 . . . . 5 (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ≠ ∅)
341, 30sge0reval 46387 . . . . . . . 8 (𝜑 → (Σ^‘(𝐹𝐴)) = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ*, < ))
3534eqcomd 2743 . . . . . . 7 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ*, < ) = (Σ^‘(𝐹𝐴)))
3635, 15eqeltrd 2841 . . . . . 6 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)), ℝ*, < ) ∈ ℝ)
37 supxrre3 45336 . . . . . . 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 232 . . . . 5 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))𝑡𝑤)
40 nelrnres 45192 . . . . . . . 8 (¬ +∞ ∈ ran 𝐹 → ¬ +∞ ∈ ran (𝐹𝐵))
4127, 40syl 17 . . . . . . 7 (𝜑 → ¬ +∞ ∈ ran (𝐹𝐵))
4221, 41fge0iccico 46385 . . . . . 6 (𝜑 → (𝐹𝐵):𝐵⟶(0[,)+∞))
4342sge0rnre 46379 . . . . 5 (𝜑 → ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) ⊆ ℝ)
44 sge0rnn0 46383 . . . . . 6 ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) ≠ ∅
4544a1i 11 . . . . 5 (𝜑 → ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) ≠ ∅)
4610, 42sge0reval 46387 . . . . . . . 8 (𝜑 → (Σ^‘(𝐹𝐵)) = sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ*, < ))
4746eqcomd 2743 . . . . . . 7 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ*, < ) = (Σ^‘(𝐹𝐵)))
4847, 22eqeltrd 2841 . . . . . 6 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)), ℝ*, < ) ∈ ℝ)
49 supxrre3 45336 . . . . . . 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 232 . . . . 5 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑡𝑤)
52 eqid 2737 . . . . 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 12236 . . . 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 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)}) → 𝜑)
55 vex 3484 . . . . . . . . . . . . 13 𝑟 ∈ V
56 eqeq1 2741 . . . . . . . . . . . . . . 15 (𝑧 = 𝑟 → (𝑧 = (𝑣 + 𝑢) ↔ 𝑟 = (𝑣 + 𝑢)))
5756rexbidv 3179 . . . . . . . . . . . . . 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 3179 . . . . . . . . . . . . 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 3679 . . . . . . . . . . . 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 216 . . . . . . . . . . 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 481 . . . . . . . . . 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 482 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) ∧ 𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)))) → 𝜑)
63 vex 3484 . . . . . . . . . . . . . . . . . . . 20 𝑣 ∈ V
64 sumeq1 15725 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → Σ𝑦𝑥 ((𝐹𝐴)‘𝑦) = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦))
6564cbvmptv 5255 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) = (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑎 ((𝐹𝐴)‘𝑦))
6665elrnmpt 5969 . . . . . . . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦))
6968adantr 480 . . . . . . . . . . . . . . . . 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 3484 . . . . . . . . . . . . . . . . . . . 20 𝑢 ∈ V
71 sumeq1 15725 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑏 → Σ𝑦𝑥 ((𝐹𝐵)‘𝑦) = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))
7271cbvmptv 5255 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) = (𝑏 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))
7372elrnmpt 5969 . . . . . . . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) → ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))
7675adantl 481 . . . . . . . . . . . . . . . . 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 511 . . . . . . . . . . . . . . . 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 3229 . . . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
82 elinel1 4201 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → 𝑎 ∈ 𝒫 𝐴)
83 elpwi 4607 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 ∈ 𝒫 𝐴𝑎𝐴)
84 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎𝐴𝑎𝐴)
8584, 6sstrdi 3996 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎𝐴𝑎𝑈)
8683, 85syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ∈ 𝒫 𝐴𝑎𝑈)
8782, 86syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → 𝑎𝑈)
8887adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎𝑈)
89 elinel1 4201 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → 𝑏 ∈ 𝒫 𝐵)
90 elpwi 4607 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏 ∈ 𝒫 𝐵𝑏𝐵)
91 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏𝐵𝑏𝐵)
9291, 19sstrdi 3996 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏𝐵𝑏𝑈)
9390, 92syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑏 ∈ 𝒫 𝐵𝑏𝑈)
9489, 93syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → 𝑏𝑈)
9594adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑏𝑈)
9688, 95unssd 4192 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑎𝑏) ⊆ 𝑈)
97 vex 3484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑎 ∈ V
98 vex 3484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑏 ∈ V
9997, 98unex 7764 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎𝑏) ∈ V
10099elpw 4604 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎𝑏) ∈ 𝒫 𝑈 ↔ (𝑎𝑏) ⊆ 𝑈)
10196, 100sylibr 234 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑎𝑏) ∈ 𝒫 𝑈)
102 elinel2 4202 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → 𝑎 ∈ Fin)
103102adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎 ∈ Fin)
104 elinel2 4202 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → 𝑏 ∈ Fin)
105104adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑏 ∈ Fin)
106 unfi 9211 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ Fin ∧ 𝑏 ∈ Fin) → (𝑎𝑏) ∈ Fin)
107103, 105, 106syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑎𝑏) ∈ Fin)
108101, 107elind 4200 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑎𝑏) ∈ (𝒫 𝑈 ∩ Fin))
109108adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝑎𝑏) ∈ (𝒫 𝑈 ∩ Fin))
110109ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)) → 𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦))
112 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)) → 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))
113111, 112oveq12d 7449 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)) → (𝑣 + 𝑢) = (Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) + Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)))
114113adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦𝑎) → 𝑦𝐴)
117 fvres 6925 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦𝐴 → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
118116, 117syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦𝑎) → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
119118sumeq2dv 15738 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) = Σ𝑦𝑎 (𝐹𝑦))
120119adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) = Σ𝑦𝑎 (𝐹𝑦))
12189, 90syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → 𝑏𝐵)
122121sselda 3983 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑏 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝑦𝑏) → 𝑦𝐵)
123 fvres 6925 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦𝐵 → ((𝐹𝐵)‘𝑦) = (𝐹𝑦))
124122, 123syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑏 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝑦𝑏) → ((𝐹𝐵)‘𝑦) = (𝐹𝑦))
125124sumeq2dv 15738 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑏 ∈ (𝒫 𝐵 ∩ Fin) → Σ𝑦𝑏 ((𝐹𝐵)‘𝑦) = Σ𝑦𝑏 (𝐹𝑦))
126125adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → Σ𝑦𝑏 ((𝐹𝐵)‘𝑦) = Σ𝑦𝑏 (𝐹𝑦))
127120, 126oveq12d 7449 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → (Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) + Σ𝑦𝑏 ((𝐹𝐵)‘𝑦)) = (Σ𝑦𝑎 (𝐹𝑦) + Σ𝑦𝑏 (𝐹𝑦)))
128127adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) ∧ (𝑣 = Σ𝑦𝑎 ((𝐹𝐴)‘𝑦) ∧ 𝑢 = Σ𝑦𝑏 ((𝐹𝐵)‘𝑦))) → (𝑣 + 𝑢) = (Σ𝑦𝑎 (𝐹𝑦) + Σ𝑦𝑏 (𝐹𝑦)))
130129ad4ant23 753 . . . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝐴𝐵) = ∅)
134115ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → 𝑎𝐴)
135121adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑏𝐵)
136135adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → 𝑏𝐵)
137 ssin0 45060 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐴𝐵) = ∅ ∧ 𝑎𝐴𝑏𝐵) → (𝑎𝑏) = ∅)
138133, 134, 136, 137syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝑎𝑏) = ∅)
139 eqidd 2738 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝑎𝑏) = (𝑎𝑏))
140107adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → (𝑎𝑏) ∈ Fin)
141 rge0ssre 13496 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0[,)+∞) ⊆ ℝ
142 ax-resscn 11212 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ℝ ⊆ ℂ
143141, 142sstri 3993 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0[,)+∞) ⊆ ℂ
1442, 27fge0iccico 46385 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐹:𝑈⟶(0[,)+∞))
145144ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎𝑏)) → 𝐹:𝑈⟶(0[,)+∞))
14696sselda 3983 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin)) ∧ 𝑦 ∈ (𝑎𝑏)) → 𝑦𝑈)
147146adantll 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎𝑏)) → 𝑦𝑈)
148145, 147ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎𝑏)) → (𝐹𝑦) ∈ (0[,)+∞))
149143, 148sselid 3981 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) ∧ 𝑦 ∈ (𝑎𝑏)) → (𝐹𝑦) ∈ ℂ)
150138, 139, 140, 149fsumsplit 15777 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑏 ∈ (𝒫 𝐵 ∩ Fin))) → Σ𝑦 ∈ (𝑎𝑏)(𝐹𝑦) = (Σ𝑦𝑎 (𝐹𝑦) + Σ𝑦𝑏 (𝐹𝑦)))
151150ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 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 2787 . . . . . . . . . . . . . . . . . . . . 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 15725 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (𝑎𝑏) → Σ𝑦𝑥 (𝐹𝑦) = Σ𝑦 ∈ (𝑎𝑏)(𝐹𝑦))
154153rspceeqv 3645 . . . . . . . . . . . . . . . . . . . . 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 5974 . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . 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 3212 . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . 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 412 . . . . . . . . . . . 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 3212 . . . . . . . . . . 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 406 . . . . . . . . . 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 412 . . . . . . . 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 5969 . . . . . . . . . . . . 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 267 . . . . . . . . . . . 12 (𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) → ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑟 = Σ𝑦𝑥 (𝐹𝑦))
171170adantl 481 . . . . . . . . . . 11 ((𝜑𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑟 = Σ𝑦𝑥 (𝐹𝑦))
172 nfv 1914 . . . . . . . . . . . . 13 𝑥𝜑
173 nfcv 2905 . . . . . . . . . . . . . 14 𝑥𝑟
174 nfmpt1 5250 . . . . . . . . . . . . . . 15 𝑥(𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
175174nfrn 5963 . . . . . . . . . . . . . 14 𝑥ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
176173, 175nfel 2920 . . . . . . . . . . . . 13 𝑥 𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
177172, 176nfan 1899 . . . . . . . . . . . 12 𝑥(𝜑𝑟 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
178 nfmpt1 5250 . . . . . . . . . . . . . 14 𝑥(𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))
179178nfrn 5963 . . . . . . . . . . . . 13 𝑥ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))
180 nfmpt1 5250 . . . . . . . . . . . . . . 15 𝑥(𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))
181180nfrn 5963 . . . . . . . . . . . . . 14 𝑥ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))
182 nfv 1914 . . . . . . . . . . . . . 14 𝑥 𝑟 = (𝑣 + 𝑢)
183181, 182nfrexw 3313 . . . . . . . . . . . . 13 𝑥𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢)
184179, 183nfrexw 3313 . . . . . . . . . . . 12 𝑥𝑣 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦))∃𝑢 ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦))𝑟 = (𝑣 + 𝑢)
185 inss2 4238 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐴) ⊆ 𝐴
186185sseli 3979 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑥𝐴) → 𝑦𝐴)
187186adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦 ∈ (𝑥𝐴)) → 𝑦𝐴)
188117eqcomd 2743 . . . . . . . . . . . . . . . . . . 19 (𝑦𝐴 → (𝐹𝑦) = ((𝐹𝐴)‘𝑦))
189187, 188syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦 ∈ (𝑥𝐴)) → (𝐹𝑦) = ((𝐹𝐴)‘𝑦))
190189sumeq2dv 15738 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) = Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦))
191 sumeq1 15725 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → Σ𝑦𝑥 ((𝐹𝐴)‘𝑦) = Σ𝑦𝑧 ((𝐹𝐴)‘𝑦))
192191cbvmptv 5255 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)) = (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑧 ((𝐹𝐴)‘𝑦))
193 vex 3484 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑥 ∈ V
194193inex1 5317 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝐴) ∈ V
195194elpw 4604 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝐴) ∈ 𝒫 𝐴 ↔ (𝑥𝐴) ⊆ 𝐴)
196185, 195mpbir 231 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐴) ∈ 𝒫 𝐴
197196a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐴) ∈ 𝒫 𝐴)
198 elinel2 4202 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ Fin)
199 inss1 4237 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝐴) ⊆ 𝑥
200199a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐴) ⊆ 𝑥)
201 ssfi 9213 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ Fin ∧ (𝑥𝐴) ⊆ 𝑥) → (𝑥𝐴) ∈ Fin)
202198, 200, 201syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐴) ∈ Fin)
203197, 202elind 4200 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐴) ∈ (𝒫 𝐴 ∩ Fin))
204 eqidd 2738 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) = Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦))
205 sumeq1 15725 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑥𝐴) → Σ𝑦𝑧 ((𝐹𝐴)‘𝑦) = Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦))
206205rspceeqv 3645 . . . . . . . . . . . . . . . . . . 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 15724 . . . . . . . . . . . . . . . . . . 19 Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) ∈ V
209208a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) ∈ V)
210192, 207, 209elrnmptd 5974 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)))
211190, 210eqeltrd 2841 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)))
2122113ad2ant2 1135 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐴)‘𝑦)))
213 sumeq1 15725 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → Σ𝑦𝑥 ((𝐹𝐵)‘𝑦) = Σ𝑦𝑧 ((𝐹𝐵)‘𝑦))
214213cbvmptv 5255 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)) = (𝑧 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑧 ((𝐹𝐵)‘𝑦))
215 inss2 4238 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐵) ⊆ 𝐵
216193inex1 5317 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝐵) ∈ V
217216elpw 4604 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝐵) ∈ 𝒫 𝐵 ↔ (𝑥𝐵) ⊆ 𝐵)
218215, 217mpbir 231 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝐵) ∈ 𝒫 𝐵
219218a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → (𝑥𝐵) ∈ 𝒫 𝐵)
220 inss1 4237 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝐵) ⊆ 𝑥
221220a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐵) ⊆ 𝑥)
222 ssfi 9213 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ Fin ∧ (𝑥𝐵) ⊆ 𝑥) → (𝑥𝐵) ∈ Fin)
223198, 221, 222syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐵) ∈ Fin)
2242233ad2ant2 1135 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → (𝑥𝐵) ∈ Fin)
225219, 224elind 4200 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → (𝑥𝐵) ∈ (𝒫 𝐵 ∩ Fin))
226215sseli 3979 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (𝑥𝐵) → 𝑦𝐵)
227123eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐵 → (𝐹𝑦) = ((𝐹𝐵)‘𝑦))
228226, 227syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝑥𝐵) → (𝐹𝑦) = ((𝐹𝐵)‘𝑦))
229228sumeq2i 15734 . . . . . . . . . . . . . . . . . . . 20 Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) = Σ𝑦 ∈ (𝑥𝐵)((𝐹𝐵)‘𝑦)
230229a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) = Σ𝑦 ∈ (𝑥𝐵)((𝐹𝐵)‘𝑦))
2312303adant3 1133 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) = Σ𝑦 ∈ (𝑥𝐵)((𝐹𝐵)‘𝑦))
232 sumeq1 15725 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑥𝐵) → Σ𝑦𝑧 ((𝐹𝐵)‘𝑦) = Σ𝑦 ∈ (𝑥𝐵)((𝐹𝐵)‘𝑦))
233232rspceeqv 3645 . . . . . . . . . . . . . . . . . 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 15724 . . . . . . . . . . . . . . . . . 18 Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ∈ V
236235a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ∈ V)
237214, 234, 236elrnmptd 5974 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ∈ ran (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑦𝑥 ((𝐹𝐵)‘𝑦)))
238 simp3 1139 . . . . . . . . . . . . . . . . 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 45060 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝐵) = ∅ ∧ (𝑥𝐴) ⊆ 𝐴 ∧ (𝑥𝐵) ⊆ 𝐵) → ((𝑥𝐴) ∩ (𝑥𝐵)) = ∅)
242132, 239, 240, 241syl3anc 1373 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑥𝐴) ∩ (𝑥𝐵)) = ∅)
243242adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ((𝑥𝐴) ∩ (𝑥𝐵)) = ∅)
244 elinel1 4201 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ 𝒫 𝑈)
245 elpwi 4607 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ 𝒫 𝑈𝑥𝑈)
246244, 245syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥𝑈)
2474ineq2i 4217 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝑈) = (𝑥 ∩ (𝐴𝐵))
248247a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑈 → (𝑥𝑈) = (𝑥 ∩ (𝐴𝐵)))
249 dfss 3970 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝑈𝑥 = (𝑥𝑈))
250249biimpi 216 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑈𝑥 = (𝑥𝑈))
251 indi 4284 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∩ (𝐴𝐵)) = ((𝑥𝐴) ∪ (𝑥𝐵))
252251eqcomi 2746 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥𝐴) ∪ (𝑥𝐵)) = (𝑥 ∩ (𝐴𝐵))
253252a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑈 → ((𝑥𝐴) ∪ (𝑥𝐵)) = (𝑥 ∩ (𝐴𝐵)))
254248, 250, 2533eqtr4d 2787 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝑈𝑥 = ((𝑥𝐴) ∪ (𝑥𝐵)))
255246, 254syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 = ((𝑥𝐴) ∪ (𝑥𝐵)))
256255adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑥 = ((𝑥𝐴) ∪ (𝑥𝐵)))
257198adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑥 ∈ Fin)
258144ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑈⟶(0[,)+∞))
259246sselda 3983 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑈)
260259adantll 714 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑈)
261258, 260ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,)+∞))
262143, 261sselid 3981 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℂ)
263243, 256, 257, 262fsumsplit 15777 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
2642633adant3 1133 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → Σ𝑦𝑥 (𝐹𝑦) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
265238, 264eqtrd 2777 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑟 = Σ𝑦𝑥 (𝐹𝑦)) → 𝑟 = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
266 oveq2 7439 . . . . . . . . . . . . . . . . 17 (𝑢 = Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) → (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + 𝑢) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
267266rspceeqv 3645 . . . . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . . . . . 18 (𝑣 = Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) → (𝑣 + 𝑢) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + 𝑢))
270269eqeq2d 2748 . . . . . . . . . . . . . . . . 17 (𝑣 = Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) → (𝑟 = (𝑣 + 𝑢) ↔ 𝑟 = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + 𝑢)))
271270rexbidv 3179 . . . . . . . . . . . . . . . 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 3622 . . . . . . . . . . . . . . 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 1120 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 3266 . . . . . . . . . . 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 234 . . . . . . . . 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 412 . . . . . . . 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 212 . . . . . . 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 1927 . . . . . 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 2730 . . . . . 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 234 . . . . 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 9486 . . . 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 2782 . . 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 46404 . . 3 (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ))
28716, 23oveq12d 7449 . . 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 2787 . 2 (𝜑 → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
289 rexadd 13274 . . 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 2777 1 (𝜑 → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087  wal 1538   = wceq 1540  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  Vcvv 3480  cun 3949  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600   class class class wbr 5143  cmpt 5225  ran crn 5686  cres 5687  wf 6557  cfv 6561  (class class class)co 7431  Fincfn 8985  supcsup 9480  cc 11153  cr 11154  0cc0 11155   + caddc 11158  +∞cpnf 11292  *cxr 11294   < clt 11295  cle 11296   +𝑒 cxad 13152  [,)cico 13389  [,]cicc 13390  Σcsu 15722  Σ^csumge0 46377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-xadd 13155  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-sumge0 46378
This theorem is referenced by:  sge0split  46424
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