Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sge0split Structured version   Visualization version   GIF version

Theorem sge0split 44640
Description: Split a sum of nonnegative extended reals into two parts. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0split.a (𝜑𝐴𝑉)
sge0split.b (𝜑𝐵𝑊)
sge0split.u 𝑈 = (𝐴𝐵)
sge0split.in0 (𝜑 → (𝐴𝐵) = ∅)
sge0split.f (𝜑𝐹:𝑈⟶(0[,]+∞))
Assertion
Ref Expression
sge0split (𝜑 → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))

Proof of Theorem sge0split
Dummy variables 𝑎 𝑏 𝑥 𝑧 𝑦 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sge0split.a . . . . 5 (𝜑𝐴𝑉)
21adantr 481 . . . 4 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → 𝐴𝑉)
3 sge0split.b . . . . 5 (𝜑𝐵𝑊)
43adantr 481 . . . 4 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → 𝐵𝑊)
5 sge0split.u . . . 4 𝑈 = (𝐴𝐵)
6 sge0split.in0 . . . . 5 (𝜑 → (𝐴𝐵) = ∅)
76adantr 481 . . . 4 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (𝐴𝐵) = ∅)
8 sge0split.f . . . . 5 (𝜑𝐹:𝑈⟶(0[,]+∞))
98adantr 481 . . . 4 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → 𝐹:𝑈⟶(0[,]+∞))
10 simpr 485 . . . 4 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (Σ^𝐹) ∈ ℝ)
112, 4, 5, 7, 9, 10sge0resplit 44637 . . 3 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))
12 unexg 7683 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
131, 3, 12syl2anc 584 . . . . . . . 8 (𝜑 → (𝐴𝐵) ∈ V)
145, 13eqeltrid 2842 . . . . . . 7 (𝜑𝑈 ∈ V)
1514adantr 481 . . . . . 6 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → 𝑈 ∈ V)
1615, 9, 10sge0ssre 44628 . . . . 5 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (Σ^‘(𝐹𝐴)) ∈ ℝ)
1715, 9, 10sge0ssre 44628 . . . . 5 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (Σ^‘(𝐹𝐵)) ∈ ℝ)
18 rexadd 13151 . . . . 5 (((Σ^‘(𝐹𝐴)) ∈ ℝ ∧ (Σ^‘(𝐹𝐵)) ∈ ℝ) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))
1916, 17, 18syl2anc 584 . . . 4 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))
2019eqcomd 2742 . . 3 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
2111, 20eqtrd 2776 . 2 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
22 simpl 483 . . 3 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → 𝜑)
23 simpr 485 . . . . 5 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → ¬ (Σ^𝐹) ∈ ℝ)
2414, 8sge0repnf 44617 . . . . . 6 (𝜑 → ((Σ^𝐹) ∈ ℝ ↔ ¬ (Σ^𝐹) = +∞))
2524adantr 481 . . . . 5 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → ((Σ^𝐹) ∈ ℝ ↔ ¬ (Σ^𝐹) = +∞))
2623, 25mtbid 323 . . . 4 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → ¬ ¬ (Σ^𝐹) = +∞)
2726notnotrd 133 . . 3 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → (Σ^𝐹) = +∞)
2814, 8sge0xrcl 44616 . . . . 5 (𝜑 → (Σ^𝐹) ∈ ℝ*)
2928adantr 481 . . . 4 ((𝜑 ∧ (Σ^𝐹) = +∞) → (Σ^𝐹) ∈ ℝ*)
30 ssun1 4132 . . . . . . . . . 10 𝐴 ⊆ (𝐴𝐵)
3130, 5sseqtrri 3981 . . . . . . . . 9 𝐴𝑈
3231a1i 11 . . . . . . . 8 (𝜑𝐴𝑈)
338, 32fssresd 6709 . . . . . . 7 (𝜑 → (𝐹𝐴):𝐴⟶(0[,]+∞))
341, 33sge0xrcl 44616 . . . . . 6 (𝜑 → (Σ^‘(𝐹𝐴)) ∈ ℝ*)
35 iccssxr 13347 . . . . . . 7 (0[,]+∞) ⊆ ℝ*
36 ssun2 4133 . . . . . . . . . . 11 𝐵 ⊆ (𝐴𝐵)
3736, 5sseqtrri 3981 . . . . . . . . . 10 𝐵𝑈
3837a1i 11 . . . . . . . . 9 (𝜑𝐵𝑈)
398, 38fssresd 6709 . . . . . . . 8 (𝜑 → (𝐹𝐵):𝐵⟶(0[,]+∞))
403, 39sge0cl 44612 . . . . . . 7 (𝜑 → (Σ^‘(𝐹𝐵)) ∈ (0[,]+∞))
4135, 40sselid 3942 . . . . . 6 (𝜑 → (Σ^‘(𝐹𝐵)) ∈ ℝ*)
4234, 41xaddcld 13220 . . . . 5 (𝜑 → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ∈ ℝ*)
4342adantr 481 . . . 4 ((𝜑 ∧ (Σ^𝐹) = +∞) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ∈ ℝ*)
44 pnfxr 11209 . . . . . . . . 9 +∞ ∈ ℝ*
45 eqid 2736 . . . . . . . . 9 +∞ = +∞
46 xreqle 43542 . . . . . . . . 9 ((+∞ ∈ ℝ* ∧ +∞ = +∞) → +∞ ≤ +∞)
4744, 45, 46mp2an 690 . . . . . . . 8 +∞ ≤ +∞
4847a1i 11 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → +∞ ≤ +∞)
4914adantr 481 . . . . . . . . 9 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → 𝑈 ∈ V)
508adantr 481 . . . . . . . . 9 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → 𝐹:𝑈⟶(0[,]+∞))
51 rnresss 5973 . . . . . . . . . . 11 ran (𝐹𝐴) ⊆ ran 𝐹
5251sseli 3940 . . . . . . . . . 10 (+∞ ∈ ran (𝐹𝐴) → +∞ ∈ ran 𝐹)
5352adantl 482 . . . . . . . . 9 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → +∞ ∈ ran 𝐹)
5449, 50, 53sge0pnfval 44604 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → (Σ^𝐹) = +∞)
55 xrge0neqmnf 13369 . . . . . . . . . . . . . 14 ((Σ^‘(𝐹𝐵)) ∈ (0[,]+∞) → (Σ^‘(𝐹𝐵)) ≠ -∞)
5640, 55syl 17 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝐹𝐵)) ≠ -∞)
57 xaddpnf2 13146 . . . . . . . . . . . . 13 (((Σ^‘(𝐹𝐵)) ∈ ℝ* ∧ (Σ^‘(𝐹𝐵)) ≠ -∞) → (+∞ +𝑒^‘(𝐹𝐵))) = +∞)
5841, 56, 57syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (+∞ +𝑒^‘(𝐹𝐵))) = +∞)
5958eqcomd 2742 . . . . . . . . . . 11 (𝜑 → +∞ = (+∞ +𝑒^‘(𝐹𝐵))))
6059adantr 481 . . . . . . . . . 10 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → +∞ = (+∞ +𝑒^‘(𝐹𝐵))))
611adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → 𝐴𝑉)
6233adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → (𝐹𝐴):𝐴⟶(0[,]+∞))
63 simpr 485 . . . . . . . . . . . 12 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → +∞ ∈ ran (𝐹𝐴))
6461, 62, 63sge0pnfval 44604 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → (Σ^‘(𝐹𝐴)) = +∞)
6564oveq1d 7372 . . . . . . . . . 10 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = (+∞ +𝑒^‘(𝐹𝐵))))
6660, 54, 653eqtr4d 2786 . . . . . . . . 9 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
6766, 54eqtr3d 2778 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = +∞)
6854, 67breq12d 5118 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → ((Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ↔ +∞ ≤ +∞))
6948, 68mpbird 256 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
7047a1i 11 . . . . . . . . 9 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → +∞ ≤ +∞)
7114adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → 𝑈 ∈ V)
728adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → 𝐹:𝑈⟶(0[,]+∞))
73 rnresss 5973 . . . . . . . . . . . . 13 ran (𝐹𝐵) ⊆ ran 𝐹
7473sseli 3940 . . . . . . . . . . . 12 (+∞ ∈ ran (𝐹𝐵) → +∞ ∈ ran 𝐹)
7574adantl 482 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → +∞ ∈ ran 𝐹)
7671, 72, 75sge0pnfval 44604 . . . . . . . . . 10 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → (Σ^𝐹) = +∞)
773adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → 𝐵𝑊)
7839adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → (𝐹𝐵):𝐵⟶(0[,]+∞))
79 simpr 485 . . . . . . . . . . . . 13 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → +∞ ∈ ran (𝐹𝐵))
8077, 78, 79sge0pnfval 44604 . . . . . . . . . . . 12 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → (Σ^‘(𝐹𝐵)) = +∞)
8180oveq2d 7373 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = ((Σ^‘(𝐹𝐴)) +𝑒 +∞))
821, 33sge0cl 44612 . . . . . . . . . . . . . 14 (𝜑 → (Σ^‘(𝐹𝐴)) ∈ (0[,]+∞))
83 xrge0neqmnf 13369 . . . . . . . . . . . . . 14 ((Σ^‘(𝐹𝐴)) ∈ (0[,]+∞) → (Σ^‘(𝐹𝐴)) ≠ -∞)
8482, 83syl 17 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝐹𝐴)) ≠ -∞)
85 xaddpnf1 13145 . . . . . . . . . . . . 13 (((Σ^‘(𝐹𝐴)) ∈ ℝ* ∧ (Σ^‘(𝐹𝐴)) ≠ -∞) → ((Σ^‘(𝐹𝐴)) +𝑒 +∞) = +∞)
8634, 84, 85syl2anc 584 . . . . . . . . . . . 12 (𝜑 → ((Σ^‘(𝐹𝐴)) +𝑒 +∞) = +∞)
8786adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → ((Σ^‘(𝐹𝐴)) +𝑒 +∞) = +∞)
8881, 87eqtrd 2776 . . . . . . . . . 10 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = +∞)
8976, 88breq12d 5118 . . . . . . . . 9 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → ((Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ↔ +∞ ≤ +∞))
9070, 89mpbird 256 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
9190adantlr 713 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ +∞ ∈ ran (𝐹𝐵)) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
92 simpr 485 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
93 vex 3449 . . . . . . . . . . . . 13 𝑧 ∈ V
94 eqid 2736 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
9594elrnmpt 5911 . . . . . . . . . . . . 13 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑧 = Σ𝑦𝑥 (𝐹𝑦)))
9693, 95ax-mp 5 . . . . . . . . . . . 12 (𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑧 = Σ𝑦𝑥 (𝐹𝑦))
9792, 96sylib 217 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑧 = Σ𝑦𝑥 (𝐹𝑦))
98 simp3 1138 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑧 = Σ𝑦𝑥 (𝐹𝑦)) → 𝑧 = Σ𝑦𝑥 (𝐹𝑦))
99 inss1 4188 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ (𝑥𝐴)
100 inss2 4189 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐴) ⊆ 𝐴
10199, 100sstri 3953 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ 𝐴
102 inss2 4189 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ (𝑥𝐵)
103 inss2 4189 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐵) ⊆ 𝐵
104102, 103sstri 3953 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ 𝐵
105101, 104ssini 4191 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ (𝐴𝐵)
106105a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ (𝐴𝐵))
107106, 6sseqtrd 3984 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ ∅)
108 ss0 4358 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ ∅ → ((𝑥𝐴) ∩ (𝑥𝐵)) = ∅)
109107, 108syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑥𝐴) ∩ (𝑥𝐵)) = ∅)
110109ad3antrrr 728 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ((𝑥𝐴) ∩ (𝑥𝐵)) = ∅)
111 indi 4233 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∩ (𝐴𝐵)) = ((𝑥𝐴) ∪ (𝑥𝐵))
112111eqcomi 2745 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝐴) ∪ (𝑥𝐵)) = (𝑥 ∩ (𝐴𝐵))
113112a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → ((𝑥𝐴) ∪ (𝑥𝐵)) = (𝑥 ∩ (𝐴𝐵)))
1145eqcomi 2745 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴𝐵) = 𝑈
115114ineq2i 4169 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∩ (𝐴𝐵)) = (𝑥𝑈)
116115a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ (𝐴𝐵)) = (𝑥𝑈))
117 elinel1 4155 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ 𝒫 𝑈)
118 elpwi 4567 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ 𝒫 𝑈𝑥𝑈)
119117, 118syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥𝑈)
120 df-ss 3927 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝑈 ↔ (𝑥𝑈) = 𝑥)
121120biimpi 215 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑈 → (𝑥𝑈) = 𝑥)
122119, 121syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝑈) = 𝑥)
123113, 116, 1223eqtrrd 2781 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 = ((𝑥𝐴) ∪ (𝑥𝐵)))
124123adantl 482 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑥 = ((𝑥𝐴) ∪ (𝑥𝐵)))
125 elinel2 4156 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ Fin)
126125adantl 482 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑥 ∈ Fin)
127 rge0ssre 13373 . . . . . . . . . . . . . . . . . . . . 21 (0[,)+∞) ⊆ ℝ
1288ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → 𝐹:𝑈⟶(0[,]+∞))
129 pm4.56 987 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((¬ +∞ ∈ ran (𝐹𝐴) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ↔ ¬ (+∞ ∈ ran (𝐹𝐴) ∨ +∞ ∈ ran (𝐹𝐵)))
130129biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((¬ +∞ ∈ ran (𝐹𝐴) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ¬ (+∞ ∈ ran (𝐹𝐴) ∨ +∞ ∈ ran (𝐹𝐵)))
131 elun 4108 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (+∞ ∈ (ran (𝐹𝐴) ∪ ran (𝐹𝐵)) ↔ (+∞ ∈ ran (𝐹𝐴) ∨ +∞ ∈ ran (𝐹𝐵)))
132130, 131sylnibr 328 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((¬ +∞ ∈ ran (𝐹𝐴) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ¬ +∞ ∈ (ran (𝐹𝐴) ∪ ran (𝐹𝐵)))
133132adantll 712 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ¬ +∞ ∈ (ran (𝐹𝐴) ∪ ran (𝐹𝐵)))
134 rnresun 43387 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ran (𝐹 ↾ (𝐴𝐵)) = (ran (𝐹𝐴) ∪ ran (𝐹𝐵))
135134eqcomi 2745 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (ran (𝐹𝐴) ∪ ran (𝐹𝐵)) = ran (𝐹 ↾ (𝐴𝐵))
136135a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (ran (𝐹𝐴) ∪ ran (𝐹𝐵)) = ran (𝐹 ↾ (𝐴𝐵)))
137114reseq2i 5934 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹 ↾ (𝐴𝐵)) = (𝐹𝑈)
138137rneqi 5892 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ran (𝐹 ↾ (𝐴𝐵)) = ran (𝐹𝑈)
139138a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ran (𝐹 ↾ (𝐴𝐵)) = ran (𝐹𝑈))
140 ffn 6668 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹:𝑈⟶(0[,]+∞) → 𝐹 Fn 𝑈)
141 fnresdm 6620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹 Fn 𝑈 → (𝐹𝑈) = 𝐹)
1428, 140, 1413syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝐹𝑈) = 𝐹)
143142rneqd 5893 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ran (𝐹𝑈) = ran 𝐹)
144136, 139, 1433eqtrd 2780 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (ran (𝐹𝐴) ∪ ran (𝐹𝐵)) = ran 𝐹)
145144ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (ran (𝐹𝐴) ∪ ran (𝐹𝐵)) = ran 𝐹)
146133, 145neleqtrd 2859 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ¬ +∞ ∈ ran 𝐹)
147128, 146fge0iccico 44601 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → 𝐹:𝑈⟶(0[,)+∞))
148147ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑈⟶(0[,)+∞))
149119adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦𝑥) → 𝑥𝑈)
150 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑥)
151149, 150sseldd 3945 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑈)
152151adantll 712 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑈)
153148, 152ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,)+∞))
154127, 153sselid 3942 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
155154recnd 11183 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℂ)
156110, 124, 126, 155fsumsplit 15626 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
157 infi 9212 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ Fin → (𝑥𝐴) ∈ Fin)
158125, 157syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐴) ∈ Fin)
159158adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥𝐴) ∈ Fin)
160 simpl 483 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐴)) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)))
161 elinel1 4155 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (𝑥𝐴) → 𝑦𝑥)
162161adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐴)) → 𝑦𝑥)
163160, 162, 154syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐴)) → (𝐹𝑦) ∈ ℝ)
164159, 163fsumrecl 15619 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ∈ ℝ)
165 infi 9212 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ Fin → (𝑥𝐵) ∈ Fin)
166125, 165syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐵) ∈ Fin)
167166adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥𝐵) ∈ Fin)
168 simpl 483 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐵)) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)))
169 elinel1 4155 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (𝑥𝐵) → 𝑦𝑥)
170169adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐵)) → 𝑦𝑥)
171168, 170, 154syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐵)) → (𝐹𝑦) ∈ ℝ)
172167, 171fsumrecl 15619 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ∈ ℝ)
173 rexadd 13151 . . . . . . . . . . . . . . . . . . . 20 ((Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ∈ ℝ ∧ Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ∈ ℝ) → (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) +𝑒 Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
174164, 172, 173syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) +𝑒 Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
175174eqcomd 2742 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) +𝑒 Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
176156, 175eqtrd 2776 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) +𝑒 Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
177 ressxr 11199 . . . . . . . . . . . . . . . . . . . 20 ℝ ⊆ ℝ*
178177, 164sselid 3942 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ∈ ℝ*)
179177, 172sselid 3942 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ∈ ℝ*)
1801adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → 𝐴𝑉)
18133adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → (𝐹𝐴):𝐴⟶(0[,]+∞))
182 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → ¬ +∞ ∈ ran (𝐹𝐴))
183181, 182fge0iccico 44601 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → (𝐹𝐴):𝐴⟶(0[,)+∞))
184180, 183sge0reval 44603 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → (Σ^‘(𝐹𝐴)) = sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ))
185184eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) = (Σ^‘(𝐹𝐴)))
18634adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → (Σ^‘(𝐹𝐴)) ∈ ℝ*)
187185, 186eqeltrd 2838 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ*)
188187adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ*)
1893adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → 𝐵𝑊)
19039adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (𝐹𝐵):𝐵⟶(0[,]+∞))
191 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ¬ +∞ ∈ ran (𝐹𝐵))
192190, 191fge0iccico 44601 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (𝐹𝐵):𝐵⟶(0[,)+∞))
193189, 192sge0reval 44603 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (Σ^‘(𝐹𝐵)) = sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))
194193eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ) = (Σ^‘(𝐹𝐵)))
19541adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (Σ^‘(𝐹𝐵)) ∈ ℝ*)
196194, 195eqeltrd 2838 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*)
197196adantlr 713 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*)
198188, 197jca 512 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ* ∧ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*))
199198adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ* ∧ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*))
200178, 179, 199jca31 515 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ((Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ∈ ℝ* ∧ Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ∈ ℝ*) ∧ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ* ∧ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*)))
201180adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝐴𝑉)
202181adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝐹𝐴):𝐴⟶(0[,]+∞))
203182adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ¬ +∞ ∈ ran (𝐹𝐴))
204202, 203fge0iccico 44601 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝐹𝐴):𝐴⟶(0[,)+∞))
205100a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥𝐴) ⊆ 𝐴)
206158adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥𝐴) ∈ Fin)
207201, 204, 205, 206fsumlesge0 44608 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) ≤ (Σ^‘(𝐹𝐴)))
208100sseli 3940 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ (𝑥𝐴) → 𝑦𝐴)
209 fvres 6861 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦𝐴 → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
210208, 209syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (𝑥𝐴) → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
211210adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐴)) → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
212211sumeq2dv 15588 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) = Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦))
213184adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ^‘(𝐹𝐴)) = sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ))
214212, 213breq12d 5118 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) ≤ (Σ^‘(𝐹𝐴)) ↔ Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ≤ sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < )))
215207, 214mpbid 231 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ≤ sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ))
216215adantlr 713 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ≤ sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ))
217189adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝐵𝑊)
218190adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝐹𝐵):𝐵⟶(0[,]+∞))
219191adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ¬ +∞ ∈ ran (𝐹𝐵))
220218, 219fge0iccico 44601 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝐹𝐵):𝐵⟶(0[,)+∞))
221103a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥𝐵) ⊆ 𝐵)
222166adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥𝐵) ∈ Fin)
223217, 220, 221, 222fsumlesge0 44608 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)((𝐹𝐵)‘𝑦) ≤ (Σ^‘(𝐹𝐵)))
224103sseli 3940 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ (𝑥𝐵) → 𝑦𝐵)
225 fvres 6861 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦𝐵 → ((𝐹𝐵)‘𝑦) = (𝐹𝑦))
226224, 225syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (𝑥𝐵) → ((𝐹𝐵)‘𝑦) = (𝐹𝑦))
227226adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐵)) → ((𝐹𝐵)‘𝑦) = (𝐹𝑦))
228227sumeq2dv 15588 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)((𝐹𝐵)‘𝑦) = Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦))
229193adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ^‘(𝐹𝐵)) = sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))
230228, 229breq12d 5118 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ𝑦 ∈ (𝑥𝐵)((𝐹𝐵)‘𝑦) ≤ (Σ^‘(𝐹𝐵)) ↔ Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ≤ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )))
231223, 230mpbid 231 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ≤ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))
232231adantllr 717 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ≤ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))
233216, 232jca 512 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ≤ sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) ∧ Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ≤ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )))
234 xle2add 13178 . . . . . . . . . . . . . . . . . 18 (((Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ∈ ℝ* ∧ Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ∈ ℝ*) ∧ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ* ∧ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*)) → ((Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ≤ sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) ∧ Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ≤ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )) → (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) +𝑒 Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)) ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))))
235200, 233, 234sylc 65 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) +𝑒 Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)) ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )))
236176, 235eqbrtrd 5127 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )))
2372363adant3 1132 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑧 = Σ𝑦𝑥 (𝐹𝑦)) → Σ𝑦𝑥 (𝐹𝑦) ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )))
23898, 237eqbrtrd 5127 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑧 = Σ𝑦𝑥 (𝐹𝑦)) → 𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )))
2392383exp 1119 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑧 = Σ𝑦𝑥 (𝐹𝑦) → 𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )))))
240239rexlimdv 3150 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑧 = Σ𝑦𝑥 (𝐹𝑦) → 𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))))
241240adantr 481 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → (∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑧 = Σ𝑦𝑥 (𝐹𝑦) → 𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))))
24297, 241mpd 15 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → 𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )))
243242ralrimiva 3143 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ∀𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )))
244147sge0rnre 44595 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ)
245177a1i 11 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ℝ ⊆ ℝ*)
246244, 245sstrd 3954 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ*)
247188, 197xaddcld 13220 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )) ∈ ℝ*)
248 supxrleub 13245 . . . . . . . . . 10 ((ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ* ∧ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )) ∈ ℝ*) → (sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )) ↔ ∀𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))))
249246, 247, 248syl2anc 584 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )) ↔ ∀𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))𝑧 ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))))
250243, 249mpbird 256 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) ≤ (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )))
25114ad2antrr 724 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → 𝑈 ∈ V)
252251, 147sge0reval 44603 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
253184adantr 481 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (Σ^‘(𝐹𝐴)) = sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ))
254193adantlr 713 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (Σ^‘(𝐹𝐵)) = sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))
255253, 254oveq12d 7375 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = (sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) +𝑒 sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < )))
256250, 252, 2553brtr4d 5137 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
25791, 256pm2.61dan 811 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
25869, 257pm2.61dan 811 . . . . 5 (𝜑 → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
259258adantr 481 . . . 4 ((𝜑 ∧ (Σ^𝐹) = +∞) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
260 pnfge 13051 . . . . . . 7 (((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ∈ ℝ* → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ≤ +∞)
26142, 260syl 17 . . . . . 6 (𝜑 → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ≤ +∞)
262261adantr 481 . . . . 5 ((𝜑 ∧ (Σ^𝐹) = +∞) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ≤ +∞)
263 id 22 . . . . . . 7 ((Σ^𝐹) = +∞ → (Σ^𝐹) = +∞)
264263eqcomd 2742 . . . . . 6 ((Σ^𝐹) = +∞ → +∞ = (Σ^𝐹))
265264adantl 482 . . . . 5 ((𝜑 ∧ (Σ^𝐹) = +∞) → +∞ = (Σ^𝐹))
266262, 265breqtrd 5131 . . . 4 ((𝜑 ∧ (Σ^𝐹) = +∞) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ≤ (Σ^𝐹))
26729, 43, 259, 266xrletrid 13074 . . 3 ((𝜑 ∧ (Σ^𝐹) = +∞) → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
26822, 27, 267syl2anc 584 . 2 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
26921, 268pm2.61dan 811 1 (𝜑 → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  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   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  Fincfn 8883  supcsup 9376  cr 11050  0cc0 11051   + caddc 11054  +∞cpnf 11186  -∞cmnf 11187  *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:  sge0splitmpt  44642
  Copyright terms: Public domain W3C validator