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Theorem sge0split 41289
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 472 . . . 4 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → 𝐴𝑉)
3 sge0split.b . . . . 5 (𝜑𝐵𝑊)
43adantr 472 . . . 4 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → 𝐵𝑊)
5 sge0split.u . . . 4 𝑈 = (𝐴𝐵)
6 sge0split.in0 . . . . 5 (𝜑 → (𝐴𝐵) = ∅)
76adantr 472 . . . 4 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (𝐴𝐵) = ∅)
8 sge0split.f . . . . 5 (𝜑𝐹:𝑈⟶(0[,]+∞))
98adantr 472 . . . 4 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → 𝐹:𝑈⟶(0[,]+∞))
10 simpr 477 . . . 4 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (Σ^𝐹) ∈ ℝ)
112, 4, 5, 7, 9, 10sge0resplit 41286 . . 3 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))
12 unexg 7161 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
131, 3, 12syl2anc 579 . . . . . . . 8 (𝜑 → (𝐴𝐵) ∈ V)
145, 13syl5eqel 2848 . . . . . . 7 (𝜑𝑈 ∈ V)
1514adantr 472 . . . . . 6 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → 𝑈 ∈ V)
1615, 9, 10sge0ssre 41277 . . . . 5 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (Σ^‘(𝐹𝐴)) ∈ ℝ)
1715, 9, 10sge0ssre 41277 . . . . 5 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (Σ^‘(𝐹𝐵)) ∈ ℝ)
18 rexadd 12272 . . . . 5 (((Σ^‘(𝐹𝐴)) ∈ ℝ ∧ (Σ^‘(𝐹𝐵)) ∈ ℝ) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))
1916, 17, 18syl2anc 579 . . . 4 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))
2019eqcomd 2771 . . 3 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
2111, 20eqtrd 2799 . 2 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
22 simpl 474 . . 3 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → 𝜑)
23 simpr 477 . . . . 5 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → ¬ (Σ^𝐹) ∈ ℝ)
2414, 8sge0repnf 41266 . . . . . 6 (𝜑 → ((Σ^𝐹) ∈ ℝ ↔ ¬ (Σ^𝐹) = +∞))
2524adantr 472 . . . . 5 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → ((Σ^𝐹) ∈ ℝ ↔ ¬ (Σ^𝐹) = +∞))
2623, 25mtbid 315 . . . 4 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → ¬ ¬ (Σ^𝐹) = +∞)
2726notnotrd 130 . . 3 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → (Σ^𝐹) = +∞)
2814, 8sge0xrcl 41265 . . . . 5 (𝜑 → (Σ^𝐹) ∈ ℝ*)
2928adantr 472 . . . 4 ((𝜑 ∧ (Σ^𝐹) = +∞) → (Σ^𝐹) ∈ ℝ*)
30 ssun1 3940 . . . . . . . . . 10 𝐴 ⊆ (𝐴𝐵)
3130, 5sseqtr4i 3800 . . . . . . . . 9 𝐴𝑈
3231a1i 11 . . . . . . . 8 (𝜑𝐴𝑈)
338, 32fssresd 6255 . . . . . . 7 (𝜑 → (𝐹𝐴):𝐴⟶(0[,]+∞))
341, 33sge0xrcl 41265 . . . . . 6 (𝜑 → (Σ^‘(𝐹𝐴)) ∈ ℝ*)
35 iccssxr 12465 . . . . . . 7 (0[,]+∞) ⊆ ℝ*
36 ssun2 3941 . . . . . . . . . . 11 𝐵 ⊆ (𝐴𝐵)
3736, 5sseqtr4i 3800 . . . . . . . . . 10 𝐵𝑈
3837a1i 11 . . . . . . . . 9 (𝜑𝐵𝑈)
398, 38fssresd 6255 . . . . . . . 8 (𝜑 → (𝐹𝐵):𝐵⟶(0[,]+∞))
403, 39sge0cl 41261 . . . . . . 7 (𝜑 → (Σ^‘(𝐹𝐵)) ∈ (0[,]+∞))
4135, 40sseldi 3761 . . . . . 6 (𝜑 → (Σ^‘(𝐹𝐵)) ∈ ℝ*)
4234, 41xaddcld 12340 . . . . 5 (𝜑 → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ∈ ℝ*)
4342adantr 472 . . . 4 ((𝜑 ∧ (Σ^𝐹) = +∞) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ∈ ℝ*)
44 pnfxr 10351 . . . . . . . . 9 +∞ ∈ ℝ*
45 eqid 2765 . . . . . . . . 9 +∞ = +∞
46 xreqle 40198 . . . . . . . . 9 ((+∞ ∈ ℝ* ∧ +∞ = +∞) → +∞ ≤ +∞)
4744, 45, 46mp2an 683 . . . . . . . 8 +∞ ≤ +∞
4847a1i 11 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → +∞ ≤ +∞)
4914adantr 472 . . . . . . . . 9 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → 𝑈 ∈ V)
508adantr 472 . . . . . . . . 9 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → 𝐹:𝑈⟶(0[,]+∞))
51 rnresss 40038 . . . . . . . . . . 11 ran (𝐹𝐴) ⊆ ran 𝐹
5251sseli 3759 . . . . . . . . . 10 (+∞ ∈ ran (𝐹𝐴) → +∞ ∈ ran 𝐹)
5352adantl 473 . . . . . . . . 9 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → +∞ ∈ ran 𝐹)
5449, 50, 53sge0pnfval 41253 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → (Σ^𝐹) = +∞)
55 xrge0neqmnf 12486 . . . . . . . . . . . . . 14 ((Σ^‘(𝐹𝐵)) ∈ (0[,]+∞) → (Σ^‘(𝐹𝐵)) ≠ -∞)
5640, 55syl 17 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝐹𝐵)) ≠ -∞)
57 xaddpnf2 12267 . . . . . . . . . . . . 13 (((Σ^‘(𝐹𝐵)) ∈ ℝ* ∧ (Σ^‘(𝐹𝐵)) ≠ -∞) → (+∞ +𝑒^‘(𝐹𝐵))) = +∞)
5841, 56, 57syl2anc 579 . . . . . . . . . . . 12 (𝜑 → (+∞ +𝑒^‘(𝐹𝐵))) = +∞)
5958eqcomd 2771 . . . . . . . . . . 11 (𝜑 → +∞ = (+∞ +𝑒^‘(𝐹𝐵))))
6059adantr 472 . . . . . . . . . 10 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → +∞ = (+∞ +𝑒^‘(𝐹𝐵))))
611adantr 472 . . . . . . . . . . . 12 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → 𝐴𝑉)
6233adantr 472 . . . . . . . . . . . 12 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → (𝐹𝐴):𝐴⟶(0[,]+∞))
63 simpr 477 . . . . . . . . . . . 12 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → +∞ ∈ ran (𝐹𝐴))
6461, 62, 63sge0pnfval 41253 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → (Σ^‘(𝐹𝐴)) = +∞)
6564oveq1d 6861 . . . . . . . . . 10 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = (+∞ +𝑒^‘(𝐹𝐵))))
6660, 54, 653eqtr4d 2809 . . . . . . . . 9 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
6766, 54eqtr3d 2801 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = +∞)
6854, 67breq12d 4824 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → ((Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ↔ +∞ ≤ +∞))
6948, 68mpbird 248 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
7047a1i 11 . . . . . . . . 9 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → +∞ ≤ +∞)
7114adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → 𝑈 ∈ V)
728adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → 𝐹:𝑈⟶(0[,]+∞))
73 rnresss 40038 . . . . . . . . . . . . 13 ran (𝐹𝐵) ⊆ ran 𝐹
7473sseli 3759 . . . . . . . . . . . 12 (+∞ ∈ ran (𝐹𝐵) → +∞ ∈ ran 𝐹)
7574adantl 473 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → +∞ ∈ ran 𝐹)
7671, 72, 75sge0pnfval 41253 . . . . . . . . . 10 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → (Σ^𝐹) = +∞)
773adantr 472 . . . . . . . . . . . . 13 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → 𝐵𝑊)
7839adantr 472 . . . . . . . . . . . . 13 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → (𝐹𝐵):𝐵⟶(0[,]+∞))
79 simpr 477 . . . . . . . . . . . . 13 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → +∞ ∈ ran (𝐹𝐵))
8077, 78, 79sge0pnfval 41253 . . . . . . . . . . . 12 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → (Σ^‘(𝐹𝐵)) = +∞)
8180oveq2d 6862 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = ((Σ^‘(𝐹𝐴)) +𝑒 +∞))
821, 33sge0cl 41261 . . . . . . . . . . . . . 14 (𝜑 → (Σ^‘(𝐹𝐴)) ∈ (0[,]+∞))
83 xrge0neqmnf 12486 . . . . . . . . . . . . . 14 ((Σ^‘(𝐹𝐴)) ∈ (0[,]+∞) → (Σ^‘(𝐹𝐴)) ≠ -∞)
8482, 83syl 17 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝐹𝐴)) ≠ -∞)
85 xaddpnf1 12266 . . . . . . . . . . . . 13 (((Σ^‘(𝐹𝐴)) ∈ ℝ* ∧ (Σ^‘(𝐹𝐴)) ≠ -∞) → ((Σ^‘(𝐹𝐴)) +𝑒 +∞) = +∞)
8634, 84, 85syl2anc 579 . . . . . . . . . . . 12 (𝜑 → ((Σ^‘(𝐹𝐴)) +𝑒 +∞) = +∞)
8786adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → ((Σ^‘(𝐹𝐴)) +𝑒 +∞) = +∞)
8881, 87eqtrd 2799 . . . . . . . . . 10 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = +∞)
8976, 88breq12d 4824 . . . . . . . . 9 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → ((Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ↔ +∞ ≤ +∞))
9070, 89mpbird 248 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
9190adantlr 706 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ +∞ ∈ ran (𝐹𝐵)) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
92 simpr 477 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
93 vex 3353 . . . . . . . . . . . . 13 𝑧 ∈ V
94 eqid 2765 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
9594elrnmpt 5543 . . . . . . . . . . . . 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 209 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑧 = Σ𝑦𝑥 (𝐹𝑦))
98 simp3 1168 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑧 = Σ𝑦𝑥 (𝐹𝑦)) → 𝑧 = Σ𝑦𝑥 (𝐹𝑦))
99 inss1 3994 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ (𝑥𝐴)
100 inss2 3995 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐴) ⊆ 𝐴
10199, 100sstri 3772 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ 𝐴
102 inss2 3995 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ (𝑥𝐵)
103 inss2 3995 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐵) ⊆ 𝐵
104102, 103sstri 3772 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ 𝐵
105101, 104ssini 3997 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ (𝐴𝐵)
106105a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ (𝐴𝐵))
107106, 6sseqtrd 3803 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ ∅)
108 ss0 4138 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ ∅ → ((𝑥𝐴) ∩ (𝑥𝐵)) = ∅)
109107, 108syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑥𝐴) ∩ (𝑥𝐵)) = ∅)
110109ad3antrrr 721 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ((𝑥𝐴) ∩ (𝑥𝐵)) = ∅)
111 indi 4040 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∩ (𝐴𝐵)) = ((𝑥𝐴) ∪ (𝑥𝐵))
112111eqcomi 2774 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝐴) ∪ (𝑥𝐵)) = (𝑥 ∩ (𝐴𝐵))
113112a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → ((𝑥𝐴) ∪ (𝑥𝐵)) = (𝑥 ∩ (𝐴𝐵)))
1145eqcomi 2774 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴𝐵) = 𝑈
115114ineq2i 3975 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∩ (𝐴𝐵)) = (𝑥𝑈)
116115a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ (𝐴𝐵)) = (𝑥𝑈))
117 elinel1 3963 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ 𝒫 𝑈)
118 elpwi 4327 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ 𝒫 𝑈𝑥𝑈)
119117, 118syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥𝑈)
120 df-ss 3748 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝑈 ↔ (𝑥𝑈) = 𝑥)
121120biimpi 207 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑈 → (𝑥𝑈) = 𝑥)
122119, 121syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝑈) = 𝑥)
123113, 116, 1223eqtrrd 2804 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 = ((𝑥𝐴) ∪ (𝑥𝐵)))
124123adantl 473 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑥 = ((𝑥𝐴) ∪ (𝑥𝐵)))
125 elinel2 3964 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ Fin)
126125adantl 473 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑥 ∈ Fin)
127 rge0ssre 12491 . . . . . . . . . . . . . . . . . . . . 21 (0[,)+∞) ⊆ ℝ
1288ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → 𝐹:𝑈⟶(0[,]+∞))
129 pm4.56 1011 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((¬ +∞ ∈ ran (𝐹𝐴) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ↔ ¬ (+∞ ∈ ran (𝐹𝐴) ∨ +∞ ∈ ran (𝐹𝐵)))
130129biimpi 207 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((¬ +∞ ∈ ran (𝐹𝐴) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ¬ (+∞ ∈ ran (𝐹𝐴) ∨ +∞ ∈ ran (𝐹𝐵)))
131 elun 3917 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (+∞ ∈ (ran (𝐹𝐴) ∪ ran (𝐹𝐵)) ↔ (+∞ ∈ ran (𝐹𝐴) ∨ +∞ ∈ ran (𝐹𝐵)))
132130, 131sylnibr 320 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((¬ +∞ ∈ ran (𝐹𝐴) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ¬ +∞ ∈ (ran (𝐹𝐴) ∪ ran (𝐹𝐵)))
133132adantll 705 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ¬ +∞ ∈ (ran (𝐹𝐴) ∪ ran (𝐹𝐵)))
134 rnresun 40035 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ran (𝐹 ↾ (𝐴𝐵)) = (ran (𝐹𝐴) ∪ ran (𝐹𝐵))
135134eqcomi 2774 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (ran (𝐹𝐴) ∪ ran (𝐹𝐵)) = ran (𝐹 ↾ (𝐴𝐵))
136135a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (ran (𝐹𝐴) ∪ ran (𝐹𝐵)) = ran (𝐹 ↾ (𝐴𝐵)))
137114reseq2i 5564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹 ↾ (𝐴𝐵)) = (𝐹𝑈)
138137rneqi 5522 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ran (𝐹 ↾ (𝐴𝐵)) = ran (𝐹𝑈)
139138a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ran (𝐹 ↾ (𝐴𝐵)) = ran (𝐹𝑈))
140 ffn 6225 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹:𝑈⟶(0[,]+∞) → 𝐹 Fn 𝑈)
141 fnresdm 6180 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹 Fn 𝑈 → (𝐹𝑈) = 𝐹)
1428, 140, 1413syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝐹𝑈) = 𝐹)
143142rneqd 5523 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ran (𝐹𝑈) = ran 𝐹)
144136, 139, 1433eqtrd 2803 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (ran (𝐹𝐴) ∪ ran (𝐹𝐵)) = ran 𝐹)
145144ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (ran (𝐹𝐴) ∪ ran (𝐹𝐵)) = ran 𝐹)
146133, 145neleqtrd 2865 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ¬ +∞ ∈ ran 𝐹)
147128, 146fge0iccico 41250 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → 𝐹:𝑈⟶(0[,)+∞))
148147ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑈⟶(0[,)+∞))
149119adantr 472 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦𝑥) → 𝑥𝑈)
150 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑥)
151149, 150sseldd 3764 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑈)
152151adantll 705 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑈)
153148, 152ffvelrnd 6554 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,)+∞))
154127, 153sseldi 3761 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
155154recnd 10326 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℂ)
156110, 124, 126, 155fsumsplit 14772 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
157 infi 8395 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ Fin → (𝑥𝐴) ∈ Fin)
158125, 157syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐴) ∈ Fin)
159158adantl 473 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥𝐴) ∈ Fin)
160 simpl 474 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐴)) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)))
161 elinel1 3963 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (𝑥𝐴) → 𝑦𝑥)
162161adantl 473 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐴)) → 𝑦𝑥)
163160, 162, 154syl2anc 579 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐴)) → (𝐹𝑦) ∈ ℝ)
164159, 163fsumrecl 14766 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ∈ ℝ)
165 infi 8395 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ Fin → (𝑥𝐵) ∈ Fin)
166125, 165syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝐵) ∈ Fin)
167166adantl 473 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥𝐵) ∈ Fin)
168 simpl 474 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐵)) → (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)))
169 elinel1 3963 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (𝑥𝐵) → 𝑦𝑥)
170169adantl 473 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐵)) → 𝑦𝑥)
171168, 170, 154syl2anc 579 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐵)) → (𝐹𝑦) ∈ ℝ)
172167, 171fsumrecl 14766 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ∈ ℝ)
173 rexadd 12272 . . . . . . . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) +𝑒 Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
175174eqcomd 2771 . . . . . . . . . . . . . . . . . 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 2799 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) +𝑒 Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
177 ressxr 10341 . . . . . . . . . . . . . . . . . . . 20 ℝ ⊆ ℝ*
178177, 164sseldi 3761 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ∈ ℝ*)
179177, 172sseldi 3761 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ∈ ℝ*)
1801adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → 𝐴𝑉)
18133adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → (𝐹𝐴):𝐴⟶(0[,]+∞))
182 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → ¬ +∞ ∈ ran (𝐹𝐴))
183181, 182fge0iccico 41250 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → (𝐹𝐴):𝐴⟶(0[,)+∞))
184180, 183sge0reval 41252 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → (Σ^‘(𝐹𝐴)) = sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ))
185184eqcomd 2771 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) = (Σ^‘(𝐹𝐴)))
18634adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → (Σ^‘(𝐹𝐴)) ∈ ℝ*)
187185, 186eqeltrd 2844 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ*)
188187adantr 472 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ) ∈ ℝ*)
1893adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → 𝐵𝑊)
19039adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (𝐹𝐵):𝐵⟶(0[,]+∞))
191 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ¬ +∞ ∈ ran (𝐹𝐵))
192190, 191fge0iccico 41250 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (𝐹𝐵):𝐵⟶(0[,)+∞))
193189, 192sge0reval 41252 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (Σ^‘(𝐹𝐵)) = sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))
194193eqcomd 2771 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ) = (Σ^‘(𝐹𝐵)))
19541adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (Σ^‘(𝐹𝐵)) ∈ ℝ*)
196194, 195eqeltrd 2844 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*)
197196adantlr 706 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*)
198188, 197jca 507 . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . 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 510 . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝐴𝑉)
202181adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝐹𝐴):𝐴⟶(0[,]+∞))
203182adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ¬ +∞ ∈ ran (𝐹𝐴))
204202, 203fge0iccico 41250 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝐹𝐴):𝐴⟶(0[,)+∞))
205100a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥𝐴) ⊆ 𝐴)
206158adantl 473 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥𝐴) ∈ Fin)
207201, 204, 205, 206fsumlesge0 41257 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) ≤ (Σ^‘(𝐹𝐴)))
208100sseli 3759 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ (𝑥𝐴) → 𝑦𝐴)
209 fvres 6398 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦𝐴 → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
210208, 209syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (𝑥𝐴) → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
211210adantl 473 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐴)) → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
212211sumeq2dv 14734 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) = Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦))
213184adantr 472 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ^‘(𝐹𝐴)) = sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ))
214212, 213breq12d 4824 . . . . . . . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ≤ sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ))
216215adantlr 706 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ≤ sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ))
217189adantr 472 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝐵𝑊)
218190adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝐹𝐵):𝐵⟶(0[,]+∞))
219191adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ¬ +∞ ∈ ran (𝐹𝐵))
220218, 219fge0iccico 41250 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝐹𝐵):𝐵⟶(0[,)+∞))
221103a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥𝐵) ⊆ 𝐵)
222166adantl 473 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑥𝐵) ∈ Fin)
223217, 220, 221, 222fsumlesge0 41257 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)((𝐹𝐵)‘𝑦) ≤ (Σ^‘(𝐹𝐵)))
224103sseli 3759 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ (𝑥𝐵) → 𝑦𝐵)
225 fvres 6398 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦𝐵 → ((𝐹𝐵)‘𝑦) = (𝐹𝑦))
226224, 225syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (𝑥𝐵) → ((𝐹𝐵)‘𝑦) = (𝐹𝑦))
227226adantl 473 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐵)) → ((𝐹𝐵)‘𝑦) = (𝐹𝑦))
228227sumeq2dv 14734 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)((𝐹𝐵)‘𝑦) = Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦))
229193adantr 472 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ^‘(𝐹𝐵)) = sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))
230228, 229breq12d 4824 . . . . . . . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ≤ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))
232231adantllr 710 . . . . . . . . . . . . . . . . . . 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 507 . . . . . . . . . . . . . . . . . 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 12298 . . . . . . . . . . . . . . . . . 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 4833 . . . . . . . . . . . . . . . 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 1162 . . . . . . . . . . . . . . 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 4833 . . . . . . . . . . . . . 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 1148 . . . . . . . . . . . . 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 3177 . . . . . . . . . . . 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 472 . . . . . . . . . . 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 3113 . . . . . . . . 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 41244 . . . . . . . . . . 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 3773 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ*)
247188, 197xaddcld 12340 . . . . . . . . . 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 12365 . . . . . . . . . 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 579 . . . . . . . . 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 248 . . . . . . . 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 717 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → 𝑈 ∈ V)
252251, 147sge0reval 41252 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
253184adantr 472 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (Σ^‘(𝐹𝐴)) = sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ))
254193adantlr 706 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (Σ^‘(𝐹𝐵)) = sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))
255253, 254oveq12d 6864 . . . . . . . 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 4843 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
25791, 256pm2.61dan 847 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
25869, 257pm2.61dan 847 . . . . 5 (𝜑 → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
259258adantr 472 . . . 4 ((𝜑 ∧ (Σ^𝐹) = +∞) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
260 pnfge 12171 . . . . . . 7 (((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ∈ ℝ* → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ≤ +∞)
26142, 260syl 17 . . . . . 6 (𝜑 → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ≤ +∞)
262261adantr 472 . . . . 5 ((𝜑 ∧ (Σ^𝐹) = +∞) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ≤ +∞)
263 id 22 . . . . . . 7 ((Σ^𝐹) = +∞ → (Σ^𝐹) = +∞)
264263eqcomd 2771 . . . . . 6 ((Σ^𝐹) = +∞ → +∞ = (Σ^𝐹))
265264adantl 473 . . . . 5 ((𝜑 ∧ (Σ^𝐹) = +∞) → +∞ = (Σ^𝐹))
266262, 265breqtrd 4837 . . . 4 ((𝜑 ∧ (Σ^𝐹) = +∞) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ≤ (Σ^𝐹))
26729, 43, 259, 266xrletrid 12195 . . 3 ((𝜑 ∧ (Σ^𝐹) = +∞) → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
26822, 27, 267syl2anc 579 . 2 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
26921, 268pm2.61dan 847 1 (𝜑 → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  Vcvv 3350  cun 3732  cin 3733  wss 3734  c0 4081  𝒫 cpw 4317   class class class wbr 4811  cmpt 4890  ran crn 5280  cres 5281   Fn wfn 6065  wf 6066  cfv 6070  (class class class)co 6846  Fincfn 8164  supcsup 8557  cr 10192  0cc0 10193   + caddc 10196  +∞cpnf 10329  -∞cmnf 10330  *cxr 10331   < clt 10332  cle 10333   +𝑒 cxad 12151  [,)cico 12386  [,]cicc 12387  Σcsu 14717  Σ^csumge0 41242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-oadd 7772  df-er 7951  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-sup 8559  df-oi 8626  df-card 9020  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10527  df-neg 10528  df-div 10944  df-nn 11280  df-2 11340  df-3 11341  df-n0 11544  df-z 11630  df-uz 11894  df-rp 12036  df-xadd 12154  df-ico 12390  df-icc 12391  df-fz 12541  df-fzo 12681  df-seq 13016  df-exp 13075  df-hash 13329  df-cj 14140  df-re 14141  df-im 14142  df-sqrt 14276  df-abs 14277  df-clim 14520  df-sum 14718  df-sumge0 41243
This theorem is referenced by:  sge0splitmpt  41291
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