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Theorem sge0split 41121
 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 479 . . . 4 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (Σ^𝐹) ∈ ℝ)
112, 4, 5, 7, 9, 10sge0resplit 41118 . . 3 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))
12 unexg 7116 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
131, 3, 12syl2anc 696 . . . . . . . 8 (𝜑 → (𝐴𝐵) ∈ V)
145, 13syl5eqel 2835 . . . . . . 7 (𝜑𝑈 ∈ V)
1514adantr 472 . . . . . 6 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → 𝑈 ∈ V)
1615, 9, 10sge0ssre 41109 . . . . 5 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (Σ^‘(𝐹𝐴)) ∈ ℝ)
1715, 9, 10sge0ssre 41109 . . . . 5 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (Σ^‘(𝐹𝐵)) ∈ ℝ)
18 rexadd 12248 . . . . 5 (((Σ^‘(𝐹𝐴)) ∈ ℝ ∧ (Σ^‘(𝐹𝐵)) ∈ ℝ) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))
1916, 17, 18syl2anc 696 . . . 4 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))
2019eqcomd 2758 . . 3 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
2111, 20eqtrd 2786 . 2 ((𝜑 ∧ (Σ^𝐹) ∈ ℝ) → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
22 simpl 474 . . 3 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → 𝜑)
23 simpr 479 . . . . 5 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → ¬ (Σ^𝐹) ∈ ℝ)
2414, 8sge0repnf 41098 . . . . . 6 (𝜑 → ((Σ^𝐹) ∈ ℝ ↔ ¬ (Σ^𝐹) = +∞))
2524adantr 472 . . . . 5 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → ((Σ^𝐹) ∈ ℝ ↔ ¬ (Σ^𝐹) = +∞))
2623, 25mtbid 313 . . . 4 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → ¬ ¬ (Σ^𝐹) = +∞)
2726notnotrd 128 . . 3 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → (Σ^𝐹) = +∞)
2814, 8sge0xrcl 41097 . . . . 5 (𝜑 → (Σ^𝐹) ∈ ℝ*)
2928adantr 472 . . . 4 ((𝜑 ∧ (Σ^𝐹) = +∞) → (Σ^𝐹) ∈ ℝ*)
30 ssun1 3911 . . . . . . . . . 10 𝐴 ⊆ (𝐴𝐵)
3130, 5sseqtr4i 3771 . . . . . . . . 9 𝐴𝑈
3231a1i 11 . . . . . . . 8 (𝜑𝐴𝑈)
338, 32fssresd 6224 . . . . . . 7 (𝜑 → (𝐹𝐴):𝐴⟶(0[,]+∞))
341, 33sge0xrcl 41097 . . . . . 6 (𝜑 → (Σ^‘(𝐹𝐴)) ∈ ℝ*)
35 iccssxr 12441 . . . . . . 7 (0[,]+∞) ⊆ ℝ*
36 ssun2 3912 . . . . . . . . . . 11 𝐵 ⊆ (𝐴𝐵)
3736, 5sseqtr4i 3771 . . . . . . . . . 10 𝐵𝑈
3837a1i 11 . . . . . . . . 9 (𝜑𝐵𝑈)
398, 38fssresd 6224 . . . . . . . 8 (𝜑 → (𝐹𝐵):𝐵⟶(0[,]+∞))
403, 39sge0cl 41093 . . . . . . 7 (𝜑 → (Σ^‘(𝐹𝐵)) ∈ (0[,]+∞))
4135, 40sseldi 3734 . . . . . 6 (𝜑 → (Σ^‘(𝐹𝐵)) ∈ ℝ*)
4234, 41xaddcld 12316 . . . . 5 (𝜑 → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ∈ ℝ*)
4342adantr 472 . . . 4 ((𝜑 ∧ (Σ^𝐹) = +∞) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ∈ ℝ*)
44 pnfxr 10276 . . . . . . . . 9 +∞ ∈ ℝ*
45 eqid 2752 . . . . . . . . 9 +∞ = +∞
46 xreqle 40024 . . . . . . . . 9 ((+∞ ∈ ℝ* ∧ +∞ = +∞) → +∞ ≤ +∞)
4744, 45, 46mp2an 710 . . . . . . . 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 39856 . . . . . . . . . . 11 ran (𝐹𝐴) ⊆ ran 𝐹
5251sseli 3732 . . . . . . . . . 10 (+∞ ∈ ran (𝐹𝐴) → +∞ ∈ ran 𝐹)
5352adantl 473 . . . . . . . . 9 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → +∞ ∈ ran 𝐹)
5449, 50, 53sge0pnfval 41085 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → (Σ^𝐹) = +∞)
55 ge0nemnf2 40250 . . . . . . . . . . . . . 14 ((Σ^‘(𝐹𝐵)) ∈ (0[,]+∞) → (Σ^‘(𝐹𝐵)) ≠ -∞)
5640, 55syl 17 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝐹𝐵)) ≠ -∞)
57 xaddpnf2 12243 . . . . . . . . . . . . 13 (((Σ^‘(𝐹𝐵)) ∈ ℝ* ∧ (Σ^‘(𝐹𝐵)) ≠ -∞) → (+∞ +𝑒^‘(𝐹𝐵))) = +∞)
5841, 56, 57syl2anc 696 . . . . . . . . . . . 12 (𝜑 → (+∞ +𝑒^‘(𝐹𝐵))) = +∞)
5958eqcomd 2758 . . . . . . . . . . 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 479 . . . . . . . . . . . 12 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → +∞ ∈ ran (𝐹𝐴))
6461, 62, 63sge0pnfval 41085 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → (Σ^‘(𝐹𝐴)) = +∞)
6564oveq1d 6820 . . . . . . . . . 10 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = (+∞ +𝑒^‘(𝐹𝐵))))
6660, 54, 653eqtr4d 2796 . . . . . . . . 9 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
6766, 54eqtr3d 2788 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = +∞)
6854, 67breq12d 4809 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐴)) → ((Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ↔ +∞ ≤ +∞))
6948, 68mpbird 247 . . . . . 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 39856 . . . . . . . . . . . . 13 ran (𝐹𝐵) ⊆ ran 𝐹
7473sseli 3732 . . . . . . . . . . . 12 (+∞ ∈ ran (𝐹𝐵) → +∞ ∈ ran 𝐹)
7574adantl 473 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → +∞ ∈ ran 𝐹)
7671, 72, 75sge0pnfval 41085 . . . . . . . . . 10 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → (Σ^𝐹) = +∞)
773adantr 472 . . . . . . . . . . . . 13 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → 𝐵𝑊)
7839adantr 472 . . . . . . . . . . . . 13 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → (𝐹𝐵):𝐵⟶(0[,]+∞))
79 simpr 479 . . . . . . . . . . . . 13 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → +∞ ∈ ran (𝐹𝐵))
8077, 78, 79sge0pnfval 41085 . . . . . . . . . . . 12 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → (Σ^‘(𝐹𝐵)) = +∞)
8180oveq2d 6821 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = ((Σ^‘(𝐹𝐴)) +𝑒 +∞))
821, 33sge0cl 41093 . . . . . . . . . . . . . 14 (𝜑 → (Σ^‘(𝐹𝐴)) ∈ (0[,]+∞))
83 ge0nemnf2 40250 . . . . . . . . . . . . . 14 ((Σ^‘(𝐹𝐴)) ∈ (0[,]+∞) → (Σ^‘(𝐹𝐴)) ≠ -∞)
8482, 83syl 17 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝐹𝐴)) ≠ -∞)
85 xaddpnf1 12242 . . . . . . . . . . . . 13 (((Σ^‘(𝐹𝐴)) ∈ ℝ* ∧ (Σ^‘(𝐹𝐴)) ≠ -∞) → ((Σ^‘(𝐹𝐴)) +𝑒 +∞) = +∞)
8634, 84, 85syl2anc 696 . . . . . . . . . . . 12 (𝜑 → ((Σ^‘(𝐹𝐴)) +𝑒 +∞) = +∞)
8786adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → ((Σ^‘(𝐹𝐴)) +𝑒 +∞) = +∞)
8881, 87eqtrd 2786 . . . . . . . . . 10 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) = +∞)
8976, 88breq12d 4809 . . . . . . . . 9 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → ((Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ↔ +∞ ≤ +∞))
9070, 89mpbird 247 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ran (𝐹𝐵)) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
9190adantlr 753 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ +∞ ∈ ran (𝐹𝐵)) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
92 simpr 479 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
93 vex 3335 . . . . . . . . . . . . 13 𝑧 ∈ V
94 eqid 2752 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
9594elrnmpt 5519 . . . . . . . . . . . . 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 208 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑧 ∈ ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → ∃𝑥 ∈ (𝒫 𝑈 ∩ Fin)𝑧 = Σ𝑦𝑥 (𝐹𝑦))
98 simp3 1132 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑧 = Σ𝑦𝑥 (𝐹𝑦)) → 𝑧 = Σ𝑦𝑥 (𝐹𝑦))
99 inss1 3968 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ (𝑥𝐴)
100 inss2 3969 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐴) ⊆ 𝐴
10199, 100sstri 3745 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ 𝐴
102 inss2 3969 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ (𝑥𝐵)
103 inss2 3969 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐵) ⊆ 𝐵
104102, 103sstri 3745 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ 𝐵
105101, 104ssini 3971 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ (𝐴𝐵)
106105a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ (𝐴𝐵))
107106, 6sseqtrd 3774 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ ∅)
108 ss0 4109 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥𝐴) ∩ (𝑥𝐵)) ⊆ ∅ → ((𝑥𝐴) ∩ (𝑥𝐵)) = ∅)
109107, 108syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑥𝐴) ∩ (𝑥𝐵)) = ∅)
110109ad3antrrr 768 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → ((𝑥𝐴) ∩ (𝑥𝐵)) = ∅)
111 indi 4008 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∩ (𝐴𝐵)) = ((𝑥𝐴) ∪ (𝑥𝐵))
112111eqcomi 2761 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝐴) ∪ (𝑥𝐵)) = (𝑥 ∩ (𝐴𝐵))
113112a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → ((𝑥𝐴) ∪ (𝑥𝐵)) = (𝑥 ∩ (𝐴𝐵)))
1145eqcomi 2761 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴𝐵) = 𝑈
115114ineq2i 3946 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∩ (𝐴𝐵)) = (𝑥𝑈)
116115a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥 ∩ (𝐴𝐵)) = (𝑥𝑈))
117 elinel1 3934 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ 𝒫 𝑈)
118 elpwi 4304 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ 𝒫 𝑈𝑥𝑈)
119117, 118syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥𝑈)
120 df-ss 3721 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝑈 ↔ (𝑥𝑈) = 𝑥)
121120biimpi 206 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑈 → (𝑥𝑈) = 𝑥)
122119, 121syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → (𝑥𝑈) = 𝑥)
123113, 116, 1223eqtrrd 2791 . . . . . . . . . . . . . . . . . . . 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 3935 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝑈 ∩ Fin) → 𝑥 ∈ Fin)
126125adantl 473 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑥 ∈ Fin)
127 rge0ssre 12465 . . . . . . . . . . . . . . . . . . . . 21 (0[,)+∞) ⊆ ℝ
1288ad2antrr 764 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → 𝐹:𝑈⟶(0[,]+∞))
129 pm4.56 517 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((¬ +∞ ∈ ran (𝐹𝐴) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ↔ ¬ (+∞ ∈ ran (𝐹𝐴) ∨ +∞ ∈ ran (𝐹𝐵)))
130129biimpi 206 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((¬ +∞ ∈ ran (𝐹𝐴) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ¬ (+∞ ∈ ran (𝐹𝐴) ∨ +∞ ∈ ran (𝐹𝐵)))
131 elun 3888 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (+∞ ∈ (ran (𝐹𝐴) ∪ ran (𝐹𝐵)) ↔ (+∞ ∈ ran (𝐹𝐴) ∨ +∞ ∈ ran (𝐹𝐵)))
132130, 131sylnibr 318 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((¬ +∞ ∈ ran (𝐹𝐴) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ¬ +∞ ∈ (ran (𝐹𝐴) ∪ ran (𝐹𝐵)))
133132adantll 752 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ¬ +∞ ∈ (ran (𝐹𝐴) ∪ ran (𝐹𝐵)))
134 rnresun 39853 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ran (𝐹 ↾ (𝐴𝐵)) = (ran (𝐹𝐴) ∪ ran (𝐹𝐵))
135134eqcomi 2761 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (ran (𝐹𝐴) ∪ ran (𝐹𝐵)) = ran (𝐹 ↾ (𝐴𝐵))
136135a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (ran (𝐹𝐴) ∪ ran (𝐹𝐵)) = ran (𝐹 ↾ (𝐴𝐵)))
137114reseq2i 5540 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹 ↾ (𝐴𝐵)) = (𝐹𝑈)
138137rneqi 5499 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ran (𝐹 ↾ (𝐴𝐵)) = ran (𝐹𝑈)
139138a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ran (𝐹 ↾ (𝐴𝐵)) = ran (𝐹𝑈))
140 ffn 6198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹:𝑈⟶(0[,]+∞) → 𝐹 Fn 𝑈)
141 fnresdm 6153 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹 Fn 𝑈 → (𝐹𝑈) = 𝐹)
1428, 140, 1413syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝐹𝑈) = 𝐹)
143142rneqd 5500 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ran (𝐹𝑈) = ran 𝐹)
144136, 139, 1433eqtrd 2790 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (ran (𝐹𝐴) ∪ ran (𝐹𝐵)) = ran 𝐹)
145144ad2antrr 764 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (ran (𝐹𝐴) ∪ ran (𝐹𝐵)) = ran 𝐹)
146133, 145neleqtrd 2852 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ¬ +∞ ∈ ran 𝐹)
147128, 146fge0iccico 41082 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → 𝐹:𝑈⟶(0[,)+∞))
148147ad2antrr 764 . . . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑥)
151149, 150sseldd 3737 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑈)
152151adantll 752 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑈)
153148, 152ffvelrnd 6515 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,)+∞))
154127, 153sseldi 3734 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
155154recnd 10252 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℂ)
156110, 124, 126, 155fsumsplit 14662 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
157 infi 8341 . . . . . . . . . . . . . . . . . . . . . . 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 3934 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (𝑥𝐴) → 𝑦𝑥)
162161adantl 473 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐴)) → 𝑦𝑥)
163160, 162, 154syl2anc 696 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐴)) → (𝐹𝑦) ∈ ℝ)
164159, 163fsumrecl 14656 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ∈ ℝ)
165 infi 8341 . . . . . . . . . . . . . . . . . . . . . . 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 3934 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (𝑥𝐵) → 𝑦𝑥)
170169adantl 473 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐵)) → 𝑦𝑥)
171168, 170, 154syl2anc 696 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) ∧ 𝑦 ∈ (𝑥𝐵)) → (𝐹𝑦) ∈ ℝ)
172167, 171fsumrecl 14656 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ∈ ℝ)
173 rexadd 12248 . . . . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) +𝑒 Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) + Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
175174eqcomd 2758 . . . . . . . . . . . . . . . . . 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 2786 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) +𝑒 Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦)))
177 ressxr 10267 . . . . . . . . . . . . . . . . . . . 20 ℝ ⊆ ℝ*
178177, 164sseldi 3734 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ∈ ℝ*)
179177, 172sseldi 3734 . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → ¬ +∞ ∈ ran (𝐹𝐴))
183181, 182fge0iccico 41082 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → (𝐹𝐴):𝐴⟶(0[,)+∞))
184180, 183sge0reval 41084 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → (Σ^‘(𝐹𝐴)) = sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ))
185184eqcomd 2758 . . . . . . . . . . . . . . . . . . . . . . 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 2831 . . . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ¬ +∞ ∈ ran (𝐹𝐵))
192190, 191fge0iccico 41082 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (𝐹𝐵):𝐵⟶(0[,)+∞))
193189, 192sge0reval 41084 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (Σ^‘(𝐹𝐵)) = sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))
194193eqcomd 2758 . . . . . . . . . . . . . . . . . . . . . . 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 2831 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*)
197196adantlr 753 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ) ∈ ℝ*)
198188, 197jca 555 . . . . . . . . . . . . . . . . . . . 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 558 . . . . . . . . . . . . . . . . . 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 41082 . . . . . . . . . . . . . . . . . . . . . 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 41089 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)((𝐹𝐴)‘𝑦) ≤ (Σ^‘(𝐹𝐴)))
208100sseli 3732 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ (𝑥𝐴) → 𝑦𝐴)
209 fvres 6360 . . . . . . . . . . . . . . . . . . . . . . . . 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 14624 . . . . . . . . . . . . . . . . . . . . . 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 4809 . . . . . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐴)(𝐹𝑦) ≤ sup(ran (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑏𝑎 ((𝐹𝐴)‘𝑏)), ℝ*, < ))
216215adantlr 753 . . . . . . . . . . . . . . . . . . 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 41082 . . . . . . . . . . . . . . . . . . . . . 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 41089 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)((𝐹𝐵)‘𝑦) ≤ (Σ^‘(𝐹𝐵)))
224103sseli 3732 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ (𝑥𝐵) → 𝑦𝐵)
225 fvres 6360 . . . . . . . . . . . . . . . . . . . . . . . . 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 14624 . . . . . . . . . . . . . . . . . . . . . 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 4809 . . . . . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐵)) ∧ 𝑥 ∈ (𝒫 𝑈 ∩ Fin)) → Σ𝑦 ∈ (𝑥𝐵)(𝐹𝑦) ≤ sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))
232231adantllr 757 . . . . . . . . . . . . . . . . . . 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 555 . . . . . . . . . . . . . . . . . 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 12274 . . . . . . . . . . . . . . . . . 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 4818 . . . . . . . . . . . . . . . 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 1126 . . . . . . . . . . . . . . 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 4818 . . . . . . . . . . . . . 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 1112 . . . . . . . . . . . . 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 3160 . . . . . . . . . . . 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 3096 . . . . . . . . 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 41076 . . . . . . . . . . 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 3746 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → ran (𝑥 ∈ (𝒫 𝑈 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ*)
247188, 197xaddcld 12316 . . . . . . . . . 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 12341 . . . . . . . . . 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 696 . . . . . . . . 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 247 . . . . . . . 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 764 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → 𝑈 ∈ V)
252251, 147sge0reval 41084 . . . . . . . 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 753 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (Σ^‘(𝐹𝐵)) = sup(ran (𝑐 ∈ (𝒫 𝐵 ∩ Fin) ↦ Σ𝑑𝑐 ((𝐹𝐵)‘𝑑)), ℝ*, < ))
255253, 254oveq12d 6823 . . . . . . . 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 4828 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) ∧ ¬ +∞ ∈ ran (𝐹𝐵)) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
25791, 256pm2.61dan 867 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran (𝐹𝐴)) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
25869, 257pm2.61dan 867 . . . . 5 (𝜑 → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
259258adantr 472 . . . 4 ((𝜑 ∧ (Σ^𝐹) = +∞) → (Σ^𝐹) ≤ ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
260 pnfge 12149 . . . . . . 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 2758 . . . . . 6 ((Σ^𝐹) = +∞ → +∞ = (Σ^𝐹))
265264adantl 473 . . . . 5 ((𝜑 ∧ (Σ^𝐹) = +∞) → +∞ = (Σ^𝐹))
266262, 265breqtrd 4822 . . . 4 ((𝜑 ∧ (Σ^𝐹) = +∞) → ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))) ≤ (Σ^𝐹))
26729, 43, 259, 266xrletrid 12171 . . 3 ((𝜑 ∧ (Σ^𝐹) = +∞) → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
26822, 27, 267syl2anc 696 . 2 ((𝜑 ∧ ¬ (Σ^𝐹) ∈ ℝ) → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
26921, 268pm2.61dan 867 1 (𝜑 → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1072   = wceq 1624   ∈ wcel 2131   ≠ wne 2924  ∀wral 3042  ∃wrex 3043  Vcvv 3332   ∪ cun 3705   ∩ cin 3706   ⊆ wss 3707  ∅c0 4050  𝒫 cpw 4294   class class class wbr 4796   ↦ cmpt 4873  ran crn 5259   ↾ cres 5260   Fn wfn 6036  ⟶wf 6037  ‘cfv 6041  (class class class)co 6805  Fincfn 8113  supcsup 8503  ℝcr 10119  0cc0 10120   + caddc 10123  +∞cpnf 10255  -∞cmnf 10256  ℝ*cxr 10257   < clt 10258   ≤ cle 10259   +𝑒 cxad 12129  [,)cico 12362  [,]cicc 12363  Σcsu 14607  Σ^csumge0 41074 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197  ax-pre-sup 10198 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8505  df-oi 8572  df-card 8947  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-div 10869  df-nn 11205  df-2 11263  df-3 11264  df-n0 11477  df-z 11562  df-uz 11872  df-rp 12018  df-xadd 12132  df-ico 12366  df-icc 12367  df-fz 12512  df-fzo 12652  df-seq 12988  df-exp 13047  df-hash 13304  df-cj 14030  df-re 14031  df-im 14032  df-sqrt 14166  df-abs 14167  df-clim 14410  df-sum 14608  df-sumge0 41075 This theorem is referenced by:  sge0splitmpt  41123
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