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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  cr 10119  0cc0 10120   + caddc 10123  +∞cpnf 10255  -∞cmnf 10256  *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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