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Theorem sge0ltfirp 44727
Description: If the sum of nonnegative extended reals is real, then it can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0ltfirp.x (𝜑𝑋𝑉)
sge0ltfirp.f (𝜑𝐹:𝑋⟶(0[,]+∞))
sge0ltfirp.y (𝜑𝑌 ∈ ℝ+)
sge0ltfirp.re (𝜑 → (Σ^𝐹) ∈ ℝ)
Assertion
Ref Expression
sge0ltfirp (𝜑 → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑋   𝑥,𝑌   𝜑,𝑥
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem sge0ltfirp
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sge0ltfirp.f . . . . 5 (𝜑𝐹:𝑋⟶(0[,]+∞))
2 sge0ltfirp.x . . . . . 6 (𝜑𝑋𝑉)
3 sge0ltfirp.re . . . . . 6 (𝜑 → (Σ^𝐹) ∈ ℝ)
42, 1, 3sge0rern 44715 . . . . 5 (𝜑 → ¬ +∞ ∈ ran 𝐹)
51, 4fge0iccico 44697 . . . 4 (𝜑𝐹:𝑋⟶(0[,)+∞))
65sge0rnre 44691 . . 3 (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ)
7 sge0rnn0 44695 . . . 4 ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ≠ ∅
87a1i 11 . . 3 (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ≠ ∅)
92, 1, 3sge0rnbnd 44720 . . 3 (𝜑 → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))𝑤𝑧)
10 sge0ltfirp.y . . 3 (𝜑𝑌 ∈ ℝ+)
116, 8, 9, 10suprltrp 43649 . 2 (𝜑 → ∃𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤)
12 nfv 1918 . . 3 𝑤𝜑
13 nfv 1918 . . 3 𝑤𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌)
14 simp1 1137 . . . . 5 ((𝜑𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤) → 𝜑)
15 vex 3448 . . . . . . . . . 10 𝑤 ∈ V
16 eqid 2733 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
1716elrnmpt 5912 . . . . . . . . . 10 (𝑤 ∈ V → (𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑤 = Σ𝑦𝑥 (𝐹𝑦)))
1815, 17ax-mp 5 . . . . . . . . 9 (𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑤 = Σ𝑦𝑥 (𝐹𝑦))
1918biimpi 215 . . . . . . . 8 (𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑤 = Σ𝑦𝑥 (𝐹𝑦))
2019adantr 482 . . . . . . 7 ((𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑤 = Σ𝑦𝑥 (𝐹𝑦))
21 nfmpt1 5214 . . . . . . . . . . . . 13 𝑥(𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
2221nfrn 5908 . . . . . . . . . . . 12 𝑥ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
23 nfcv 2904 . . . . . . . . . . . 12 𝑥
24 nfcv 2904 . . . . . . . . . . . 12 𝑥 <
2522, 23, 24nfsup 9392 . . . . . . . . . . 11 𝑥sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < )
26 nfcv 2904 . . . . . . . . . . 11 𝑥
27 nfcv 2904 . . . . . . . . . . 11 𝑥𝑌
2825, 26, 27nfov 7388 . . . . . . . . . 10 𝑥(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌)
29 nfcv 2904 . . . . . . . . . 10 𝑥𝑤
3028, 24, 29nfbr 5153 . . . . . . . . 9 𝑥(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤
31 simpl 484 . . . . . . . . . . . 12 (((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤𝑤 = Σ𝑦𝑥 (𝐹𝑦)) → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤)
32 simpr 486 . . . . . . . . . . . 12 (((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤𝑤 = Σ𝑦𝑥 (𝐹𝑦)) → 𝑤 = Σ𝑦𝑥 (𝐹𝑦))
3331, 32breqtrd 5132 . . . . . . . . . . 11 (((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤𝑤 = Σ𝑦𝑥 (𝐹𝑦)) → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
3433ex 414 . . . . . . . . . 10 ((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤 → (𝑤 = Σ𝑦𝑥 (𝐹𝑦) → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)))
3534a1d 25 . . . . . . . . 9 ((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤 → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → (𝑤 = Σ𝑦𝑥 (𝐹𝑦) → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))))
3630, 35reximdai 3243 . . . . . . . 8 ((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤 → (∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑤 = Σ𝑦𝑥 (𝐹𝑦) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)))
3736adantl 483 . . . . . . 7 ((𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤) → (∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑤 = Σ𝑦𝑥 (𝐹𝑦) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)))
3820, 37mpd 15 . . . . . 6 ((𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
39383adant1 1131 . . . . 5 ((𝜑𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
40 simpl 484 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)))
412, 1, 3sge0supre 44716 . . . . . . . . . . . . 13 (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ))
4241oveq1d 7373 . . . . . . . . . . . 12 (𝜑 → ((Σ^𝐹) − 𝑌) = (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌))
4342adantr 482 . . . . . . . . . . 11 ((𝜑 ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → ((Σ^𝐹) − 𝑌) = (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌))
44 simpr 486 . . . . . . . . . . 11 ((𝜑 ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
4543, 44eqbrtrd 5128 . . . . . . . . . 10 ((𝜑 ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
4645adantlr 714 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
47 simpr 486 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
483adantr 482 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^𝐹) ∈ ℝ)
4910rpred 12962 . . . . . . . . . . . . . 14 (𝜑𝑌 ∈ ℝ)
5049adantr 482 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑌 ∈ ℝ)
51 elinel2 4157 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
5251adantl 483 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
53 rge0ssre 13379 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ ℝ
545adantr 482 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,)+∞))
5554adantr 482 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑋⟶(0[,)+∞))
56 elpwinss 43345 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
5756adantl 483 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
5857sselda 3945 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
5955, 58ffvelcdmd 7037 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,)+∞))
6053, 59sselid 3943 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
6152, 60fsumrecl 15624 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ)
6248, 50, 61ltsubaddd 11756 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦) ↔ (Σ^𝐹) < (Σ𝑦𝑥 (𝐹𝑦) + 𝑌)))
6362adantr 482 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦) ↔ (Σ^𝐹) < (Σ𝑦𝑥 (𝐹𝑦) + 𝑌)))
6447, 63mpbid 231 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ^𝐹) < (Σ𝑦𝑥 (𝐹𝑦) + 𝑌))
6554, 57fssresd 6710 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,)+∞))
6652, 65sge0fsum 44714 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) = Σ𝑦𝑥 ((𝐹𝑥)‘𝑦))
67 fvres 6862 . . . . . . . . . . . . . . 15 (𝑦𝑥 → ((𝐹𝑥)‘𝑦) = (𝐹𝑦))
6867sumeq2i 15589 . . . . . . . . . . . . . 14 Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦)
6968a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦))
7066, 69eqtr2d 2774 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (Σ^‘(𝐹𝑥)))
7170oveq1d 7373 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ𝑦𝑥 (𝐹𝑦) + 𝑌) = ((Σ^‘(𝐹𝑥)) + 𝑌))
7271adantr 482 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ𝑦𝑥 (𝐹𝑦) + 𝑌) = ((Σ^‘(𝐹𝑥)) + 𝑌))
7364, 72breqtrd 5132 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))
7440, 46, 73syl2anc 585 . . . . . . . 8 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))
7574ex 414 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦) → (Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌)))
7675reximdva 3162 . . . . . 6 (𝜑 → (∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌)))
7776imp 408 . . . . 5 ((𝜑 ∧ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))
7814, 39, 77syl2anc 585 . . . 4 ((𝜑𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))
79783exp 1120 . . 3 (𝜑 → (𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) → ((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤 → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))))
8012, 13, 79rexlimd 3248 . 2 (𝜑 → (∃𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤 → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌)))
8111, 80mpd 15 1 (𝜑 → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wrex 3070  Vcvv 3444  cin 3910  wss 3911  c0 4283  𝒫 cpw 4561   class class class wbr 5106  cmpt 5189  ran crn 5635  cres 5636  wf 6493  cfv 6497  (class class class)co 7358  Fincfn 8886  supcsup 9381  cr 11055  0cc0 11056   + caddc 11059  +∞cpnf 11191   < clt 11194  cmin 11390  +crp 12920  [,)cico 13272  [,]cicc 13273  Σcsu 15576  Σ^csumge0 44689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-z 12505  df-uz 12769  df-rp 12921  df-ico 13276  df-icc 13277  df-fz 13431  df-fzo 13574  df-seq 13913  df-exp 13974  df-hash 14237  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-clim 15376  df-sum 15577  df-sumge0 44690
This theorem is referenced by:  sge0ltfirpmpt  44735  sge0ltfirpmpt2  44753
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