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Theorem sge0ltfirp 45415
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 45403 . . . . 5 (𝜑 → ¬ +∞ ∈ ran 𝐹)
51, 4fge0iccico 45385 . . . 4 (𝜑𝐹:𝑋⟶(0[,)+∞))
65sge0rnre 45379 . . 3 (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ)
7 sge0rnn0 45383 . . . 4 ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ≠ ∅
87a1i 11 . . 3 (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ≠ ∅)
92, 1, 3sge0rnbnd 45408 . . 3 (𝜑 → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))𝑤𝑧)
10 sge0ltfirp.y . . 3 (𝜑𝑌 ∈ ℝ+)
116, 8, 9, 10suprltrp 44337 . 2 (𝜑 → ∃𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤)
12 nfv 1916 . . 3 𝑤𝜑
13 nfv 1916 . . 3 𝑤𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌)
14 simp1 1135 . . . . 5 ((𝜑𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤) → 𝜑)
15 vex 3477 . . . . . . . . . 10 𝑤 ∈ V
16 eqid 2731 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
1716elrnmpt 5955 . . . . . . . . . 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 480 . . . . . . 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 5256 . . . . . . . . . . . . 13 𝑥(𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
2221nfrn 5951 . . . . . . . . . . . 12 𝑥ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
23 nfcv 2902 . . . . . . . . . . . 12 𝑥
24 nfcv 2902 . . . . . . . . . . . 12 𝑥 <
2522, 23, 24nfsup 9450 . . . . . . . . . . 11 𝑥sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < )
26 nfcv 2902 . . . . . . . . . . 11 𝑥
27 nfcv 2902 . . . . . . . . . . 11 𝑥𝑌
2825, 26, 27nfov 7442 . . . . . . . . . 10 𝑥(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌)
29 nfcv 2902 . . . . . . . . . 10 𝑥𝑤
3028, 24, 29nfbr 5195 . . . . . . . . 9 𝑥(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤
31 simpl 482 . . . . . . . . . . . 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 484 . . . . . . . . . . . 12 (((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤𝑤 = Σ𝑦𝑥 (𝐹𝑦)) → 𝑤 = Σ𝑦𝑥 (𝐹𝑦))
3331, 32breqtrd 5174 . . . . . . . . . . 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 412 . . . . . . . . . 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 3257 . . . . . . . 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 481 . . . . . . 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 1129 . . . . 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 482 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)))
412, 1, 3sge0supre 45404 . . . . . . . . . . . . 13 (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ))
4241oveq1d 7427 . . . . . . . . . . . 12 (𝜑 → ((Σ^𝐹) − 𝑌) = (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌))
4342adantr 480 . . . . . . . . . . 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 484 . . . . . . . . . . 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 5170 . . . . . . . . . 10 ((𝜑 ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
4645adantlr 712 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
47 simpr 484 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
483adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^𝐹) ∈ ℝ)
4910rpred 13021 . . . . . . . . . . . . . 14 (𝜑𝑌 ∈ ℝ)
5049adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑌 ∈ ℝ)
51 elinel2 4196 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
5251adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
53 rge0ssre 13438 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ ℝ
545adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,)+∞))
5554adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑋⟶(0[,)+∞))
56 elpwinss 44038 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
5756adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
5857sselda 3982 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
5955, 58ffvelcdmd 7087 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,)+∞))
6053, 59sselid 3980 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
6152, 60fsumrecl 15685 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ)
6248, 50, 61ltsubaddd 11815 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦) ↔ (Σ^𝐹) < (Σ𝑦𝑥 (𝐹𝑦) + 𝑌)))
6362adantr 480 . . . . . . . . . . 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 6758 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,)+∞))
6652, 65sge0fsum 45402 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) = Σ𝑦𝑥 ((𝐹𝑥)‘𝑦))
67 fvres 6910 . . . . . . . . . . . . . . 15 (𝑦𝑥 → ((𝐹𝑥)‘𝑦) = (𝐹𝑦))
6867sumeq2i 15650 . . . . . . . . . . . . . 14 Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦)
6968a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦))
7066, 69eqtr2d 2772 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (Σ^‘(𝐹𝑥)))
7170oveq1d 7427 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ𝑦𝑥 (𝐹𝑦) + 𝑌) = ((Σ^‘(𝐹𝑥)) + 𝑌))
7271adantr 480 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ𝑦𝑥 (𝐹𝑦) + 𝑌) = ((Σ^‘(𝐹𝑥)) + 𝑌))
7364, 72breqtrd 5174 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))
7440, 46, 73syl2anc 583 . . . . . . . 8 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))
7574ex 412 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦) → (Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌)))
7675reximdva 3167 . . . . . 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 406 . . . . 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 583 . . . 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 1118 . . 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 3262 . 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 395  w3a 1086   = wceq 1540  wcel 2105  wne 2939  wrex 3069  Vcvv 3473  cin 3947  wss 3948  c0 4322  𝒫 cpw 4602   class class class wbr 5148  cmpt 5231  ran crn 5677  cres 5678  wf 6539  cfv 6543  (class class class)co 7412  Fincfn 8943  supcsup 9439  cr 11113  0cc0 11114   + caddc 11117  +∞cpnf 11250   < clt 11253  cmin 11449  +crp 12979  [,)cico 13331  [,]cicc 13332  Σcsu 15637  Σ^csumge0 45377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9640  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9441  df-oi 9509  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-n0 12478  df-z 12564  df-uz 12828  df-rp 12980  df-ico 13335  df-icc 13336  df-fz 13490  df-fzo 13633  df-seq 13972  df-exp 14033  df-hash 14296  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  df-clim 15437  df-sum 15638  df-sumge0 45378
This theorem is referenced by:  sge0ltfirpmpt  45423  sge0ltfirpmpt2  45441
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