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Theorem sge0ltfirp 42969
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 42957 . . . . 5 (𝜑 → ¬ +∞ ∈ ran 𝐹)
51, 4fge0iccico 42939 . . . 4 (𝜑𝐹:𝑋⟶(0[,)+∞))
65sge0rnre 42933 . . 3 (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ)
7 sge0rnn0 42937 . . . 4 ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ≠ ∅
87a1i 11 . . 3 (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ≠ ∅)
92, 1, 3sge0rnbnd 42962 . . 3 (𝜑 → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))𝑤𝑧)
10 sge0ltfirp.y . . 3 (𝜑𝑌 ∈ ℝ+)
116, 8, 9, 10suprltrp 41890 . 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 1133 . . . . 5 ((𝜑𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤) → 𝜑)
15 vex 3483 . . . . . . . . . 10 𝑤 ∈ V
16 eqid 2824 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
1716elrnmpt 5815 . . . . . . . . . 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 219 . . . . . . . 8 (𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑤 = Σ𝑦𝑥 (𝐹𝑦))
2019adantr 484 . . . . . . 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 5150 . . . . . . . . . . . . 13 𝑥(𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
2221nfrn 5811 . . . . . . . . . . . 12 𝑥ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
23 nfcv 2982 . . . . . . . . . . . 12 𝑥
24 nfcv 2982 . . . . . . . . . . . 12 𝑥 <
2522, 23, 24nfsup 8912 . . . . . . . . . . 11 𝑥sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < )
26 nfcv 2982 . . . . . . . . . . 11 𝑥
27 nfcv 2982 . . . . . . . . . . 11 𝑥𝑌
2825, 26, 27nfov 7179 . . . . . . . . . 10 𝑥(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌)
29 nfcv 2982 . . . . . . . . . 10 𝑥𝑤
3028, 24, 29nfbr 5099 . . . . . . . . 9 𝑥(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤
31 simpl 486 . . . . . . . . . . . 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 488 . . . . . . . . . . . 12 (((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤𝑤 = Σ𝑦𝑥 (𝐹𝑦)) → 𝑤 = Σ𝑦𝑥 (𝐹𝑦))
3331, 32breqtrd 5078 . . . . . . . . . . 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 416 . . . . . . . . . 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 3303 . . . . . . . 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 485 . . . . . . 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 1127 . . . . 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 486 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)))
412, 1, 3sge0supre 42958 . . . . . . . . . . . . 13 (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ))
4241oveq1d 7164 . . . . . . . . . . . 12 (𝜑 → ((Σ^𝐹) − 𝑌) = (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌))
4342adantr 484 . . . . . . . . . . 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 488 . . . . . . . . . . 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 5074 . . . . . . . . . 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 488 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
483adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^𝐹) ∈ ℝ)
4910rpred 12428 . . . . . . . . . . . . . 14 (𝜑𝑌 ∈ ℝ)
5049adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑌 ∈ ℝ)
51 elinel2 4158 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
5251adantl 485 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
53 rge0ssre 12843 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ ℝ
545adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,)+∞))
5554adantr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑋⟶(0[,)+∞))
56 elpwinss 41607 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
5756adantl 485 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
5857sselda 3953 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
5955, 58ffvelrnd 6843 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,)+∞))
6053, 59sseldi 3951 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
6152, 60fsumrecl 15091 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ)
6248, 50, 61ltsubaddd 11234 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦) ↔ (Σ^𝐹) < (Σ𝑦𝑥 (𝐹𝑦) + 𝑌)))
6362adantr 484 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦) ↔ (Σ^𝐹) < (Σ𝑦𝑥 (𝐹𝑦) + 𝑌)))
6447, 63mpbid 235 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ^𝐹) < (Σ𝑦𝑥 (𝐹𝑦) + 𝑌))
6554, 57fssresd 6535 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,)+∞))
6652, 65sge0fsum 42956 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) = Σ𝑦𝑥 ((𝐹𝑥)‘𝑦))
67 fvres 6680 . . . . . . . . . . . . . . 15 (𝑦𝑥 → ((𝐹𝑥)‘𝑦) = (𝐹𝑦))
6867sumeq2i 15056 . . . . . . . . . . . . . 14 Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦)
6968a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦))
7066, 69eqtr2d 2860 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (Σ^‘(𝐹𝑥)))
7170oveq1d 7164 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ𝑦𝑥 (𝐹𝑦) + 𝑌) = ((Σ^‘(𝐹𝑥)) + 𝑌))
7271adantr 484 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ𝑦𝑥 (𝐹𝑦) + 𝑌) = ((Σ^‘(𝐹𝑥)) + 𝑌))
7364, 72breqtrd 5078 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))
7440, 46, 73syl2anc 587 . . . . . . . 8 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))
7574ex 416 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦) → (Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌)))
7675reximdva 3266 . . . . . 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 410 . . . . 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 587 . . . 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 1116 . . 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 3309 . 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wrex 3134  Vcvv 3480  cin 3918  wss 3919  c0 4276  𝒫 cpw 4522   class class class wbr 5052  cmpt 5132  ran crn 5543  cres 5544  wf 6339  cfv 6343  (class class class)co 7149  Fincfn 8505  supcsup 8901  cr 10534  0cc0 10535   + caddc 10538  +∞cpnf 10670   < clt 10673  cmin 10868  +crp 12386  [,)cico 12737  [,]cicc 12738  Σcsu 15042  Σ^csumge0 42931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-rp 12387  df-ico 12741  df-icc 12742  df-fz 12895  df-fzo 13038  df-seq 13374  df-exp 13435  df-hash 13696  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-sum 15043  df-sumge0 42932
This theorem is referenced by:  sge0ltfirpmpt  42977  sge0ltfirpmpt2  42995
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