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Theorem sge0ltfirp 42145
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 42133 . . . . 5 (𝜑 → ¬ +∞ ∈ ran 𝐹)
51, 4fge0iccico 42115 . . . 4 (𝜑𝐹:𝑋⟶(0[,)+∞))
65sge0rnre 42109 . . 3 (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ)
7 sge0rnn0 42113 . . . 4 ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ≠ ∅
87a1i 11 . . 3 (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ≠ ∅)
92, 1, 3sge0rnbnd 42138 . . 3 (𝜑 → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))𝑤𝑧)
10 sge0ltfirp.y . . 3 (𝜑𝑌 ∈ ℝ+)
116, 8, 9, 10suprltrp 41057 . 2 (𝜑 → ∃𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤)
12 nfv 1874 . . 3 𝑤𝜑
13 nfv 1874 . . 3 𝑤𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌)
14 simp1 1117 . . . . 5 ((𝜑𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤) → 𝜑)
15 vex 3411 . . . . . . . . . 10 𝑤 ∈ V
16 eqid 2771 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
1716elrnmpt 5668 . . . . . . . . . 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 208 . . . . . . . 8 (𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑤 = Σ𝑦𝑥 (𝐹𝑦))
2019adantr 473 . . . . . . 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 5021 . . . . . . . . . . . . 13 𝑥(𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
2221nfrn 5664 . . . . . . . . . . . 12 𝑥ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
23 nfcv 2925 . . . . . . . . . . . 12 𝑥
24 nfcv 2925 . . . . . . . . . . . 12 𝑥 <
2522, 23, 24nfsup 8708 . . . . . . . . . . 11 𝑥sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < )
26 nfcv 2925 . . . . . . . . . . 11 𝑥
27 nfcv 2925 . . . . . . . . . . 11 𝑥𝑌
2825, 26, 27nfov 7004 . . . . . . . . . 10 𝑥(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌)
29 nfcv 2925 . . . . . . . . . 10 𝑥𝑤
3028, 24, 29nfbr 4972 . . . . . . . . 9 𝑥(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤
31 simpl 475 . . . . . . . . . . . 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 477 . . . . . . . . . . . 12 (((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤𝑤 = Σ𝑦𝑥 (𝐹𝑦)) → 𝑤 = Σ𝑦𝑥 (𝐹𝑦))
3331, 32breqtrd 4951 . . . . . . . . . . 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 405 . . . . . . . . . 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 3247 . . . . . . . 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 474 . . . . . . 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 1111 . . . . 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 475 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)))
412, 1, 3sge0supre 42134 . . . . . . . . . . . . 13 (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ))
4241oveq1d 6989 . . . . . . . . . . . 12 (𝜑 → ((Σ^𝐹) − 𝑌) = (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌))
4342adantr 473 . . . . . . . . . . 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 477 . . . . . . . . . . 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 4947 . . . . . . . . . 10 ((𝜑 ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
4645adantlr 703 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
47 simpr 477 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
483adantr 473 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^𝐹) ∈ ℝ)
4910rpred 12246 . . . . . . . . . . . . . 14 (𝜑𝑌 ∈ ℝ)
5049adantr 473 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑌 ∈ ℝ)
51 elinel2 4055 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
5251adantl 474 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
53 rge0ssre 12658 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ ℝ
545adantr 473 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,)+∞))
5554adantr 473 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑋⟶(0[,)+∞))
56 elpwinss 40767 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
5756adantl 474 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
5857sselda 3851 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
5955, 58ffvelrnd 6675 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,)+∞))
6053, 59sseldi 3849 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
6152, 60fsumrecl 14949 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ)
6248, 50, 61ltsubaddd 11035 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦) ↔ (Σ^𝐹) < (Σ𝑦𝑥 (𝐹𝑦) + 𝑌)))
6362adantr 473 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦) ↔ (Σ^𝐹) < (Σ𝑦𝑥 (𝐹𝑦) + 𝑌)))
6447, 63mpbid 224 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ^𝐹) < (Σ𝑦𝑥 (𝐹𝑦) + 𝑌))
6554, 57fssresd 6371 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,)+∞))
6652, 65sge0fsum 42132 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) = Σ𝑦𝑥 ((𝐹𝑥)‘𝑦))
67 fvres 6515 . . . . . . . . . . . . . . 15 (𝑦𝑥 → ((𝐹𝑥)‘𝑦) = (𝐹𝑦))
6867sumeq2i 14914 . . . . . . . . . . . . . 14 Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦)
6968a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦))
7066, 69eqtr2d 2808 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (Σ^‘(𝐹𝑥)))
7170oveq1d 6989 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ𝑦𝑥 (𝐹𝑦) + 𝑌) = ((Σ^‘(𝐹𝑥)) + 𝑌))
7271adantr 473 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ𝑦𝑥 (𝐹𝑦) + 𝑌) = ((Σ^‘(𝐹𝑥)) + 𝑌))
7364, 72breqtrd 4951 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))
7440, 46, 73syl2anc 576 . . . . . . . 8 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))
7574ex 405 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦) → (Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌)))
7675reximdva 3212 . . . . . 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 398 . . . . 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 576 . . . 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 1100 . . 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 3253 . 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 198  wa 387  w3a 1069   = wceq 1508  wcel 2051  wne 2960  wrex 3082  Vcvv 3408  cin 3821  wss 3822  c0 4172  𝒫 cpw 4416   class class class wbr 4925  cmpt 5004  ran crn 5404  cres 5405  wf 6181  cfv 6185  (class class class)co 6974  Fincfn 8304  supcsup 8697  cr 10332  0cc0 10333   + caddc 10336  +∞cpnf 10469   < clt 10472  cmin 10668  +crp 12202  [,)cico 12554  [,]cicc 12555  Σcsu 14901  Σ^csumge0 42107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-inf2 8896  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410  ax-pre-sup 10411
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-nel 3067  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-se 5363  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-isom 6194  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-oadd 7907  df-er 8087  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-sup 8699  df-oi 8767  df-card 9160  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-div 11097  df-nn 11438  df-2 11501  df-3 11502  df-n0 11706  df-z 11792  df-uz 12057  df-rp 12203  df-ico 12558  df-icc 12559  df-fz 12707  df-fzo 12848  df-seq 13183  df-exp 13243  df-hash 13504  df-cj 14317  df-re 14318  df-im 14319  df-sqrt 14453  df-abs 14454  df-clim 14704  df-sum 14902  df-sumge0 42108
This theorem is referenced by:  sge0ltfirpmpt  42153  sge0ltfirpmpt2  42171
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