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Theorem sge0ltfirp 45926
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 45914 . . . . 5 (𝜑 → ¬ +∞ ∈ ran 𝐹)
51, 4fge0iccico 45896 . . . 4 (𝜑𝐹:𝑋⟶(0[,)+∞))
65sge0rnre 45890 . . 3 (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ)
7 sge0rnn0 45894 . . . 4 ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ≠ ∅
87a1i 11 . . 3 (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ≠ ∅)
92, 1, 3sge0rnbnd 45919 . . 3 (𝜑 → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))𝑤𝑧)
10 sge0ltfirp.y . . 3 (𝜑𝑌 ∈ ℝ+)
116, 8, 9, 10suprltrp 44848 . 2 (𝜑 → ∃𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤)
12 nfv 1909 . . 3 𝑤𝜑
13 nfv 1909 . . 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 3465 . . . . . . . . . 10 𝑤 ∈ V
16 eqid 2725 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
1716elrnmpt 5958 . . . . . . . . . 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 479 . . . . . . 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 5257 . . . . . . . . . . . . 13 𝑥(𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
2221nfrn 5954 . . . . . . . . . . . 12 𝑥ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
23 nfcv 2891 . . . . . . . . . . . 12 𝑥
24 nfcv 2891 . . . . . . . . . . . 12 𝑥 <
2522, 23, 24nfsup 9476 . . . . . . . . . . 11 𝑥sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < )
26 nfcv 2891 . . . . . . . . . . 11 𝑥
27 nfcv 2891 . . . . . . . . . . 11 𝑥𝑌
2825, 26, 27nfov 7449 . . . . . . . . . 10 𝑥(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌)
29 nfcv 2891 . . . . . . . . . 10 𝑥𝑤
3028, 24, 29nfbr 5196 . . . . . . . . 9 𝑥(sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤
31 simpl 481 . . . . . . . . . . . 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 483 . . . . . . . . . . . 12 (((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < 𝑤𝑤 = Σ𝑦𝑥 (𝐹𝑦)) → 𝑤 = Σ𝑦𝑥 (𝐹𝑦))
3331, 32breqtrd 5175 . . . . . . . . . . 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 411 . . . . . . . . . 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 3248 . . . . . . . 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 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) → ∃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 481 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)))
412, 1, 3sge0supre 45915 . . . . . . . . . . . . 13 (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ))
4241oveq1d 7434 . . . . . . . . . . . 12 (𝜑 → ((Σ^𝐹) − 𝑌) = (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌))
4342adantr 479 . . . . . . . . . . 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 483 . . . . . . . . . . 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 5171 . . . . . . . . . 10 ((𝜑 ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
4645adantlr 713 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
47 simpr 483 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦))
483adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^𝐹) ∈ ℝ)
4910rpred 13051 . . . . . . . . . . . . . 14 (𝜑𝑌 ∈ ℝ)
5049adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑌 ∈ ℝ)
51 elinel2 4194 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
5251adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
53 rge0ssre 13468 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ ℝ
545adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,)+∞))
5554adantr 479 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑋⟶(0[,)+∞))
56 elpwinss 44555 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
5756adantl 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
5857sselda 3976 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
5955, 58ffvelcdmd 7094 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,)+∞))
6053, 59sselid 3974 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
6152, 60fsumrecl 15716 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ)
6248, 50, 61ltsubaddd 11842 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦) ↔ (Σ^𝐹) < (Σ𝑦𝑥 (𝐹𝑦) + 𝑌)))
6362adantr 479 . . . . . . . . . . 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 6764 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,)+∞))
6652, 65sge0fsum 45913 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) = Σ𝑦𝑥 ((𝐹𝑥)‘𝑦))
67 fvres 6915 . . . . . . . . . . . . . . 15 (𝑦𝑥 → ((𝐹𝑥)‘𝑦) = (𝐹𝑦))
6867sumeq2i 15681 . . . . . . . . . . . . . 14 Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦)
6968a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦))
7066, 69eqtr2d 2766 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (Σ^‘(𝐹𝑥)))
7170oveq1d 7434 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ𝑦𝑥 (𝐹𝑦) + 𝑌) = ((Σ^‘(𝐹𝑥)) + 𝑌))
7271adantr 479 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ𝑦𝑥 (𝐹𝑦) + 𝑌) = ((Σ^‘(𝐹𝑥)) + 𝑌))
7364, 72breqtrd 5175 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ ((Σ^𝐹) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))
7440, 46, 73syl2anc 582 . . . . . . . 8 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦)) → (Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))
7574ex 411 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ) − 𝑌) < Σ𝑦𝑥 (𝐹𝑦) → (Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌)))
7675reximdva 3157 . . . . . 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 405 . . . . 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 582 . . . 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 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 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wrex 3059  Vcvv 3461  cin 3943  wss 3944  c0 4322  𝒫 cpw 4604   class class class wbr 5149  cmpt 5232  ran crn 5679  cres 5680  wf 6545  cfv 6549  (class class class)co 7419  Fincfn 8964  supcsup 9465  cr 11139  0cc0 11140   + caddc 11143  +∞cpnf 11277   < clt 11280  cmin 11476  +crp 13009  [,)cico 13361  [,]cicc 13362  Σcsu 15668  Σ^csumge0 45888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-z 12592  df-uz 12856  df-rp 13010  df-ico 13365  df-icc 13366  df-fz 13520  df-fzo 13663  df-seq 14003  df-exp 14063  df-hash 14326  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-clim 15468  df-sum 15669  df-sumge0 45889
This theorem is referenced by:  sge0ltfirpmpt  45934  sge0ltfirpmpt2  45952
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