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Theorem mbfi1flimlem 25087
Description: Lemma for mbfi1flim 25088. (Contributed by Mario Carneiro, 5-Sep-2014.)
Hypotheses
Ref Expression
mbfi1flim.1 (𝜑𝐹 ∈ MblFn)
mbfi1flimlem.2 (𝜑𝐹:ℝ⟶ℝ)
Assertion
Ref Expression
mbfi1flimlem (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
Distinct variable groups:   𝑔,𝑛,𝑥,𝐹   𝜑,𝑔,𝑛,𝑥

Proof of Theorem mbfi1flimlem
Dummy variables 𝑦 𝑓 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mbfi1flimlem.2 . . . . 5 (𝜑𝐹:ℝ⟶ℝ)
21ffvelcdmda 7035 . . . 4 ((𝜑𝑦 ∈ ℝ) → (𝐹𝑦) ∈ ℝ)
31feqmptd 6910 . . . . 5 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
4 mbfi1flim.1 . . . . 5 (𝜑𝐹 ∈ MblFn)
53, 4eqeltrrd 2839 . . . 4 (𝜑 → (𝑦 ∈ ℝ ↦ (𝐹𝑦)) ∈ MblFn)
62, 5mbfpos 25015 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)) ∈ MblFn)
7 0re 11157 . . . . . 6 0 ∈ ℝ
8 ifcl 4531 . . . . . 6 (((𝐹𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ ℝ)
92, 7, 8sylancl 586 . . . . 5 ((𝜑𝑦 ∈ ℝ) → if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ ℝ)
10 max1 13104 . . . . . 6 ((0 ∈ ℝ ∧ (𝐹𝑦) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))
117, 2, 10sylancr 587 . . . . 5 ((𝜑𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))
12 elrege0 13371 . . . . 5 (if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ (0[,)+∞) ↔ (if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)))
139, 11, 12sylanbrc 583 . . . 4 ((𝜑𝑦 ∈ ℝ) → if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ (0[,)+∞))
1413fmpttd 7063 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)):ℝ⟶(0[,)+∞))
156, 14mbfi1fseq 25086 . 2 (𝜑 → ∃𝑓(𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑓𝑛) ∧ (𝑓𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥)))
162renegcld 11582 . . . 4 ((𝜑𝑦 ∈ ℝ) → -(𝐹𝑦) ∈ ℝ)
172, 5mbfneg 25014 . . . 4 (𝜑 → (𝑦 ∈ ℝ ↦ -(𝐹𝑦)) ∈ MblFn)
1816, 17mbfpos 25015 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)) ∈ MblFn)
19 ifcl 4531 . . . . . 6 ((-(𝐹𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ ℝ)
2016, 7, 19sylancl 586 . . . . 5 ((𝜑𝑦 ∈ ℝ) → if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ ℝ)
21 max1 13104 . . . . . 6 ((0 ∈ ℝ ∧ -(𝐹𝑦) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))
227, 16, 21sylancr 587 . . . . 5 ((𝜑𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))
23 elrege0 13371 . . . . 5 (if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ (0[,)+∞) ↔ (if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)))
2420, 22, 23sylanbrc 583 . . . 4 ((𝜑𝑦 ∈ ℝ) → if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ (0[,)+∞))
2524fmpttd 7063 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)):ℝ⟶(0[,)+∞))
2618, 25mbfi1fseq 25086 . 2 (𝜑 → ∃(:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑛) ∧ (𝑛) ∘r ≤ (‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥)))
27 exdistrv 1959 . . 3 (∃𝑓((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑓𝑛) ∧ (𝑓𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥)) ∧ (:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑛) ∧ (𝑛) ∘r ≤ (‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥))) ↔ (∃𝑓(𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑓𝑛) ∧ (𝑓𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥)) ∧ ∃(:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑛) ∧ (𝑛) ∘r ≤ (‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥))))
28 3simpb 1149 . . . . . . 7 ((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑓𝑛) ∧ (𝑓𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥)) → (𝑓:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥)))
29 3simpb 1149 . . . . . . 7 ((:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑛) ∧ (𝑛) ∘r ≤ (‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥)) → (:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥)))
3028, 29anim12i 613 . . . . . 6 (((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑓𝑛) ∧ (𝑓𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥)) ∧ (:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑛) ∧ (𝑛) ∘r ≤ (‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥))) → ((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥)) ∧ (:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥))))
31 an4 654 . . . . . 6 (((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥)) ∧ (:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥))) ↔ ((𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1) ∧ (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥))))
3230, 31sylib 217 . . . . 5 (((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑓𝑛) ∧ (𝑓𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥)) ∧ (:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑛) ∧ (𝑛) ∘r ≤ (‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥))) → ((𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1) ∧ (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥))))
33 r19.26 3114 . . . . . . 7 (∀𝑥 ∈ ℝ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥)) ↔ (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥)))
34 i1fsub 25073 . . . . . . . . . 10 ((𝑥 ∈ dom ∫1𝑦 ∈ dom ∫1) → (𝑥f𝑦) ∈ dom ∫1)
3534adantl 482 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ (𝑥 ∈ dom ∫1𝑦 ∈ dom ∫1)) → (𝑥f𝑦) ∈ dom ∫1)
36 simprl 769 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → 𝑓:ℕ⟶dom ∫1)
37 simprr 771 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → :ℕ⟶dom ∫1)
38 nnex 12159 . . . . . . . . . 10 ℕ ∈ V
3938a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → ℕ ∈ V)
40 inidm 4178 . . . . . . . . 9 (ℕ ∩ ℕ) = ℕ
4135, 36, 37, 39, 39, 40off 7635 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → (𝑓ff):ℕ⟶dom ∫1)
42 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
4342breq2d 5117 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (0 ≤ (𝐹𝑦) ↔ 0 ≤ (𝐹𝑥)))
4443, 42ifbieq1d 4510 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) = if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))
45 eqid 2736 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)) = (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))
46 fvex 6855 . . . . . . . . . . . . . . 15 (𝐹𝑥) ∈ V
47 c0ex 11149 . . . . . . . . . . . . . . 15 0 ∈ V
4846, 47ifex 4536 . . . . . . . . . . . . . 14 if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∈ V
4944, 45, 48fvmpt 6948 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) = if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))
5049breq2d 5117 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)))
5142negeqd 11395 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → -(𝐹𝑦) = -(𝐹𝑥))
5251breq2d 5117 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (0 ≤ -(𝐹𝑦) ↔ 0 ≤ -(𝐹𝑥)))
5352, 51ifbieq1d 4510 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) = if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))
54 eqid 2736 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)) = (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))
55 negex 11399 . . . . . . . . . . . . . . 15 -(𝐹𝑥) ∈ V
5655, 47ifex 4536 . . . . . . . . . . . . . 14 if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0) ∈ V
5753, 54, 56fvmpt 6948 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥) = if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))
5857breq2d 5117 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)))
5950, 58anbi12d 631 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥)) ↔ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))))
6059adantl 482 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥)) ↔ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))))
61 nnuz 12806 . . . . . . . . . . . . 13 ℕ = (ℤ‘1)
62 1zzd 12534 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) → 1 ∈ ℤ)
63 simprl 769 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) → (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))
6438mptex 7173 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) ∈ V
6564a1i 11 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) → (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) ∈ V)
66 simprr 771 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) → (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))
6736ffvelcdmda 7035 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ dom ∫1)
68 i1ff 25040 . . . . . . . . . . . . . . . . . . . 20 ((𝑓𝑛) ∈ dom ∫1 → (𝑓𝑛):ℝ⟶ℝ)
6967, 68syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛):ℝ⟶ℝ)
7069ffvelcdmda 7035 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑓𝑛)‘𝑥) ∈ ℝ)
7170an32s 650 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑓𝑛)‘𝑥) ∈ ℝ)
7271recnd 11183 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑓𝑛)‘𝑥) ∈ ℂ)
7372fmpttd 7063 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)):ℕ⟶ℂ)
7473adantr 481 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) → (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)):ℕ⟶ℂ)
7574ffvelcdmda 7035 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))‘𝑘) ∈ ℂ)
7637ffvelcdmda 7035 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑛) ∈ dom ∫1)
77 i1ff 25040 . . . . . . . . . . . . . . . . . . . 20 ((𝑛) ∈ dom ∫1 → (𝑛):ℝ⟶ℝ)
7876, 77syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑛):ℝ⟶ℝ)
7978ffvelcdmda 7035 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛)‘𝑥) ∈ ℝ)
8079an32s 650 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛)‘𝑥) ∈ ℝ)
8180recnd 11183 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛)‘𝑥) ∈ ℂ)
8281fmpttd 7063 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)):ℕ⟶ℂ)
8382adantr 481 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) → (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)):ℕ⟶ℂ)
8483ffvelcdmda 7035 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))‘𝑘) ∈ ℂ)
8536ffnd 6669 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → 𝑓 Fn ℕ)
8637ffnd 6669 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → Fn ℕ)
87 eqidd 2737 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) = (𝑓𝑘))
88 eqidd 2737 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑘) = (𝑘))
8985, 86, 39, 39, 40, 87, 88ofval 7628 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → ((𝑓ff)‘𝑘) = ((𝑓𝑘) ∘f − (𝑘)))
9089fveq1d 6844 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (((𝑓ff)‘𝑘)‘𝑥) = (((𝑓𝑘) ∘f − (𝑘))‘𝑥))
9190adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓ff)‘𝑘)‘𝑥) = (((𝑓𝑘) ∘f − (𝑘))‘𝑥))
9236ffvelcdmda 7035 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ dom ∫1)
93 i1ff 25040 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑘) ∈ dom ∫1 → (𝑓𝑘):ℝ⟶ℝ)
94 ffn 6668 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑘):ℝ⟶ℝ → (𝑓𝑘) Fn ℝ)
9592, 93, 943syl 18 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) Fn ℝ)
9637ffvelcdmda 7035 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑘) ∈ dom ∫1)
97 i1ff 25040 . . . . . . . . . . . . . . . . . . 19 ((𝑘) ∈ dom ∫1 → (𝑘):ℝ⟶ℝ)
98 ffn 6668 . . . . . . . . . . . . . . . . . . 19 ((𝑘):ℝ⟶ℝ → (𝑘) Fn ℝ)
9996, 97, 983syl 18 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑘) Fn ℝ)
100 reex 11142 . . . . . . . . . . . . . . . . . . 19 ℝ ∈ V
101100a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → ℝ ∈ V)
102 inidm 4178 . . . . . . . . . . . . . . . . . 18 (ℝ ∩ ℝ) = ℝ
103 eqidd 2737 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑓𝑘)‘𝑥) = ((𝑓𝑘)‘𝑥))
104 eqidd 2737 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑘)‘𝑥) = ((𝑘)‘𝑥))
10595, 99, 101, 101, 102, 103, 104ofval 7628 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓𝑘) ∘f − (𝑘))‘𝑥) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
10691, 105eqtrd 2776 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓ff)‘𝑘)‘𝑥) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
107106an32s 650 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (((𝑓ff)‘𝑘)‘𝑥) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
108 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝑓ff)‘𝑛) = ((𝑓ff)‘𝑘))
109108fveq1d 6844 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (((𝑓ff)‘𝑛)‘𝑥) = (((𝑓ff)‘𝑘)‘𝑥))
110 eqid 2736 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥))
111 fvex 6855 . . . . . . . . . . . . . . . . 17 (((𝑓ff)‘𝑘)‘𝑥) ∈ V
112109, 110, 111fvmpt 6948 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥))‘𝑘) = (((𝑓ff)‘𝑘)‘𝑥))
113112adantl 482 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥))‘𝑘) = (((𝑓ff)‘𝑘)‘𝑥))
114 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝑓𝑛) = (𝑓𝑘))
115114fveq1d 6844 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝑓𝑛)‘𝑥) = ((𝑓𝑘)‘𝑥))
116 eqid 2736 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))
117 fvex 6855 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑘)‘𝑥) ∈ V
118115, 116, 117fvmpt 6948 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))‘𝑘) = ((𝑓𝑘)‘𝑥))
119 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝑛) = (𝑘))
120119fveq1d 6844 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝑛)‘𝑥) = ((𝑘)‘𝑥))
121 eqid 2736 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))
122 fvex 6855 . . . . . . . . . . . . . . . . . 18 ((𝑘)‘𝑥) ∈ V
123120, 121, 122fvmpt 6948 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))‘𝑘) = ((𝑘)‘𝑥))
124118, 123oveq12d 7375 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))‘𝑘) − ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))‘𝑘)) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
125124adantl 482 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))‘𝑘) − ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))‘𝑘)) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
126107, 113, 1253eqtr4d 2786 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥))‘𝑘) = (((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))‘𝑘) − ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))‘𝑘)))
127126adantlr 713 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥))‘𝑘) = (((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))‘𝑘) − ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))‘𝑘)))
12861, 62, 63, 65, 66, 75, 84, 127climsub 15516 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) → (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) ⇝ (if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) − if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)))
1291adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → 𝐹:ℝ⟶ℝ)
130129ffvelcdmda 7035 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
131 max0sub 13115 . . . . . . . . . . . . . 14 ((𝐹𝑥) ∈ ℝ → (if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) − if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) = (𝐹𝑥))
132130, 131syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) − if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) = (𝐹𝑥))
133132adantr 481 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) → (if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) − if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) = (𝐹𝑥))
134128, 133breqtrd 5131 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))) → (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥))
135134ex 413 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) → (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
13660, 135sylbid 239 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥)) → (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
137136ralimdva 3164 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → (∀𝑥 ∈ ℝ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥)) → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
138 ovex 7390 . . . . . . . . 9 (𝑓ff) ∈ V
139 feq1 6649 . . . . . . . . . 10 (𝑔 = (𝑓ff) → (𝑔:ℕ⟶dom ∫1 ↔ (𝑓ff):ℕ⟶dom ∫1))
140 fveq1 6841 . . . . . . . . . . . . . 14 (𝑔 = (𝑓ff) → (𝑔𝑛) = ((𝑓ff)‘𝑛))
141140fveq1d 6844 . . . . . . . . . . . . 13 (𝑔 = (𝑓ff) → ((𝑔𝑛)‘𝑥) = (((𝑓ff)‘𝑛)‘𝑥))
142141mpteq2dv 5207 . . . . . . . . . . . 12 (𝑔 = (𝑓ff) → (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)))
143142breq1d 5115 . . . . . . . . . . 11 (𝑔 = (𝑓ff) → ((𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
144143ralbidv 3174 . . . . . . . . . 10 (𝑔 = (𝑓ff) → (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
145139, 144anbi12d 631 . . . . . . . . 9 (𝑔 = (𝑓ff) → ((𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)) ↔ ((𝑓ff):ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥))))
146138, 145spcev 3565 . . . . . . . 8 (((𝑓ff):ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥)) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
14741, 137, 146syl6an 682 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → (∀𝑥 ∈ ℝ ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) ∧ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥)) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥))))
14833, 147biimtrrid 242 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → ((∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥)) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥))))
149148expimpd 454 . . . . 5 (𝜑 → (((𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1) ∧ (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥))) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥))))
15032, 149syl5 34 . . . 4 (𝜑 → (((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑓𝑛) ∧ (𝑓𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥)) ∧ (:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑛) ∧ (𝑛) ∘r ≤ (‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥))) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥))))
151150exlimdvv 1937 . . 3 (𝜑 → (∃𝑓((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑓𝑛) ∧ (𝑓𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥)) ∧ (:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑛) ∧ (𝑛) ∘r ≤ (‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥))) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥))))
15227, 151biimtrrid 242 . 2 (𝜑 → ((∃𝑓(𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑓𝑛) ∧ (𝑓𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥)) ∧ ∃(:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑛) ∧ (𝑛) ∘r ≤ (‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥))) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥))))
15315, 26, 152mp2and 697 1 (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wral 3064  Vcvv 3445  ifcif 4486   class class class wbr 5105  cmpt 5188  dom cdm 5633   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  r cofr 7616  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054  +∞cpnf 11186  cle 11190  cmin 11385  -cneg 11386  cn 12153  [,)cico 13266  cli 15366  MblFncmbf 24978  1citg1 24979  0𝑝c0p 25033
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cmp 22738  df-ovol 24828  df-vol 24829  df-mbf 24983  df-itg1 24984  df-0p 25034
This theorem is referenced by:  mbfi1flim  25088
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