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Theorem mbfi1flimlem 25758
Description: Lemma for mbfi1flim 25759. (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 7103 . . . 4 ((𝜑𝑦 ∈ ℝ) → (𝐹𝑦) ∈ ℝ)
31feqmptd 6976 . . . . 5 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
4 mbfi1flim.1 . . . . 5 (𝜑𝐹 ∈ MblFn)
53, 4eqeltrrd 2841 . . . 4 (𝜑 → (𝑦 ∈ ℝ ↦ (𝐹𝑦)) ∈ MblFn)
62, 5mbfpos 25687 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)) ∈ MblFn)
7 0re 11264 . . . . . 6 0 ∈ ℝ
8 ifcl 4570 . . . . . 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 13228 . . . . . 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 13495 . . . . 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 7134 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)):ℝ⟶(0[,)+∞))
156, 14mbfi1fseq 25757 . 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 11691 . . . 4 ((𝜑𝑦 ∈ ℝ) → -(𝐹𝑦) ∈ ℝ)
172, 5mbfneg 25686 . . . 4 (𝜑 → (𝑦 ∈ ℝ ↦ -(𝐹𝑦)) ∈ MblFn)
1816, 17mbfpos 25687 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)) ∈ MblFn)
19 ifcl 4570 . . . . . 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 13228 . . . . . 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 13495 . . . . 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 7134 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)):ℝ⟶(0[,)+∞))
2618, 25mbfi1fseq 25757 . 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 1954 . . 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 656 . . . . . 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 218 . . . . 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 3110 . . . . . . 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 25744 . . . . . . . . . 10 ((𝑥 ∈ dom ∫1𝑦 ∈ dom ∫1) → (𝑥f𝑦) ∈ dom ∫1)
3534adantl 481 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ (𝑥 ∈ dom ∫1𝑦 ∈ dom ∫1)) → (𝑥f𝑦) ∈ dom ∫1)
36 simprl 770 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → 𝑓:ℕ⟶dom ∫1)
37 simprr 772 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → :ℕ⟶dom ∫1)
38 nnex 12273 . . . . . . . . . 10 ℕ ∈ V
3938a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → ℕ ∈ V)
40 inidm 4226 . . . . . . . . 9 (ℕ ∩ ℕ) = ℕ
4135, 36, 37, 39, 39, 40off 7716 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → (𝑓ff):ℕ⟶dom ∫1)
42 fveq2 6905 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
4342breq2d 5154 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (0 ≤ (𝐹𝑦) ↔ 0 ≤ (𝐹𝑥)))
4443, 42ifbieq1d 4549 . . . . . . . . . . . . . 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 6918 . . . . . . . . . . . . . . 15 (𝐹𝑥) ∈ V
47 c0ex 11256 . . . . . . . . . . . . . . 15 0 ∈ V
4846, 47ifex 4575 . . . . . . . . . . . . . 14 if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∈ V
4944, 45, 48fvmpt 7015 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) = if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))
5049breq2d 5154 . . . . . . . . . . . 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 11503 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → -(𝐹𝑦) = -(𝐹𝑥))
5251breq2d 5154 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (0 ≤ -(𝐹𝑦) ↔ 0 ≤ -(𝐹𝑥)))
5352, 51ifbieq1d 4549 . . . . . . . . . . . . . 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 11507 . . . . . . . . . . . . . . 15 -(𝐹𝑥) ∈ V
5655, 47ifex 4575 . . . . . . . . . . . . . 14 if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0) ∈ V
5753, 54, 56fvmpt 7015 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥) = if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))
5857breq2d 5154 . . . . . . . . . . . 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 632 . . . . . . . . . . 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 481 . . . . . . . . . 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 12922 . . . . . . . . . . . . 13 ℕ = (ℤ‘1)
62 1zzd 12650 . . . . . . . . . . . . 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 770 . . . . . . . . . . . . 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 7244 . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . 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 7103 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ dom ∫1)
68 i1ff 25712 . . . . . . . . . . . . . . . . . . . 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 7103 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑓𝑛)‘𝑥) ∈ ℝ)
7170an32s 652 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑓𝑛)‘𝑥) ∈ ℝ)
7271recnd 11290 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑓𝑛)‘𝑥) ∈ ℂ)
7372fmpttd 7134 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)):ℕ⟶ℂ)
7473adantr 480 . . . . . . . . . . . . . 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 7103 . . . . . . . . . . . . 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 7103 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑛) ∈ dom ∫1)
77 i1ff 25712 . . . . . . . . . . . . . . . . . . . 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 7103 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛)‘𝑥) ∈ ℝ)
8079an32s 652 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛)‘𝑥) ∈ ℝ)
8180recnd 11290 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛)‘𝑥) ∈ ℂ)
8281fmpttd 7134 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)):ℕ⟶ℂ)
8382adantr 480 . . . . . . . . . . . . . 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 7103 . . . . . . . . . . . . 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 6736 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → 𝑓 Fn ℕ)
8637ffnd 6736 . . . . . . . . . . . . . . . . . . . 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 7709 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → ((𝑓ff)‘𝑘) = ((𝑓𝑘) ∘f − (𝑘)))
9089fveq1d 6907 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (((𝑓ff)‘𝑘)‘𝑥) = (((𝑓𝑘) ∘f − (𝑘))‘𝑥))
9190adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓ff)‘𝑘)‘𝑥) = (((𝑓𝑘) ∘f − (𝑘))‘𝑥))
9236ffvelcdmda 7103 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ dom ∫1)
93 i1ff 25712 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑘) ∈ dom ∫1 → (𝑓𝑘):ℝ⟶ℝ)
94 ffn 6735 . . . . . . . . . . . . . . . . . . 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 7103 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑘) ∈ dom ∫1)
97 i1ff 25712 . . . . . . . . . . . . . . . . . . 19 ((𝑘) ∈ dom ∫1 → (𝑘):ℝ⟶ℝ)
98 ffn 6735 . . . . . . . . . . . . . . . . . . 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 11247 . . . . . . . . . . . . . . . . . . 19 ℝ ∈ V
101100a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → ℝ ∈ V)
102 inidm 4226 . . . . . . . . . . . . . . . . . 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 7709 . . . . . . . . . . . . . . . . 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 652 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (((𝑓ff)‘𝑘)‘𝑥) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
108 fveq2 6905 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝑓ff)‘𝑛) = ((𝑓ff)‘𝑘))
109108fveq1d 6907 . . . . . . . . . . . . . . . . 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 6918 . . . . . . . . . . . . . . . . 17 (((𝑓ff)‘𝑘)‘𝑥) ∈ V
112109, 110, 111fvmpt 7015 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥))‘𝑘) = (((𝑓ff)‘𝑘)‘𝑥))
113112adantl 481 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥))‘𝑘) = (((𝑓ff)‘𝑘)‘𝑥))
114 fveq2 6905 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝑓𝑛) = (𝑓𝑘))
115114fveq1d 6907 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝑓𝑛)‘𝑥) = ((𝑓𝑘)‘𝑥))
116 eqid 2736 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))
117 fvex 6918 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑘)‘𝑥) ∈ V
118115, 116, 117fvmpt 7015 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))‘𝑘) = ((𝑓𝑘)‘𝑥))
119 fveq2 6905 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝑛) = (𝑘))
120119fveq1d 6907 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝑛)‘𝑥) = ((𝑘)‘𝑥))
121 eqid 2736 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))
122 fvex 6918 . . . . . . . . . . . . . . . . . 18 ((𝑘)‘𝑥) ∈ V
123120, 121, 122fvmpt 7015 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))‘𝑘) = ((𝑘)‘𝑥))
124118, 123oveq12d 7450 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))‘𝑘) − ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))‘𝑘)) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
125124adantl 481 . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . 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 15671 . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → 𝐹:ℝ⟶ℝ)
130129ffvelcdmda 7103 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
131 max0sub 13239 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 5168 . . . . . . . . . . 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 412 . . . . . . . . . 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 240 . . . . . . . . 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 3166 . . . . . . . 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 7465 . . . . . . . . 9 (𝑓ff) ∈ V
139 feq1 6715 . . . . . . . . . 10 (𝑔 = (𝑓ff) → (𝑔:ℕ⟶dom ∫1 ↔ (𝑓ff):ℕ⟶dom ∫1))
140 fveq1 6904 . . . . . . . . . . . . . 14 (𝑔 = (𝑓ff) → (𝑔𝑛) = ((𝑓ff)‘𝑛))
141140fveq1d 6907 . . . . . . . . . . . . 13 (𝑔 = (𝑓ff) → ((𝑔𝑛)‘𝑥) = (((𝑓ff)‘𝑛)‘𝑥))
142141mpteq2dv 5243 . . . . . . . . . . . 12 (𝑔 = (𝑓ff) → (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)))
143142breq1d 5152 . . . . . . . . . . 11 (𝑔 = (𝑓ff) → ((𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
144143ralbidv 3177 . . . . . . . . . 10 (𝑔 = (𝑓ff) → (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
145139, 144anbi12d 632 . . . . . . . . 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 3605 . . . . . . . 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 684 . . . . . . 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 243 . . . . . 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 453 . . . . 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 1933 . . 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 243 . 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 699 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 206  wa 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  wral 3060  Vcvv 3479  ifcif 4524   class class class wbr 5142  cmpt 5224  dom cdm 5684   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  f cof 7696  r cofr 7697  cc 11154  cr 11155  0cc0 11156  1c1 11157   + caddc 11159  +∞cpnf 11293  cle 11297  cmin 11493  -cneg 11494  cn 12267  [,)cico 13390  cli 15521  MblFncmbf 25650  1citg1 25651  0𝑝c0p 25705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-ofr 7699  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fi 9452  df-sup 9483  df-inf 9484  df-oi 9551  df-dju 9942  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-q 12992  df-rp 13036  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-ioo 13392  df-ico 13394  df-icc 13395  df-fz 13549  df-fzo 13696  df-fl 13833  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-clim 15525  df-rlim 15526  df-sum 15724  df-rest 17468  df-topgen 17489  df-psmet 21357  df-xmet 21358  df-met 21359  df-bl 21360  df-mopn 21361  df-top 22901  df-topon 22918  df-bases 22954  df-cmp 23396  df-ovol 25500  df-vol 25501  df-mbf 25655  df-itg1 25656  df-0p 25706
This theorem is referenced by:  mbfi1flim  25759
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