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Theorem mbfi1flimlem 24422
 Description: Lemma for mbfi1flim 24423. (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)
21ffvelrnda 6842 . . . 4 ((𝜑𝑦 ∈ ℝ) → (𝐹𝑦) ∈ ℝ)
31feqmptd 6721 . . . . 5 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
4 mbfi1flim.1 . . . . 5 (𝜑𝐹 ∈ MblFn)
53, 4eqeltrrd 2853 . . . 4 (𝜑 → (𝑦 ∈ ℝ ↦ (𝐹𝑦)) ∈ MblFn)
62, 5mbfpos 24351 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)) ∈ MblFn)
7 0re 10681 . . . . . 6 0 ∈ ℝ
8 ifcl 4465 . . . . . 6 (((𝐹𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ ℝ)
92, 7, 8sylancl 589 . . . . 5 ((𝜑𝑦 ∈ ℝ) → if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ ℝ)
10 max1 12619 . . . . . 6 ((0 ∈ ℝ ∧ (𝐹𝑦) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))
117, 2, 10sylancr 590 . . . . 5 ((𝜑𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))
12 elrege0 12886 . . . . 5 (if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ (0[,)+∞) ↔ (if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)))
139, 11, 12sylanbrc 586 . . . 4 ((𝜑𝑦 ∈ ℝ) → if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ (0[,)+∞))
1413fmpttd 6870 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)):ℝ⟶(0[,)+∞))
156, 14mbfi1fseq 24421 . 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 11105 . . . 4 ((𝜑𝑦 ∈ ℝ) → -(𝐹𝑦) ∈ ℝ)
172, 5mbfneg 24350 . . . 4 (𝜑 → (𝑦 ∈ ℝ ↦ -(𝐹𝑦)) ∈ MblFn)
1816, 17mbfpos 24351 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)) ∈ MblFn)
19 ifcl 4465 . . . . . 6 ((-(𝐹𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ ℝ)
2016, 7, 19sylancl 589 . . . . 5 ((𝜑𝑦 ∈ ℝ) → if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ ℝ)
21 max1 12619 . . . . . 6 ((0 ∈ ℝ ∧ -(𝐹𝑦) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))
227, 16, 21sylancr 590 . . . . 5 ((𝜑𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))
23 elrege0 12886 . . . . 5 (if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ (0[,)+∞) ↔ (if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)))
2420, 22, 23sylanbrc 586 . . . 4 ((𝜑𝑦 ∈ ℝ) → if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ (0[,)+∞))
2524fmpttd 6870 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)):ℝ⟶(0[,)+∞))
2618, 25mbfi1fseq 24421 . 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 1956 . . 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 1146 . . . . . . 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 1146 . . . . . . 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 615 . . . . . 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 655 . . . . . 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 221 . . . . 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 3101 . . . . . . 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 24408 . . . . . . . . . 10 ((𝑥 ∈ dom ∫1𝑦 ∈ dom ∫1) → (𝑥f𝑦) ∈ dom ∫1)
3534adantl 485 . . . . . . . . 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 11680 . . . . . . . . . 10 ℕ ∈ V
3938a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → ℕ ∈ V)
40 inidm 4123 . . . . . . . . 9 (ℕ ∩ ℕ) = ℕ
4135, 36, 37, 39, 39, 40off 7422 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → (𝑓ff):ℕ⟶dom ∫1)
42 fveq2 6658 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
4342breq2d 5044 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (0 ≤ (𝐹𝑦) ↔ 0 ≤ (𝐹𝑥)))
4443, 42ifbieq1d 4444 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) = if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))
45 eqid 2758 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)) = (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))
46 fvex 6671 . . . . . . . . . . . . . . 15 (𝐹𝑥) ∈ V
47 c0ex 10673 . . . . . . . . . . . . . . 15 0 ∈ V
4846, 47ifex 4470 . . . . . . . . . . . . . 14 if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∈ V
4944, 45, 48fvmpt 6759 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) = if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))
5049breq2d 5044 . . . . . . . . . . . 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 10918 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → -(𝐹𝑦) = -(𝐹𝑥))
5251breq2d 5044 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (0 ≤ -(𝐹𝑦) ↔ 0 ≤ -(𝐹𝑥)))
5352, 51ifbieq1d 4444 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) = if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))
54 eqid 2758 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)) = (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))
55 negex 10922 . . . . . . . . . . . . . . 15 -(𝐹𝑥) ∈ V
5655, 47ifex 4470 . . . . . . . . . . . . . 14 if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0) ∈ V
5753, 54, 56fvmpt 6759 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥) = if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))
5857breq2d 5044 . . . . . . . . . . . 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 633 . . . . . . . . . . 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 485 . . . . . . . . . 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 12321 . . . . . . . . . . . . 13 ℕ = (ℤ‘1)
62 1zzd 12052 . . . . . . . . . . . . 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 6977 . . . . . . . . . . . . . 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))
6736ffvelrnda 6842 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ dom ∫1)
68 i1ff 24376 . . . . . . . . . . . . . . . . . . . 20 ((𝑓𝑛) ∈ dom ∫1 → (𝑓𝑛):ℝ⟶ℝ)
6967, 68syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛):ℝ⟶ℝ)
7069ffvelrnda 6842 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑓𝑛)‘𝑥) ∈ ℝ)
7170an32s 651 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑓𝑛)‘𝑥) ∈ ℝ)
7271recnd 10707 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑓𝑛)‘𝑥) ∈ ℂ)
7372fmpttd 6870 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)):ℕ⟶ℂ)
7473adantr 484 . . . . . . . . . . . . . 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)
7574ffvelrnda 6842 . . . . . . . . . . . . 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)
7637ffvelrnda 6842 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑛) ∈ dom ∫1)
77 i1ff 24376 . . . . . . . . . . . . . . . . . . . 20 ((𝑛) ∈ dom ∫1 → (𝑛):ℝ⟶ℝ)
7876, 77syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑛):ℝ⟶ℝ)
7978ffvelrnda 6842 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛)‘𝑥) ∈ ℝ)
8079an32s 651 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛)‘𝑥) ∈ ℝ)
8180recnd 10707 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛)‘𝑥) ∈ ℂ)
8281fmpttd 6870 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)):ℕ⟶ℂ)
8382adantr 484 . . . . . . . . . . . . . 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)
8483ffvelrnda 6842 . . . . . . . . . . . . 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 6499 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → 𝑓 Fn ℕ)
8637ffnd 6499 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → Fn ℕ)
87 eqidd 2759 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) = (𝑓𝑘))
88 eqidd 2759 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑘) = (𝑘))
8985, 86, 39, 39, 40, 87, 88ofval 7415 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → ((𝑓ff)‘𝑘) = ((𝑓𝑘) ∘f − (𝑘)))
9089fveq1d 6660 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (((𝑓ff)‘𝑘)‘𝑥) = (((𝑓𝑘) ∘f − (𝑘))‘𝑥))
9190adantr 484 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓ff)‘𝑘)‘𝑥) = (((𝑓𝑘) ∘f − (𝑘))‘𝑥))
9236ffvelrnda 6842 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ dom ∫1)
93 i1ff 24376 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑘) ∈ dom ∫1 → (𝑓𝑘):ℝ⟶ℝ)
94 ffn 6498 . . . . . . . . . . . . . . . . . . 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)
9637ffvelrnda 6842 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑘) ∈ dom ∫1)
97 i1ff 24376 . . . . . . . . . . . . . . . . . . 19 ((𝑘) ∈ dom ∫1 → (𝑘):ℝ⟶ℝ)
98 ffn 6498 . . . . . . . . . . . . . . . . . . 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 10666 . . . . . . . . . . . . . . . . . . 19 ℝ ∈ V
101100a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → ℝ ∈ V)
102 inidm 4123 . . . . . . . . . . . . . . . . . 18 (ℝ ∩ ℝ) = ℝ
103 eqidd 2759 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑓𝑘)‘𝑥) = ((𝑓𝑘)‘𝑥))
104 eqidd 2759 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑘)‘𝑥) = ((𝑘)‘𝑥))
10595, 99, 101, 101, 102, 103, 104ofval 7415 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓𝑘) ∘f − (𝑘))‘𝑥) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
10691, 105eqtrd 2793 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓ff)‘𝑘)‘𝑥) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
107106an32s 651 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (((𝑓ff)‘𝑘)‘𝑥) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
108 fveq2 6658 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝑓ff)‘𝑛) = ((𝑓ff)‘𝑘))
109108fveq1d 6660 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (((𝑓ff)‘𝑛)‘𝑥) = (((𝑓ff)‘𝑘)‘𝑥))
110 eqid 2758 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥))
111 fvex 6671 . . . . . . . . . . . . . . . . 17 (((𝑓ff)‘𝑘)‘𝑥) ∈ V
112109, 110, 111fvmpt 6759 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥))‘𝑘) = (((𝑓ff)‘𝑘)‘𝑥))
113112adantl 485 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥))‘𝑘) = (((𝑓ff)‘𝑘)‘𝑥))
114 fveq2 6658 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝑓𝑛) = (𝑓𝑘))
115114fveq1d 6660 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝑓𝑛)‘𝑥) = ((𝑓𝑘)‘𝑥))
116 eqid 2758 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))
117 fvex 6671 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑘)‘𝑥) ∈ V
118115, 116, 117fvmpt 6759 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))‘𝑘) = ((𝑓𝑘)‘𝑥))
119 fveq2 6658 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝑛) = (𝑘))
120119fveq1d 6660 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝑛)‘𝑥) = ((𝑘)‘𝑥))
121 eqid 2758 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))
122 fvex 6671 . . . . . . . . . . . . . . . . . 18 ((𝑘)‘𝑥) ∈ V
123120, 121, 122fvmpt 6759 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))‘𝑘) = ((𝑘)‘𝑥))
124118, 123oveq12d 7168 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))‘𝑘) − ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))‘𝑘)) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
125124adantl 485 . . . . . . . . . . . . . . 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 2803 . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . 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 15038 . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → 𝐹:ℝ⟶ℝ)
130129ffvelrnda 6842 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
131 max0sub 12630 . . . . . . . . . . . . . 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 484 . . . . . . . . . . . 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 5058 . . . . . . . . . . 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 416 . . . . . . . . . 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 243 . . . . . . . . 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 3108 . . . . . . . 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 7183 . . . . . . . . 9 (𝑓ff) ∈ V
139 feq1 6479 . . . . . . . . . 10 (𝑔 = (𝑓ff) → (𝑔:ℕ⟶dom ∫1 ↔ (𝑓ff):ℕ⟶dom ∫1))
140 fveq1 6657 . . . . . . . . . . . . . 14 (𝑔 = (𝑓ff) → (𝑔𝑛) = ((𝑓ff)‘𝑛))
141140fveq1d 6660 . . . . . . . . . . . . 13 (𝑔 = (𝑓ff) → ((𝑔𝑛)‘𝑥) = (((𝑓ff)‘𝑛)‘𝑥))
142141mpteq2dv 5128 . . . . . . . . . . . 12 (𝑔 = (𝑓ff) → (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)))
143142breq1d 5042 . . . . . . . . . . 11 (𝑔 = (𝑓ff) → ((𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
144143ralbidv 3126 . . . . . . . . . 10 (𝑔 = (𝑓ff) → (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ (((𝑓ff)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
145139, 144anbi12d 633 . . . . . . . . 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 3525 . . . . . . . 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 683 . . . . . . 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, 147syl5bir 246 . . . . . 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 457 . . . . 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 1935 . . 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, 151syl5bir 246 . 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 698 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3070  Vcvv 3409  ifcif 4420   class class class wbr 5032   ↦ cmpt 5112  dom cdm 5524   Fn wfn 6330  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150   ∘f cof 7403   ∘r cofr 7404  ℂcc 10573  ℝcr 10574  0cc0 10575  1c1 10576   + caddc 10578  +∞cpnf 10710   ≤ cle 10714   − cmin 10908  -cneg 10909  ℕcn 11674  [,)cico 12781   ⇝ cli 14889  MblFncmbf 24314  ∫1citg1 24315  0𝑝c0p 24369 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-inf2 9137  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652  ax-pre-sup 10653 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 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-se 5484  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-isom 6344  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7405  df-ofr 7406  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-2o 8113  df-er 8299  df-map 8418  df-pm 8419  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-fi 8908  df-sup 8939  df-inf 8940  df-oi 9007  df-dju 9363  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-div 11336  df-nn 11675  df-2 11737  df-3 11738  df-n0 11935  df-z 12021  df-uz 12283  df-q 12389  df-rp 12431  df-xneg 12548  df-xadd 12549  df-xmul 12550  df-ioo 12783  df-ico 12785  df-icc 12786  df-fz 12940  df-fzo 13083  df-fl 13211  df-seq 13419  df-exp 13480  df-hash 13741  df-cj 14506  df-re 14507  df-im 14508  df-sqrt 14642  df-abs 14643  df-clim 14893  df-rlim 14894  df-sum 15091  df-rest 16754  df-topgen 16775  df-psmet 20158  df-xmet 20159  df-met 20160  df-bl 20161  df-mopn 20162  df-top 21594  df-topon 21611  df-bases 21646  df-cmp 22087  df-ovol 24164  df-vol 24165  df-mbf 24319  df-itg1 24320  df-0p 24370 This theorem is referenced by:  mbfi1flim  24423
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