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Theorem mbfi1flimlem 23295
 Description: Lemma for mbfi1flim 23296. (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 6267 . . . 4 ((𝜑𝑦 ∈ ℝ) → (𝐹𝑦) ∈ ℝ)
31feqmptd 6159 . . . . 5 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
4 mbfi1flim.1 . . . . 5 (𝜑𝐹 ∈ MblFn)
53, 4eqeltrrd 2689 . . . 4 (𝜑 → (𝑦 ∈ ℝ ↦ (𝐹𝑦)) ∈ MblFn)
62, 5mbfpos 23224 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)) ∈ MblFn)
7 0re 9919 . . . . . 6 0 ∈ ℝ
8 ifcl 4080 . . . . . 6 (((𝐹𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ ℝ)
92, 7, 8sylancl 693 . . . . 5 ((𝜑𝑦 ∈ ℝ) → if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ ℝ)
10 max1 11890 . . . . . 6 ((0 ∈ ℝ ∧ (𝐹𝑦) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))
117, 2, 10sylancr 694 . . . . 5 ((𝜑𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))
12 elrege0 12149 . . . . 5 (if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ (0[,)+∞) ↔ (if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)))
139, 11, 12sylanbrc 695 . . . 4 ((𝜑𝑦 ∈ ℝ) → if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ (0[,)+∞))
14 eqid 2610 . . . 4 (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)) = (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))
1513, 14fmptd 6292 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)):ℝ⟶(0[,)+∞))
166, 15mbfi1fseq 23294 . 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)))
172renegcld 10336 . . . 4 ((𝜑𝑦 ∈ ℝ) → -(𝐹𝑦) ∈ ℝ)
182, 5mbfneg 23223 . . . 4 (𝜑 → (𝑦 ∈ ℝ ↦ -(𝐹𝑦)) ∈ MblFn)
1917, 18mbfpos 23224 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)) ∈ MblFn)
20 ifcl 4080 . . . . . 6 ((-(𝐹𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ ℝ)
2117, 7, 20sylancl 693 . . . . 5 ((𝜑𝑦 ∈ ℝ) → if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ ℝ)
22 max1 11890 . . . . . 6 ((0 ∈ ℝ ∧ -(𝐹𝑦) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))
237, 17, 22sylancr 694 . . . . 5 ((𝜑𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))
24 elrege0 12149 . . . . 5 (if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ (0[,)+∞) ↔ (if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)))
2521, 23, 24sylanbrc 695 . . . 4 ((𝜑𝑦 ∈ ℝ) → if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ (0[,)+∞))
26 eqid 2610 . . . 4 (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)) = (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))
2725, 26fmptd 6292 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)):ℝ⟶(0[,)+∞))
2819, 27mbfi1fseq 23294 . 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)))
29 eeanv 2170 . . 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))))
30 3simpb 1052 . . . . . . 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)))
31 3simpb 1052 . . . . . . 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)))
3230, 31anim12i 588 . . . . . 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))))
33 an4 861 . . . . . 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))))
3432, 33sylib 207 . . . . 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))))
35 r19.26 3046 . . . . . . 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)))
36 i1fsub 23281 . . . . . . . . . 10 ((𝑥 ∈ dom ∫1𝑦 ∈ dom ∫1) → (𝑥𝑓𝑦) ∈ dom ∫1)
3736adantl 481 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ (𝑥 ∈ dom ∫1𝑦 ∈ dom ∫1)) → (𝑥𝑓𝑦) ∈ dom ∫1)
38 simprl 790 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → 𝑓:ℕ⟶dom ∫1)
39 simprr 792 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → :ℕ⟶dom ∫1)
40 nnex 10903 . . . . . . . . . 10 ℕ ∈ V
4140a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → ℕ ∈ V)
42 inidm 3784 . . . . . . . . 9 (ℕ ∩ ℕ) = ℕ
4337, 38, 39, 41, 41, 42off 6810 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → (𝑓𝑓𝑓):ℕ⟶dom ∫1)
44 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
4544breq2d 4595 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (0 ≤ (𝐹𝑦) ↔ 0 ≤ (𝐹𝑥)))
4645, 44ifbieq1d 4059 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) = if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))
47 fvex 6113 . . . . . . . . . . . . . . 15 (𝐹𝑥) ∈ V
48 c0ex 9913 . . . . . . . . . . . . . . 15 0 ∈ V
4947, 48ifex 4106 . . . . . . . . . . . . . 14 if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∈ V
5046, 14, 49fvmpt 6191 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) = if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))
5150breq2d 4595 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0)))
5244negeqd 10154 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → -(𝐹𝑦) = -(𝐹𝑥))
5352breq2d 4595 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (0 ≤ -(𝐹𝑦) ↔ 0 ≤ -(𝐹𝑥)))
5453, 52ifbieq1d 4059 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) = if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))
55 negex 10158 . . . . . . . . . . . . . . 15 -(𝐹𝑥) ∈ V
5655, 48ifex 4106 . . . . . . . . . . . . . 14 if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0) ∈ V
5754, 26, 56fvmpt 6191 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥) = if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))
5857breq2d 4595 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)))
5951, 58anbi12d 743 . . . . . . . . . . 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 11599 . . . . . . . . . . . . 13 ℕ = (ℤ‘1)
62 1zzd 11285 . . . . . . . . . . . . 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 790 . . . . . . . . . . . . 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))
6440mptex 6390 . . . . . . . . . . . . . 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 792 . . . . . . . . . . . . 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))
6738ffvelrnda 6267 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ dom ∫1)
68 i1ff 23249 . . . . . . . . . . . . . . . . . . . 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 6267 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑓𝑛)‘𝑥) ∈ ℝ)
7170an32s 842 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑓𝑛)‘𝑥) ∈ ℝ)
7271recnd 9947 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑓𝑛)‘𝑥) ∈ ℂ)
73 eqid 2610 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))
7472, 73fmptd 6292 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)):ℕ⟶ℂ)
7574adantr 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)
7675ffvelrnda 6267 . . . . . . . . . . . . 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)
7739ffvelrnda 6267 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑛) ∈ dom ∫1)
78 i1ff 23249 . . . . . . . . . . . . . . . . . . . 20 ((𝑛) ∈ dom ∫1 → (𝑛):ℝ⟶ℝ)
7977, 78syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑛):ℝ⟶ℝ)
8079ffvelrnda 6267 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛)‘𝑥) ∈ ℝ)
8180an32s 842 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛)‘𝑥) ∈ ℝ)
8281recnd 9947 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛)‘𝑥) ∈ ℂ)
83 eqid 2610 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))
8482, 83fmptd 6292 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)):ℕ⟶ℂ)
8584adantr 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)
8685ffvelrnda 6267 . . . . . . . . . . . . 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)
87 ffn 5958 . . . . . . . . . . . . . . . . . . . . 21 (𝑓:ℕ⟶dom ∫1𝑓 Fn ℕ)
8838, 87syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → 𝑓 Fn ℕ)
89 ffn 5958 . . . . . . . . . . . . . . . . . . . . 21 (:ℕ⟶dom ∫1 Fn ℕ)
9039, 89syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → Fn ℕ)
91 eqidd 2611 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) = (𝑓𝑘))
92 eqidd 2611 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑘) = (𝑘))
9388, 90, 41, 41, 42, 91, 92ofval 6804 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → ((𝑓𝑓𝑓)‘𝑘) = ((𝑓𝑘) ∘𝑓 − (𝑘)))
9493fveq1d 6105 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (((𝑓𝑓𝑓)‘𝑘)‘𝑥) = (((𝑓𝑘) ∘𝑓 − (𝑘))‘𝑥))
9594adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓𝑓𝑓)‘𝑘)‘𝑥) = (((𝑓𝑘) ∘𝑓 − (𝑘))‘𝑥))
9638ffvelrnda 6267 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ dom ∫1)
97 i1ff 23249 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑘) ∈ dom ∫1 → (𝑓𝑘):ℝ⟶ℝ)
98 ffn 5958 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑘):ℝ⟶ℝ → (𝑓𝑘) Fn ℝ)
9996, 97, 983syl 18 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) Fn ℝ)
10039ffvelrnda 6267 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑘) ∈ dom ∫1)
101 i1ff 23249 . . . . . . . . . . . . . . . . . . 19 ((𝑘) ∈ dom ∫1 → (𝑘):ℝ⟶ℝ)
102 ffn 5958 . . . . . . . . . . . . . . . . . . 19 ((𝑘):ℝ⟶ℝ → (𝑘) Fn ℝ)
103100, 101, 1023syl 18 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑘) Fn ℝ)
104 reex 9906 . . . . . . . . . . . . . . . . . . 19 ℝ ∈ V
105104a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → ℝ ∈ V)
106 inidm 3784 . . . . . . . . . . . . . . . . . 18 (ℝ ∩ ℝ) = ℝ
107 eqidd 2611 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑓𝑘)‘𝑥) = ((𝑓𝑘)‘𝑥))
108 eqidd 2611 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑘)‘𝑥) = ((𝑘)‘𝑥))
10999, 103, 105, 105, 106, 107, 108ofval 6804 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓𝑘) ∘𝑓 − (𝑘))‘𝑥) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
11095, 109eqtrd 2644 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓𝑓𝑓)‘𝑘)‘𝑥) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
111110an32s 842 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (((𝑓𝑓𝑓)‘𝑘)‘𝑥) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
112 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝑓𝑓𝑓)‘𝑛) = ((𝑓𝑓𝑓)‘𝑘))
113112fveq1d 6105 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (((𝑓𝑓𝑓)‘𝑛)‘𝑥) = (((𝑓𝑓𝑓)‘𝑘)‘𝑥))
114 eqid 2610 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ ↦ (((𝑓𝑓𝑓)‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ (((𝑓𝑓𝑓)‘𝑛)‘𝑥))
115 fvex 6113 . . . . . . . . . . . . . . . . 17 (((𝑓𝑓𝑓)‘𝑘)‘𝑥) ∈ V
116113, 114, 115fvmpt 6191 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((𝑓𝑓𝑓)‘𝑛)‘𝑥))‘𝑘) = (((𝑓𝑓𝑓)‘𝑘)‘𝑥))
117116adantl 481 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((𝑓𝑓𝑓)‘𝑛)‘𝑥))‘𝑘) = (((𝑓𝑓𝑓)‘𝑘)‘𝑥))
118 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝑓𝑛) = (𝑓𝑘))
119118fveq1d 6105 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝑓𝑛)‘𝑥) = ((𝑓𝑘)‘𝑥))
120 fvex 6113 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑘)‘𝑥) ∈ V
121119, 73, 120fvmpt 6191 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))‘𝑘) = ((𝑓𝑘)‘𝑥))
122 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝑛) = (𝑘))
123122fveq1d 6105 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝑛)‘𝑥) = ((𝑘)‘𝑥))
124 fvex 6113 . . . . . . . . . . . . . . . . . 18 ((𝑘)‘𝑥) ∈ V
125123, 83, 124fvmpt 6191 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))‘𝑘) = ((𝑘)‘𝑥))
126121, 125oveq12d 6567 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))‘𝑘) − ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))‘𝑘)) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
127126adantl 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)))
128111, 117, 1273eqtr4d 2654 . . . . . . . . . . . . . 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)))
129128adantlr 747 . . . . . . . . . . . . 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)))
13061, 62, 63, 65, 66, 76, 86, 129climsub 14212 . . . . . . . . . . . 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)))
1311adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → 𝐹:ℝ⟶ℝ)
132131ffvelrnda 6267 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
133 max0sub 11901 . . . . . . . . . . . . . 14 ((𝐹𝑥) ∈ ℝ → (if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) − if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) = (𝐹𝑥))
134132, 133syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) − if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0)) = (𝐹𝑥))
135134adantr 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))
136130, 135breqtrd 4609 . . . . . . . . . . 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))
137136ex 449 . . . . . . . . . 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)))
13860, 137sylbid 229 . . . . . . . . 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)))
139138ralimdva 2945 . . . . . . . 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)))
140 ovex 6577 . . . . . . . . 9 (𝑓𝑓𝑓) ∈ V
141 feq1 5939 . . . . . . . . . 10 (𝑔 = (𝑓𝑓𝑓) → (𝑔:ℕ⟶dom ∫1 ↔ (𝑓𝑓𝑓):ℕ⟶dom ∫1))
142 fveq1 6102 . . . . . . . . . . . . . 14 (𝑔 = (𝑓𝑓𝑓) → (𝑔𝑛) = ((𝑓𝑓𝑓)‘𝑛))
143142fveq1d 6105 . . . . . . . . . . . . 13 (𝑔 = (𝑓𝑓𝑓) → ((𝑔𝑛)‘𝑥) = (((𝑓𝑓𝑓)‘𝑛)‘𝑥))
144143mpteq2dv 4673 . . . . . . . . . . . 12 (𝑔 = (𝑓𝑓𝑓) → (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ (((𝑓𝑓𝑓)‘𝑛)‘𝑥)))
145144breq1d 4593 . . . . . . . . . . 11 (𝑔 = (𝑓𝑓𝑓) → ((𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ (((𝑓𝑓𝑓)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
146145ralbidv 2969 . . . . . . . . . 10 (𝑔 = (𝑓𝑓𝑓) → (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ (((𝑓𝑓𝑓)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
147141, 146anbi12d 743 . . . . . . . . 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))))
148140, 147spcev 3273 . . . . . . . 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)))
14943, 139, 148syl6an 566 . . . . . . 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))))
15035, 149syl5bir 232 . . . . . 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))))
151150expimpd 627 . . . . 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))))
15234, 151syl5 33 . . . 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))))
153152exlimdvv 1849 . . 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))))
15429, 153syl5bir 232 . 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))))
15516, 28, 154mp2and 711 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793   ∘𝑟 cofr 6794  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818  +∞cpnf 9950   ≤ cle 9954   − cmin 10145  -cneg 10146  ℕcn 10897  [,)cico 12048   ⇝ cli 14063  MblFncmbf 23189  ∫1citg1 23190  0𝑝c0p 23242 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-mbf 23194  df-itg1 23195  df-0p 23243 This theorem is referenced by:  mbfi1flim  23296
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