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Theorem mbfi1flimlem 23429
 Description: Lemma for mbfi1flim 23430. (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 6325 . . . 4 ((𝜑𝑦 ∈ ℝ) → (𝐹𝑦) ∈ ℝ)
31feqmptd 6216 . . . . 5 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
4 mbfi1flim.1 . . . . 5 (𝜑𝐹 ∈ MblFn)
53, 4eqeltrrd 2699 . . . 4 (𝜑 → (𝑦 ∈ ℝ ↦ (𝐹𝑦)) ∈ MblFn)
62, 5mbfpos 23358 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)) ∈ MblFn)
7 0re 10000 . . . . . 6 0 ∈ ℝ
8 ifcl 4108 . . . . . 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 11975 . . . . . 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 12236 . . . . 5 (if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ (0[,)+∞) ↔ (if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)))
139, 11, 12sylanbrc 697 . . . 4 ((𝜑𝑦 ∈ ℝ) → if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) ∈ (0[,)+∞))
14 eqid 2621 . . . 4 (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)) = (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))
1513, 14fmptd 6351 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0)):ℝ⟶(0[,)+∞))
166, 15mbfi1fseq 23428 . 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 10417 . . . 4 ((𝜑𝑦 ∈ ℝ) → -(𝐹𝑦) ∈ ℝ)
182, 5mbfneg 23357 . . . 4 (𝜑 → (𝑦 ∈ ℝ ↦ -(𝐹𝑦)) ∈ MblFn)
1917, 18mbfpos 23358 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)) ∈ MblFn)
20 ifcl 4108 . . . . . 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 11975 . . . . . 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 12236 . . . . 5 (if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ (0[,)+∞) ↔ (if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)))
2521, 23, 24sylanbrc 697 . . . 4 ((𝜑𝑦 ∈ ℝ) → if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) ∈ (0[,)+∞))
26 eqid 2621 . . . 4 (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)) = (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))
2725, 26fmptd 6351 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0)):ℝ⟶(0[,)+∞))
2819, 27mbfi1fseq 23428 . 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 2181 . . 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 1057 . . . . . . 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 1057 . . . . . . 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 589 . . . . . 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 864 . . . . . 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 208 . . . . 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 3059 . . . . . . 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 23415 . . . . . . . . . 10 ((𝑥 ∈ dom ∫1𝑦 ∈ dom ∫1) → (𝑥𝑓𝑦) ∈ dom ∫1)
3736adantl 482 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ (𝑥 ∈ dom ∫1𝑦 ∈ dom ∫1)) → (𝑥𝑓𝑦) ∈ dom ∫1)
38 simprl 793 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → 𝑓:ℕ⟶dom ∫1)
39 simprr 795 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → :ℕ⟶dom ∫1)
40 nnex 10986 . . . . . . . . . 10 ℕ ∈ V
4140a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → ℕ ∈ V)
42 inidm 3806 . . . . . . . . 9 (ℕ ∩ ℕ) = ℕ
4337, 38, 39, 41, 41, 42off 6877 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → (𝑓𝑓𝑓):ℕ⟶dom ∫1)
44 fveq2 6158 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
4544breq2d 4635 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (0 ≤ (𝐹𝑦) ↔ 0 ≤ (𝐹𝑥)))
4645, 44ifbieq1d 4087 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0) = if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))
47 fvex 6168 . . . . . . . . . . . . . . 15 (𝐹𝑥) ∈ V
48 c0ex 9994 . . . . . . . . . . . . . . 15 0 ∈ V
4947, 48ifex 4134 . . . . . . . . . . . . . 14 if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0) ∈ V
5046, 14, 49fvmpt 6249 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹𝑦), (𝐹𝑦), 0))‘𝑥) = if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))
5150breq2d 4635 . . . . . . . . . . . 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 10235 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → -(𝐹𝑦) = -(𝐹𝑥))
5352breq2d 4635 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (0 ≤ -(𝐹𝑦) ↔ 0 ≤ -(𝐹𝑥)))
5453, 52ifbieq1d 4087 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0) = if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))
55 negex 10239 . . . . . . . . . . . . . . 15 -(𝐹𝑥) ∈ V
5655, 48ifex 4134 . . . . . . . . . . . . . 14 if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0) ∈ V
5754, 26, 56fvmpt 6249 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹𝑦), -(𝐹𝑦), 0))‘𝑥) = if(0 ≤ -(𝐹𝑥), -(𝐹𝑥), 0))
5857breq2d 4635 . . . . . . . . . . . 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 746 . . . . . . . . . . 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 11683 . . . . . . . . . . . . 13 ℕ = (ℤ‘1)
62 1zzd 11368 . . . . . . . . . . . . 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 793 . . . . . . . . . . . . 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 6451 . . . . . . . . . . . . . 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 795 . . . . . . . . . . . . 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 6325 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ dom ∫1)
68 i1ff 23383 . . . . . . . . . . . . . . . . . . . 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 6325 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑓𝑛)‘𝑥) ∈ ℝ)
7170an32s 845 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑓𝑛)‘𝑥) ∈ ℝ)
7271recnd 10028 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑓𝑛)‘𝑥) ∈ ℂ)
73 eqid 2621 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))
7472, 73fmptd 6351 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥)):ℕ⟶ℂ)
7574adantr 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)
7675ffvelrnda 6325 . . . . . . . . . . . . 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 6325 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑛) ∈ dom ∫1)
78 i1ff 23383 . . . . . . . . . . . . . . . . . . . 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 6325 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛)‘𝑥) ∈ ℝ)
8180an32s 845 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛)‘𝑥) ∈ ℝ)
8281recnd 10028 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛)‘𝑥) ∈ ℂ)
83 eqid 2621 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))
8482, 83fmptd 6351 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥)):ℕ⟶ℂ)
8584adantr 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)
8685ffvelrnda 6325 . . . . . . . . . . . . 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 6012 . . . . . . . . . . . . . . . . . . . . 21 (𝑓:ℕ⟶dom ∫1𝑓 Fn ℕ)
8838, 87syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → 𝑓 Fn ℕ)
89 ffn 6012 . . . . . . . . . . . . . . . . . . . . 21 (:ℕ⟶dom ∫1 Fn ℕ)
9039, 89syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → Fn ℕ)
91 eqidd 2622 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) = (𝑓𝑘))
92 eqidd 2622 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑘) = (𝑘))
9388, 90, 41, 41, 42, 91, 92ofval 6871 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → ((𝑓𝑓𝑓)‘𝑘) = ((𝑓𝑘) ∘𝑓 − (𝑘)))
9493fveq1d 6160 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (((𝑓𝑓𝑓)‘𝑘)‘𝑥) = (((𝑓𝑘) ∘𝑓 − (𝑘))‘𝑥))
9594adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓𝑓𝑓)‘𝑘)‘𝑥) = (((𝑓𝑘) ∘𝑓 − (𝑘))‘𝑥))
9638ffvelrnda 6325 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ dom ∫1)
97 i1ff 23383 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑘) ∈ dom ∫1 → (𝑓𝑘):ℝ⟶ℝ)
98 ffn 6012 . . . . . . . . . . . . . . . . . . 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 6325 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑘) ∈ dom ∫1)
101 i1ff 23383 . . . . . . . . . . . . . . . . . . 19 ((𝑘) ∈ dom ∫1 → (𝑘):ℝ⟶ℝ)
102 ffn 6012 . . . . . . . . . . . . . . . . . . 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 9987 . . . . . . . . . . . . . . . . . . 19 ℝ ∈ V
105104a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → ℝ ∈ V)
106 inidm 3806 . . . . . . . . . . . . . . . . . 18 (ℝ ∩ ℝ) = ℝ
107 eqidd 2622 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑓𝑘)‘𝑥) = ((𝑓𝑘)‘𝑥))
108 eqidd 2622 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑘)‘𝑥) = ((𝑘)‘𝑥))
10999, 103, 105, 105, 106, 107, 108ofval 6871 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓𝑘) ∘𝑓 − (𝑘))‘𝑥) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
11095, 109eqtrd 2655 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓𝑓𝑓)‘𝑘)‘𝑥) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
111110an32s 845 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (((𝑓𝑓𝑓)‘𝑘)‘𝑥) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
112 fveq2 6158 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝑓𝑓𝑓)‘𝑛) = ((𝑓𝑓𝑓)‘𝑘))
113112fveq1d 6160 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (((𝑓𝑓𝑓)‘𝑛)‘𝑥) = (((𝑓𝑓𝑓)‘𝑘)‘𝑥))
114 eqid 2621 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ ↦ (((𝑓𝑓𝑓)‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ (((𝑓𝑓𝑓)‘𝑛)‘𝑥))
115 fvex 6168 . . . . . . . . . . . . . . . . 17 (((𝑓𝑓𝑓)‘𝑘)‘𝑥) ∈ V
116113, 114, 115fvmpt 6249 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((𝑓𝑓𝑓)‘𝑛)‘𝑥))‘𝑘) = (((𝑓𝑓𝑓)‘𝑘)‘𝑥))
117116adantl 482 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((𝑓𝑓𝑓)‘𝑛)‘𝑥))‘𝑘) = (((𝑓𝑓𝑓)‘𝑘)‘𝑥))
118 fveq2 6158 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝑓𝑛) = (𝑓𝑘))
119118fveq1d 6160 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝑓𝑛)‘𝑥) = ((𝑓𝑘)‘𝑥))
120 fvex 6168 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑘)‘𝑥) ∈ V
121119, 73, 120fvmpt 6249 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))‘𝑘) = ((𝑓𝑘)‘𝑥))
122 fveq2 6158 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝑛) = (𝑘))
123122fveq1d 6160 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((𝑛)‘𝑥) = ((𝑘)‘𝑥))
124 fvex 6168 . . . . . . . . . . . . . . . . . 18 ((𝑘)‘𝑥) ∈ V
125123, 83, 124fvmpt 6249 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))‘𝑘) = ((𝑘)‘𝑥))
126121, 125oveq12d 6633 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑓𝑛)‘𝑥))‘𝑘) − ((𝑛 ∈ ℕ ↦ ((𝑛)‘𝑥))‘𝑘)) = (((𝑓𝑘)‘𝑥) − ((𝑘)‘𝑥)))
127126adantl 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)))
128111, 117, 1273eqtr4d 2665 . . . . . . . . . . . . . 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 750 . . . . . . . . . . . . 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 14314 . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) → 𝐹:ℝ⟶ℝ)
132131ffvelrnda 6325 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
133 max0sub 11986 . . . . . . . . . . . . . 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 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))
136130, 135breqtrd 4649 . . . . . . . . . . 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 450 . . . . . . . . . 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 230 . . . . . . . . 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 2958 . . . . . . . 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 6643 . . . . . . . . 9 (𝑓𝑓𝑓) ∈ V
141 feq1 5993 . . . . . . . . . 10 (𝑔 = (𝑓𝑓𝑓) → (𝑔:ℕ⟶dom ∫1 ↔ (𝑓𝑓𝑓):ℕ⟶dom ∫1))
142 fveq1 6157 . . . . . . . . . . . . . 14 (𝑔 = (𝑓𝑓𝑓) → (𝑔𝑛) = ((𝑓𝑓𝑓)‘𝑛))
143142fveq1d 6160 . . . . . . . . . . . . 13 (𝑔 = (𝑓𝑓𝑓) → ((𝑔𝑛)‘𝑥) = (((𝑓𝑓𝑓)‘𝑛)‘𝑥))
144143mpteq2dv 4715 . . . . . . . . . . . 12 (𝑔 = (𝑓𝑓𝑓) → (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ (((𝑓𝑓𝑓)‘𝑛)‘𝑥)))
145144breq1d 4633 . . . . . . . . . . 11 (𝑔 = (𝑓𝑓𝑓) → ((𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ (𝑛 ∈ ℕ ↦ (((𝑓𝑓𝑓)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
146145ralbidv 2982 . . . . . . . . . 10 (𝑔 = (𝑓𝑓𝑓) → (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥) ↔ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ (((𝑓𝑓𝑓)‘𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
147141, 146anbi12d 746 . . . . . . . . 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 3290 . . . . . . . 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 567 . . . . . . 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 233 . . . . . 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 628 . . . . 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 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))))
153152exlimdvv 1859 . . 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 233 . 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 714 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 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ∀wral 2908  Vcvv 3190  ifcif 4064   class class class wbr 4623   ↦ cmpt 4683  dom cdm 5084   Fn wfn 5852  ⟶wf 5853  ‘cfv 5857  (class class class)co 6615   ∘𝑓 cof 6860   ∘𝑟 cofr 6861  ℂcc 9894  ℝcr 9895  0cc0 9896  1c1 9897   + caddc 9899  +∞cpnf 10031   ≤ cle 10035   − cmin 10226  -cneg 10227  ℕcn 10980  [,)cico 12135   ⇝ cli 14165  MblFncmbf 23323  ∫1citg1 23324  0𝑝c0p 23376 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-ofr 6863  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fi 8277  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-z 11338  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-ioo 12137  df-ico 12139  df-icc 12140  df-fz 12285  df-fzo 12423  df-fl 12549  df-seq 12758  df-exp 12817  df-hash 13074  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-rlim 14170  df-sum 14367  df-rest 16023  df-topgen 16044  df-psmet 19678  df-xmet 19679  df-met 19680  df-bl 19681  df-mopn 19682  df-top 20639  df-topon 20656  df-bases 20690  df-cmp 21130  df-ovol 23173  df-vol 23174  df-mbf 23328  df-itg1 23329  df-0p 23377 This theorem is referenced by:  mbfi1flim  23430
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