Theorem mbfi1flimlem 25691

Description: Lemma for mbfi1flim 25692. (Contributed by Mario Carneiro, 5-Sep-2014.)

Hypotheses

Ref	Expression
mbfi1flim.1	⊢ (𝜑 → 𝐹 ∈ MblFn)
mbfi1flimlem.2	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)

Assertion

Ref	Expression
mbfi1flimlem	⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))

Distinct variable groups:   𝑔,𝑛,𝑥,𝐹   𝜑,𝑔,𝑛,𝑥

Proof of Theorem mbfi1flimlem

Dummy variables 𝑦 𝑓 ℎ 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfi1flimlem.2	. . . . 5 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
2	1	ffvelcdmda 7038	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℝ)
3	1	feqmptd 6910	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)))
4		mbfi1flim.1	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ MblFn)
5	3, 4	eqeltrrd 2838	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ∈ MblFn)
6	2, 5	mbfpos 25620	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0)) ∈ MblFn)
7		0re 11146	. . . . . 6 ⊢ 0 ∈ ℝ
8		ifcl 4527	. . . . . 6 ⊢ (((𝐹‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0) ∈ ℝ)
9	2, 7, 8	sylancl 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0) ∈ ℝ)
10		max1 13112	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ (𝐹‘𝑦) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))
11	7, 2, 10	sylancr 588	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))
12		elrege0 13382	. . . . 5 ⊢ (if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0) ∈ (0[,)+∞) ↔ (if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0)))
13	9, 11, 12	sylanbrc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0) ∈ (0[,)+∞))
14	13	fmpttd 7069	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0)):ℝ⟶(0[,)+∞))
15	6, 14	mbfi1fseq 25690	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥)))
16	2	renegcld 11576	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → -(𝐹‘𝑦) ∈ ℝ)
17	2, 5	mbfneg 25619	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ -(𝐹‘𝑦)) ∈ MblFn)
18	16, 17	mbfpos 25620	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0)) ∈ MblFn)
19		ifcl 4527	. . . . . 6 ⊢ ((-(𝐹‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0) ∈ ℝ)
20	16, 7, 19	sylancl 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0) ∈ ℝ)
21		max1 13112	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ -(𝐹‘𝑦) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))
22	7, 16, 21	sylancr 588	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))
23		elrege0 13382	. . . . 5 ⊢ (if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0) ∈ (0[,)+∞) ↔ (if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0)))
24	20, 22, 23	sylanbrc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0) ∈ (0[,)+∞))
25	24	fmpttd 7069	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0)):ℝ⟶(0[,)+∞))
26	18, 25	mbfi1fseq 25690	. 2 ⊢ (𝜑 → ∃ℎ(ℎ:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (ℎ‘𝑛) ∧ (ℎ‘𝑛) ∘r ≤ (ℎ‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥)))
27		exdistrv 1957	. . 3 ⊢ (∃𝑓∃ℎ((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥)) ∧ (ℎ:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (ℎ‘𝑛) ∧ (ℎ‘𝑛) ∘r ≤ (ℎ‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥))) ↔ (∃𝑓(𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥)) ∧ ∃ℎ(ℎ:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (ℎ‘𝑛) ∧ (ℎ‘𝑛) ∘r ≤ (ℎ‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥))))
28		3simpb 1150	. . . . . . 7 ⊢ ((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥)) → (𝑓:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥)))
29		3simpb 1150	. . . . . . 7 ⊢ ((ℎ:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (ℎ‘𝑛) ∧ (ℎ‘𝑛) ∘r ≤ (ℎ‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥)) → (ℎ:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥)))
30	28, 29	anim12i 614	. . . . . 6 ⊢ (((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥)) ∧ (ℎ:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (ℎ‘𝑛) ∧ (ℎ‘𝑛) ∘r ≤ (ℎ‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥))) → ((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥)) ∧ (ℎ:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥))))
31		an4 657	. . . . . 6 ⊢ (((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥)) ∧ (ℎ:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥))) ↔ ((𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1) ∧ (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥))))
32	30, 31	sylib 218	. . . . 5 ⊢ (((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥)) ∧ (ℎ:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (ℎ‘𝑛) ∧ (ℎ‘𝑛) ∘r ≤ (ℎ‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥))) → ((𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1) ∧ (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥))))
33		r19.26 3098	. . . . . . 7 ⊢ (∀𝑥 ∈ ℝ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥)) ↔ (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥)))
34		i1fsub 25677	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom ∫1 ∧ 𝑦 ∈ dom ∫1) → (𝑥 ∘f − 𝑦) ∈ dom ∫1)
35	34	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ (𝑥 ∈ dom ∫1 ∧ 𝑦 ∈ dom ∫1)) → (𝑥 ∘f − 𝑦) ∈ dom ∫1)
36		simprl 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → 𝑓:ℕ⟶dom ∫1)
37		simprr 773	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → ℎ:ℕ⟶dom ∫1)
38		nnex 12163	. . . . . . . . . 10 ⊢ ℕ ∈ V
39	38	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → ℕ ∈ V)
40		inidm 4181	. . . . . . . . 9 ⊢ (ℕ ∩ ℕ) = ℕ
41	35, 36, 37, 39, 39, 40	off 7650	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → (𝑓 ∘f ∘f − ℎ):ℕ⟶dom ∫1)
42		fveq2 6842	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
43	42	breq2d 5112	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (0 ≤ (𝐹‘𝑦) ↔ 0 ≤ (𝐹‘𝑥)))
44	43, 42	ifbieq1d 4506	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0) = if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
45		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0)) = (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))
46		fvex 6855	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑥) ∈ V
47		c0ex 11138	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
48	46, 47	ifex 4532	. . . . . . . . . . . . . 14 ⊢ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∈ V
49	44, 45, 48	fvmpt 6949	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥) = if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
50	49	breq2d 5112	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥) ↔ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)))
51	42	negeqd 11386	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → -(𝐹‘𝑦) = -(𝐹‘𝑥))
52	51	breq2d 5112	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (0 ≤ -(𝐹‘𝑦) ↔ 0 ≤ -(𝐹‘𝑥)))
53	52, 51	ifbieq1d 4506	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0) = if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
54		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0)) = (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))
55		negex 11390	. . . . . . . . . . . . . . 15 ⊢ -(𝐹‘𝑥) ∈ V
56	55, 47	ifex 4532	. . . . . . . . . . . . . 14 ⊢ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) ∈ V
57	53, 54, 56	fvmpt 6949	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥) = if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
58	57	breq2d 5112	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → ((𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥) ↔ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))
59	50, 58	anbi12d 633	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥)) ↔ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))))
60	59	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥)) ↔ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))))
61		nnuz 12802	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ≥‘1)
62		1zzd 12534	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) → 1 ∈ ℤ)
63		simprl 771	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) → (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
64	38	mptex 7179	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥)) ∈ V
65	64	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) → (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥)) ∈ V)
66		simprr 773	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) → (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
67	36	ffvelcdmda 7038	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ dom ∫1)
68		i1ff 25645	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑛) ∈ dom ∫1 → (𝑓‘𝑛):ℝ⟶ℝ)
69	67, 68	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛):ℝ⟶ℝ)
70	69	ffvelcdmda 7038	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑓‘𝑛)‘𝑥) ∈ ℝ)
71	70	an32s 653	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑓‘𝑛)‘𝑥) ∈ ℝ)
72	71	recnd 11172	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑓‘𝑛)‘𝑥) ∈ ℂ)
73	72	fmpttd 7069	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)):ℕ⟶ℂ)
74	73	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) → (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)):ℕ⟶ℂ)
75	74	ffvelcdmda 7038	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥))‘𝑘) ∈ ℂ)
76	37	ffvelcdmda 7038	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (ℎ‘𝑛) ∈ dom ∫1)
77		i1ff 25645	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ‘𝑛) ∈ dom ∫1 → (ℎ‘𝑛):ℝ⟶ℝ)
78	76, 77	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (ℎ‘𝑛):ℝ⟶ℝ)
79	78	ffvelcdmda 7038	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((ℎ‘𝑛)‘𝑥) ∈ ℝ)
80	79	an32s 653	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((ℎ‘𝑛)‘𝑥) ∈ ℝ)
81	80	recnd 11172	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((ℎ‘𝑛)‘𝑥) ∈ ℂ)
82	81	fmpttd 7069	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)):ℕ⟶ℂ)
83	82	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) → (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)):ℕ⟶ℂ)
84	83	ffvelcdmda 7038	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥))‘𝑘) ∈ ℂ)
85	36	ffnd 6671	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → 𝑓 Fn ℕ)
86	37	ffnd 6671	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → ℎ Fn ℕ)
87		eqidd 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) = (𝑓‘𝑘))
88		eqidd 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (ℎ‘𝑘) = (ℎ‘𝑘))
89	85, 86, 39, 39, 40, 87, 88	ofval 7643	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → ((𝑓 ∘f ∘f − ℎ)‘𝑘) = ((𝑓‘𝑘) ∘f − (ℎ‘𝑘)))
90	89	fveq1d 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (((𝑓 ∘f ∘f − ℎ)‘𝑘)‘𝑥) = (((𝑓‘𝑘) ∘f − (ℎ‘𝑘))‘𝑥))
91	90	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓 ∘f ∘f − ℎ)‘𝑘)‘𝑥) = (((𝑓‘𝑘) ∘f − (ℎ‘𝑘))‘𝑥))
92	36	ffvelcdmda 7038	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ dom ∫1)
93		i1ff 25645	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑘) ∈ dom ∫1 → (𝑓‘𝑘):ℝ⟶ℝ)
94		ffn 6670	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑘):ℝ⟶ℝ → (𝑓‘𝑘) Fn ℝ)
95	92, 93, 94	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) Fn ℝ)
96	37	ffvelcdmda 7038	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (ℎ‘𝑘) ∈ dom ∫1)
97		i1ff 25645	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ‘𝑘) ∈ dom ∫1 → (ℎ‘𝑘):ℝ⟶ℝ)
98		ffn 6670	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ‘𝑘):ℝ⟶ℝ → (ℎ‘𝑘) Fn ℝ)
99	96, 97, 98	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (ℎ‘𝑘) Fn ℝ)
100		reex 11129	. . . . . . . . . . . . . . . . . . 19 ⊢ ℝ ∈ V
101	100	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → ℝ ∈ V)
102		inidm 4181	. . . . . . . . . . . . . . . . . 18 ⊢ (ℝ ∩ ℝ) = ℝ
103		eqidd 2738	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑓‘𝑘)‘𝑥) = ((𝑓‘𝑘)‘𝑥))
104		eqidd 2738	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((ℎ‘𝑘)‘𝑥) = ((ℎ‘𝑘)‘𝑥))
105	95, 99, 101, 101, 102, 103, 104	ofval 7643	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓‘𝑘) ∘f − (ℎ‘𝑘))‘𝑥) = (((𝑓‘𝑘)‘𝑥) − ((ℎ‘𝑘)‘𝑥)))
106	91, 105	eqtrd 2772	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (((𝑓 ∘f ∘f − ℎ)‘𝑘)‘𝑥) = (((𝑓‘𝑘)‘𝑥) − ((ℎ‘𝑘)‘𝑥)))
107	106	an32s 653	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (((𝑓 ∘f ∘f − ℎ)‘𝑘)‘𝑥) = (((𝑓‘𝑘)‘𝑥) − ((ℎ‘𝑘)‘𝑥)))
108		fveq2 6842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((𝑓 ∘f ∘f − ℎ)‘𝑛) = ((𝑓 ∘f ∘f − ℎ)‘𝑘))
109	108	fveq1d 6844	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥) = (((𝑓 ∘f ∘f − ℎ)‘𝑘)‘𝑥))
110		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥))
111		fvex 6855	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∘f ∘f − ℎ)‘𝑘)‘𝑥) ∈ V
112	109, 110, 111	fvmpt 6949	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥))‘𝑘) = (((𝑓 ∘f ∘f − ℎ)‘𝑘)‘𝑥))
113	112	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥))‘𝑘) = (((𝑓 ∘f ∘f − ℎ)‘𝑘)‘𝑥))
114		fveq2 6842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → (𝑓‘𝑛) = (𝑓‘𝑘))
115	114	fveq1d 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((𝑓‘𝑛)‘𝑥) = ((𝑓‘𝑘)‘𝑥))
116		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥))
117		fvex 6855	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑘)‘𝑥) ∈ V
118	115, 116, 117	fvmpt 6949	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥))‘𝑘) = ((𝑓‘𝑘)‘𝑥))
119		fveq2 6842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → (ℎ‘𝑛) = (ℎ‘𝑘))
120	119	fveq1d 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((ℎ‘𝑛)‘𝑥) = ((ℎ‘𝑘)‘𝑥))
121		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥))
122		fvex 6855	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℎ‘𝑘)‘𝑥) ∈ V
123	120, 121, 122	fvmpt 6949	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥))‘𝑘) = ((ℎ‘𝑘)‘𝑥))
124	118, 123	oveq12d 7386	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥))‘𝑘) − ((𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥))‘𝑘)) = (((𝑓‘𝑘)‘𝑥) − ((ℎ‘𝑘)‘𝑥)))
125	124	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥))‘𝑘) − ((𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥))‘𝑘)) = (((𝑓‘𝑘)‘𝑥) − ((ℎ‘𝑘)‘𝑥)))
126	107, 113, 125	3eqtr4d 2782	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥))‘𝑘) = (((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥))‘𝑘) − ((𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥))‘𝑘)))
127	126	adantlr 716	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥))‘𝑘) = (((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥))‘𝑘) − ((𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥))‘𝑘)))
128	61, 62, 63, 65, 66, 75, 84, 127	climsub 15569	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) → (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥)) ⇝ (if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) − if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))
129	1	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → 𝐹:ℝ⟶ℝ)
130	129	ffvelcdmda 7038	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
131		max0sub 13123	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑥) ∈ ℝ → (if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) − if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) = (𝐹‘𝑥))
132	130, 131	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) − if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) = (𝐹‘𝑥))
133	132	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) → (if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) − if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) = (𝐹‘𝑥))
134	128, 133	breqtrd 5126	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) → (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))
135	134	ex 412	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) → (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))
136	60, 135	sylbid 240	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥)) → (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))
137	136	ralimdva 3150	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → (∀𝑥 ∈ ℝ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥)) → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))
138		ovex 7401	. . . . . . . . 9 ⊢ (𝑓 ∘f ∘f − ℎ) ∈ V
139		feq1 6648	. . . . . . . . . 10 ⊢ (𝑔 = (𝑓 ∘f ∘f − ℎ) → (𝑔:ℕ⟶dom ∫1 ↔ (𝑓 ∘f ∘f − ℎ):ℕ⟶dom ∫1))
140		fveq1 6841	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑓 ∘f ∘f − ℎ) → (𝑔‘𝑛) = ((𝑓 ∘f ∘f − ℎ)‘𝑛))
141	140	fveq1d 6844	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑓 ∘f ∘f − ℎ) → ((𝑔‘𝑛)‘𝑥) = (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥))
142	141	mpteq2dv 5194	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ∘f ∘f − ℎ) → (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥)))
143	142	breq1d 5110	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑓 ∘f ∘f − ℎ) → ((𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥) ↔ (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))
144	143	ralbidv 3161	. . . . . . . . . 10 ⊢ (𝑔 = (𝑓 ∘f ∘f − ℎ) → (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥) ↔ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))
145	139, 144	anbi12d 633	. . . . . . . . 9 ⊢ (𝑔 = (𝑓 ∘f ∘f − ℎ) → ((𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ↔ ((𝑓 ∘f ∘f − ℎ):ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))))
146	138, 145	spcev 3562	. . . . . . . 8 ⊢ (((𝑓 ∘f ∘f − ℎ):ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))
147	41, 137, 146	syl6an 685	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → (∀𝑥 ∈ ℝ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥) ∧ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥)) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))))
148	33, 147	biimtrrid 243	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → ((∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥)) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))))
149	148	expimpd 453	. . . . 5 ⊢ (𝜑 → (((𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1) ∧ (∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥))) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))))
150	32, 149	syl5 34	. . . 4 ⊢ (𝜑 → (((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥)) ∧ (ℎ:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (ℎ‘𝑛) ∧ (ℎ‘𝑛) ∘r ≤ (ℎ‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥))) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))))
151	150	exlimdvv 1936	. . 3 ⊢ (𝜑 → (∃𝑓∃ℎ((𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥)) ∧ (ℎ:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (ℎ‘𝑛) ∧ (ℎ‘𝑛) ∘r ≤ (ℎ‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥))) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))))
152	27, 151	biimtrrid 243	. 2 ⊢ (𝜑 → ((∃𝑓(𝑓:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥)) ∧ ∃ℎ(ℎ:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (ℎ‘𝑛) ∧ (ℎ‘𝑛) ∘r ≤ (ℎ‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) ⇝ ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥))) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))))
153	15, 26, 152	mp2and 700	1 ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  Vcvv 3442  ifcif 4481   class class class wbr 5100   ↦ cmpt 5181  dom cdm 5632   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ∘_f cof 7630   ∘_r cofr 7631  ℂcc 11036  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041  +∞cpnf 11175   ≤ cle 11179   − cmin 11376  -cneg 11377  ℕcn 12157  [,)cico 13275   ⇝ cli 15419  MblFncmbf 25583  ∫₁citg1 25584  0_𝑝c0p 25638

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632  df-ofr 7633  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-map 8777  df-pm 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fi 9326  df-sup 9357  df-inf 9358  df-oi 9427  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-z 12501  df-uz 12764  df-q 12874  df-rp 12918  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-ioo 13277  df-ico 13279  df-icc 13280  df-fz 13436  df-fzo 13583  df-fl 13724  df-seq 13937  df-exp 13997  df-hash 14266  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-clim 15423  df-rlim 15424  df-sum 15622  df-rest 17354  df-topgen 17375  df-psmet 21313  df-xmet 21314  df-met 21315  df-bl 21316  df-mopn 21317  df-top 22850  df-topon 22867  df-bases 22902  df-cmp 23343  df-ovol 25433  df-vol 25434  df-mbf 25588  df-itg1 25589  df-0p 25639

This theorem is referenced by:  mbfi1flim  25692