Theorem mbfi1flimlem 25777

Description: Lemma for mbfi1flim 25778. (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 7118	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℝ)
3	1	feqmptd 6990	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)))
4		mbfi1flim.1	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ MblFn)
5	3, 4	eqeltrrd 2845	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ∈ MblFn)
6	2, 5	mbfpos 25705	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0)) ∈ MblFn)
7		0re 11292	. . . . . 6 ⊢ 0 ∈ ℝ
8		ifcl 4593	. . . . . 6 ⊢ (((𝐹‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0) ∈ ℝ)
9	2, 7, 8	sylancl 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0) ∈ ℝ)
10		max1 13247	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ (𝐹‘𝑦) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))
11	7, 2, 10	sylancr 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))
12		elrege0 13514	. . . . 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 582	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0) ∈ (0[,)+∞))
14	13	fmpttd 7149	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0)):ℝ⟶(0[,)+∞))
15	6, 14	mbfi1fseq 25776	. 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 11717	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → -(𝐹‘𝑦) ∈ ℝ)
17	2, 5	mbfneg 25704	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ -(𝐹‘𝑦)) ∈ MblFn)
18	16, 17	mbfpos 25705	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0)) ∈ MblFn)
19		ifcl 4593	. . . . . 6 ⊢ ((-(𝐹‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0) ∈ ℝ)
20	16, 7, 19	sylancl 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0) ∈ ℝ)
21		max1 13247	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ -(𝐹‘𝑦) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))
22	7, 16, 21	sylancr 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))
23		elrege0 13514	. . . . 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 582	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0) ∈ (0[,)+∞))
25	24	fmpttd 7149	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0)):ℝ⟶(0[,)+∞))
26	18, 25	mbfi1fseq 25776	. 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 1955	. . 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 1149	. . . . . . 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 1149	. . . . . . 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 612	. . . . . 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 655	. . . . . 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 3117	. . . . . . 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 25763	. . . . . . . . . 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 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → 𝑓:ℕ⟶dom ∫1)
37		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → ℎ:ℕ⟶dom ∫1)
38		nnex 12299	. . . . . . . . . 10 ⊢ ℕ ∈ V
39	38	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → ℕ ∈ V)
40		inidm 4248	. . . . . . . . 9 ⊢ (ℕ ∩ ℕ) = ℕ
41	35, 36, 37, 39, 39, 40	off 7732	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → (𝑓 ∘f ∘f − ℎ):ℕ⟶dom ∫1)
42		fveq2 6920	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
43	42	breq2d 5178	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (0 ≤ (𝐹‘𝑦) ↔ 0 ≤ (𝐹‘𝑥)))
44	43, 42	ifbieq1d 4572	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0) = if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
45		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0)) = (𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))
46		fvex 6933	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑥) ∈ V
47		c0ex 11284	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
48	46, 47	ifex 4598	. . . . . . . . . . . . . 14 ⊢ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∈ V
49	44, 45, 48	fvmpt 7029	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → ((𝑦 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑦), (𝐹‘𝑦), 0))‘𝑥) = if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
50	49	breq2d 5178	. . . . . . . . . . . 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 11530	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → -(𝐹‘𝑦) = -(𝐹‘𝑥))
52	51	breq2d 5178	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (0 ≤ -(𝐹‘𝑦) ↔ 0 ≤ -(𝐹‘𝑥)))
53	52, 51	ifbieq1d 4572	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0) = if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
54		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0)) = (𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))
55		negex 11534	. . . . . . . . . . . . . . 15 ⊢ -(𝐹‘𝑥) ∈ V
56	55, 47	ifex 4598	. . . . . . . . . . . . . 14 ⊢ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) ∈ V
57	53, 54, 56	fvmpt 7029	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → ((𝑦 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑦), -(𝐹‘𝑦), 0))‘𝑥) = if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
58	57	breq2d 5178	. . . . . . . . . . . 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 631	. . . . . . . . . . 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 12946	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ≥‘1)
62		1zzd 12674	. . . . . . . . . . . . 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 770	. . . . . . . . . . . . 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 7260	. . . . . . . . . . . . . 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 772	. . . . . . . . . . . . 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 7118	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ dom ∫1)
68		i1ff 25730	. . . . . . . . . . . . . . . . . . . 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 7118	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑓‘𝑛)‘𝑥) ∈ ℝ)
71	70	an32s 651	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑓‘𝑛)‘𝑥) ∈ ℝ)
72	71	recnd 11318	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑓‘𝑛)‘𝑥) ∈ ℂ)
73	72	fmpttd 7149	. . . . . . . . . . . . . . 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 7118	. . . . . . . . . . . . 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 7118	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) → (ℎ‘𝑛) ∈ dom ∫1)
77		i1ff 25730	. . . . . . . . . . . . . . . . . . . 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 7118	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((ℎ‘𝑛)‘𝑥) ∈ ℝ)
80	79	an32s 651	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((ℎ‘𝑛)‘𝑥) ∈ ℝ)
81	80	recnd 11318	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((ℎ‘𝑛)‘𝑥) ∈ ℂ)
82	81	fmpttd 7149	. . . . . . . . . . . . . . 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 7118	. . . . . . . . . . . . 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 6748	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → 𝑓 Fn ℕ)
86	37	ffnd 6748	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) → ℎ Fn ℕ)
87		eqidd 2741	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) = (𝑓‘𝑘))
88		eqidd 2741	. . . . . . . . . . . . . . . . . . . 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 7725	. . . . . . . . . . . . . . . . . . 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 6922	. . . . . . . . . . . . . . . . . 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 7118	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ dom ∫1)
93		i1ff 25730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑘) ∈ dom ∫1 → (𝑓‘𝑘):ℝ⟶ℝ)
94		ffn 6747	. . . . . . . . . . . . . . . . . . 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 7118	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → (ℎ‘𝑘) ∈ dom ∫1)
97		i1ff 25730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ‘𝑘) ∈ dom ∫1 → (ℎ‘𝑘):ℝ⟶ℝ)
98		ffn 6747	. . . . . . . . . . . . . . . . . . 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 11275	. . . . . . . . . . . . . . . . . . 19 ⊢ ℝ ∈ V
101	100	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) → ℝ ∈ V)
102		inidm 4248	. . . . . . . . . . . . . . . . . 18 ⊢ (ℝ ∩ ℝ) = ℝ
103		eqidd 2741	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑓‘𝑘)‘𝑥) = ((𝑓‘𝑘)‘𝑥))
104		eqidd 2741	. . . . . . . . . . . . . . . . . 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 7725	. . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . 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 651	. . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((𝑓 ∘f ∘f − ℎ)‘𝑛) = ((𝑓 ∘f ∘f − ℎ)‘𝑘))
109	108	fveq1d 6922	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥) = (((𝑓 ∘f ∘f − ℎ)‘𝑘)‘𝑥))
110		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥))
111		fvex 6933	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∘f ∘f − ℎ)‘𝑘)‘𝑥) ∈ V
112	109, 110, 111	fvmpt 7029	. . . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → (𝑓‘𝑛) = (𝑓‘𝑘))
115	114	fveq1d 6922	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((𝑓‘𝑛)‘𝑥) = ((𝑓‘𝑘)‘𝑥))
116		eqid 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥))
117		fvex 6933	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑘)‘𝑥) ∈ V
118	115, 116, 117	fvmpt 7029	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥))‘𝑘) = ((𝑓‘𝑘)‘𝑥))
119		fveq2 6920	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → (ℎ‘𝑛) = (ℎ‘𝑘))
120	119	fveq1d 6922	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((ℎ‘𝑛)‘𝑥) = ((ℎ‘𝑘)‘𝑥))
121		eqid 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥))
122		fvex 6933	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℎ‘𝑘)‘𝑥) ∈ V
123	120, 121, 122	fvmpt 7029	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((ℎ‘𝑛)‘𝑥))‘𝑘) = ((ℎ‘𝑘)‘𝑥))
124	118, 123	oveq12d 7466	. . . . . . . . . . . . . . . 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 2790	. . . . . . . . . . . . . 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 714	. . . . . . . . . . . . 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 15680	. . . . . . . . . . . 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 7118	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶dom ∫1 ∧ ℎ:ℕ⟶dom ∫1)) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
131		max0sub 13258	. . . . . . . . . . . . . 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 5192	. . . . . . . . . . 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 3173	. . . . . . . 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 7481	. . . . . . . . 9 ⊢ (𝑓 ∘f ∘f − ℎ) ∈ V
139		feq1 6728	. . . . . . . . . 10 ⊢ (𝑔 = (𝑓 ∘f ∘f − ℎ) → (𝑔:ℕ⟶dom ∫1 ↔ (𝑓 ∘f ∘f − ℎ):ℕ⟶dom ∫1))
140		fveq1 6919	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑓 ∘f ∘f − ℎ) → (𝑔‘𝑛) = ((𝑓 ∘f ∘f − ℎ)‘𝑛))
141	140	fveq1d 6922	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑓 ∘f ∘f − ℎ) → ((𝑔‘𝑛)‘𝑥) = (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥))
142	141	mpteq2dv 5268	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ∘f ∘f − ℎ) → (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ (((𝑓 ∘f ∘f − ℎ)‘𝑛)‘𝑥)))
143	142	breq1d 5176	. . . . . . . . . . 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 3184	. . . . . . . . . 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 631	. . . . . . . . 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 3619	. . . . . . . 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 683	. . . . . . 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 1933	. . 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 698	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 1537  ∃wex 1777   ∈ wcel 2108  ∀wral 3067  Vcvv 3488  ifcif 4548   class class class wbr 5166   ↦ cmpt 5249  dom cdm 5700   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∘_f cof 7712   ∘_r cofr 7713  ℂcc 11182  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187  +∞cpnf 11321   ≤ cle 11325   − cmin 11520  -cneg 11521  ℕcn 12293  [,)cico 13409   ⇝ cli 15530  MblFncmbf 25668  ∫₁citg1 25669  0_𝑝c0p 25723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-ofr 7715  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fi 9480  df-sup 9511  df-inf 9512  df-oi 9579  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-ioo 13411  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-fl 13843  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-rlim 15535  df-sum 15735  df-rest 17482  df-topgen 17503  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-top 22921  df-topon 22938  df-bases 22974  df-cmp 23416  df-ovol 25518  df-vol 25519  df-mbf 25673  df-itg1 25674  df-0p 25724

This theorem is referenced by:  mbfi1flim  25778