ftc1anclem4 - Mathbox for Brendan Leahy

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Theorem ftc1anclem4 36656

Description: Lemma for ftc1anc 36661. (Contributed by Brendan Leahy, 17-Jun-2018.)

**Assertion**
Ref	Expression
ftc1anclem4	⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) ∈ ℝ)

Distinct variable groups: 𝑡,𝐹 𝑡,𝐺

Proof of Theorem ftc1anclem4 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffvelcdm 7083	. . . . . . . . . 10 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → (𝐺‘𝑡) ∈ ℝ)
2	1	recnd 11244	. . . . . . . . 9 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → (𝐺‘𝑡) ∈ ℂ)
3		i1ff 25200	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹:ℝ⟶ℝ)
4	3	ffvelcdmda 7086	. . . . . . . . . 10 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) → (𝐹‘𝑡) ∈ ℝ)
5	4	recnd 11244	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) → (𝐹‘𝑡) ∈ ℂ)
6		subcl 11461	. . . . . . . . 9 ⊢ (((𝐺‘𝑡) ∈ ℂ ∧ (𝐹‘𝑡) ∈ ℂ) → ((𝐺‘𝑡) − (𝐹‘𝑡)) ∈ ℂ)
7	2, 5, 6	syl2anr 597	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) ∧ (𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ)) → ((𝐺‘𝑡) − (𝐹‘𝑡)) ∈ ℂ)
8	7	anandirs 677	. . . . . . 7 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → ((𝐺‘𝑡) − (𝐹‘𝑡)) ∈ ℂ)
9	8	abscld 15385	. . . . . 6 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ ℝ)
10	9	rexrd 11266	. . . . 5 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ ℝ^*)
11	8	absge0d 15393	. . . . 5 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))
12		elxrge0 13436	. . . . 5 ⊢ ((abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ (0[,]+∞) ↔ ((abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ ℝ^* ∧ 0 ≤ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))))
13	10, 11, 12	sylanbrc 583	. . . 4 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ (0[,]+∞))
14	13	fmpttd 7116	. . 3 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))):ℝ⟶(0[,]+∞))
15	14	3adant2 1131	. 2 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))):ℝ⟶(0[,]+∞))
16		reex 11203	. . . . . . 7 ⊢ ℝ ∈ V
17	16	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → ℝ ∈ V)
18		fvexd 6906	. . . . . 6 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘(𝐺‘𝑡)) ∈ V)
19		fvexd 6906	. . . . . 6 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘(𝐹‘𝑡)) ∈ V)
20		eqidd 2733	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) = (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))))
21		eqidd 2733	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))) = (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))))
22	17, 18, 19, 20, 21	offval2 7692	. . . . 5 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → ((𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) ∘_f + (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡)))) = (𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))))
23	22	fveq2d 6895	. . . 4 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘((𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) ∘_f + (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))))) = (∫₂‘(𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))))
24		id 22	. . . . . . . . . 10 ⊢ (𝐺:ℝ⟶ℝ → 𝐺:ℝ⟶ℝ)
25	24	feqmptd 6960	. . . . . . . . 9 ⊢ (𝐺:ℝ⟶ℝ → 𝐺 = (𝑡 ∈ ℝ ↦ (𝐺‘𝑡)))
26		absf 15286	. . . . . . . . . . 11 ⊢ abs:ℂ⟶ℝ
27	26	a1i 11	. . . . . . . . . 10 ⊢ (𝐺:ℝ⟶ℝ → abs:ℂ⟶ℝ)
28	27	feqmptd 6960	. . . . . . . . 9 ⊢ (𝐺:ℝ⟶ℝ → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)))
29		fveq2 6891	. . . . . . . . 9 ⊢ (𝑥 = (𝐺‘𝑡) → (abs‘𝑥) = (abs‘(𝐺‘𝑡)))
30	2, 25, 28, 29	fmptco 7129	. . . . . . . 8 ⊢ (𝐺:ℝ⟶ℝ → (abs ∘ 𝐺) = (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))))
31	30	adantl 482	. . . . . . 7 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (abs ∘ 𝐺) = (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))))
32		iblmbf 25292	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝐿¹ → 𝐺 ∈ MblFn)
33		ftc1anclem1 36653	. . . . . . . . 9 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝐺 ∈ MblFn) → (abs ∘ 𝐺) ∈ MblFn)
34	32, 33	sylan2 593	. . . . . . . 8 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝐺 ∈ 𝐿¹) → (abs ∘ 𝐺) ∈ MblFn)
35	34	ancoms 459	. . . . . . 7 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (abs ∘ 𝐺) ∈ MblFn)
36	31, 35	eqeltrrd 2834	. . . . . 6 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) ∈ MblFn)
37	36	3adant1 1130	. . . . 5 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) ∈ MblFn)
38	2	abscld 15385	. . . . . . . 8 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → (abs‘(𝐺‘𝑡)) ∈ ℝ)
39	2	absge0d 15393	. . . . . . . 8 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(𝐺‘𝑡)))
40		elrege0 13433	. . . . . . . 8 ⊢ ((abs‘(𝐺‘𝑡)) ∈ (0[,)+∞) ↔ ((abs‘(𝐺‘𝑡)) ∈ ℝ ∧ 0 ≤ (abs‘(𝐺‘𝑡))))
41	38, 39, 40	sylanbrc 583	. . . . . . 7 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → (abs‘(𝐺‘𝑡)) ∈ (0[,)+∞))
42	41	fmpttd 7116	. . . . . 6 ⊢ (𝐺:ℝ⟶ℝ → (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))):ℝ⟶(0[,)+∞))
43	42	3ad2ant3 1135	. . . . 5 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))):ℝ⟶(0[,)+∞))
44		iftrue 4534	. . . . . . . . 9 ⊢ (𝑡 ∈ ℝ → if(𝑡 ∈ ℝ, (abs‘(𝐺‘𝑡)), 0) = (abs‘(𝐺‘𝑡)))
45	44	mpteq2ia 5251	. . . . . . . 8 ⊢ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐺‘𝑡)), 0)) = (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡)))
46	45	fveq2i 6894	. . . . . . 7 ⊢ (∫₂‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐺‘𝑡)), 0))) = (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))))
47	1	adantll 712	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (𝐺‘𝑡) ∈ ℝ)
48		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → 𝐺:ℝ⟶ℝ)
49	48	feqmptd 6960	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → 𝐺 = (𝑡 ∈ ℝ ↦ (𝐺‘𝑡)))
50		simpl 483	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → 𝐺 ∈ 𝐿¹)
51	49, 50	eqeltrrd 2834	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (𝐺‘𝑡)) ∈ 𝐿¹)
52	47, 51, 36	iblabsnc 36644	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) ∈ 𝐿¹)
53	38	adantll 712	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘(𝐺‘𝑡)) ∈ ℝ)
54	39	adantll 712	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(𝐺‘𝑡)))
55	53, 54	iblpos 25317	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → ((𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) ∈ 𝐿¹ ↔ ((𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) ∈ MblFn ∧ (∫₂‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐺‘𝑡)), 0))) ∈ ℝ)))
56	52, 55	mpbid 231	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → ((𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) ∈ MblFn ∧ (∫₂‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐺‘𝑡)), 0))) ∈ ℝ))
57	56	simprd 496	. . . . . . 7 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐺‘𝑡)), 0))) ∈ ℝ)
58	46, 57	eqeltrrid 2838	. . . . . 6 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡)))) ∈ ℝ)
59	58	3adant1 1130	. . . . 5 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡)))) ∈ ℝ)
60	5	abscld 15385	. . . . . . . 8 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) → (abs‘(𝐹‘𝑡)) ∈ ℝ)
61	5	absge0d 15393	. . . . . . . 8 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(𝐹‘𝑡)))
62		elrege0 13433	. . . . . . . 8 ⊢ ((abs‘(𝐹‘𝑡)) ∈ (0[,)+∞) ↔ ((abs‘(𝐹‘𝑡)) ∈ ℝ ∧ 0 ≤ (abs‘(𝐹‘𝑡))))
63	60, 61, 62	sylanbrc 583	. . . . . . 7 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) → (abs‘(𝐹‘𝑡)) ∈ (0[,)+∞))
64	63	fmpttd 7116	. . . . . 6 ⊢ (𝐹 ∈ dom ∫₁ → (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))):ℝ⟶(0[,)+∞))
65	64	3ad2ant1 1133	. . . . 5 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))):ℝ⟶(0[,)+∞))
66		iftrue 4534	. . . . . . . . 9 ⊢ (𝑡 ∈ ℝ → if(𝑡 ∈ ℝ, (abs‘(𝐹‘𝑡)), 0) = (abs‘(𝐹‘𝑡)))
67	66	mpteq2ia 5251	. . . . . . . 8 ⊢ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐹‘𝑡)), 0)) = (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡)))
68	67	fveq2i 6894	. . . . . . 7 ⊢ (∫₂‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐹‘𝑡)), 0))) = (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))))
69	3	feqmptd 6960	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹 = (𝑡 ∈ ℝ ↦ (𝐹‘𝑡)))
70		i1fibl 25332	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹 ∈ 𝐿¹)
71	69, 70	eqeltrrd 2834	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫₁ → (𝑡 ∈ ℝ ↦ (𝐹‘𝑡)) ∈ 𝐿¹)
72	26	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ dom ∫₁ → abs:ℂ⟶ℝ)
73	72	feqmptd 6960	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ dom ∫₁ → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)))
74		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐹‘𝑡) → (abs‘𝑥) = (abs‘(𝐹‘𝑡)))
75	5, 69, 73, 74	fmptco 7129	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫₁ → (abs ∘ 𝐹) = (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))))
76		i1fmbf 25199	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹 ∈ MblFn)
77		ftc1anclem1 36653	. . . . . . . . . . . 12 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝐹 ∈ MblFn) → (abs ∘ 𝐹) ∈ MblFn)
78	3, 76, 77	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫₁ → (abs ∘ 𝐹) ∈ MblFn)
79	75, 78	eqeltrrd 2834	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫₁ → (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))) ∈ MblFn)
80	4, 71, 79	iblabsnc 36644	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫₁ → (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))) ∈ 𝐿¹)
81	60, 61	iblpos 25317	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫₁ → ((𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))) ∈ 𝐿¹ ↔ ((𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))) ∈ MblFn ∧ (∫₂‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐹‘𝑡)), 0))) ∈ ℝ)))
82	80, 81	mpbid 231	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫₁ → ((𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))) ∈ MblFn ∧ (∫₂‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐹‘𝑡)), 0))) ∈ ℝ))
83	82	simprd 496	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫₁ → (∫₂‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐹‘𝑡)), 0))) ∈ ℝ)
84	68, 83	eqeltrrid 2838	. . . . . 6 ⊢ (𝐹 ∈ dom ∫₁ → (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡)))) ∈ ℝ)
85	84	3ad2ant1 1133	. . . . 5 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡)))) ∈ ℝ)
86	37, 43, 59, 65, 85	itg2addnc 36634	. . . 4 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘((𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) ∘_f + (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))))) = ((∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡)))) + (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))))))
87	23, 86	eqtr3d 2774	. . 3 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘(𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))) = ((∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡)))) + (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))))))
88	59, 85	readdcld 11245	. . 3 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → ((∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡)))) + (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))))) ∈ ℝ)
89	87, 88	eqeltrd 2833	. 2 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘(𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))) ∈ ℝ)
90		readdcl 11195	. . . . . . . . 9 ⊢ (((abs‘(𝐺‘𝑡)) ∈ ℝ ∧ (abs‘(𝐹‘𝑡)) ∈ ℝ) → ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ ℝ)
91	38, 60, 90	syl2anr 597	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) ∧ (𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ)) → ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ ℝ)
92	91	anandirs 677	. . . . . . 7 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ ℝ)
93	92	rexrd 11266	. . . . . 6 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ ℝ^*)
94	38	adantll 712	. . . . . . 7 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘(𝐺‘𝑡)) ∈ ℝ)
95	60	adantlr 713	. . . . . . 7 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘(𝐹‘𝑡)) ∈ ℝ)
96	39	adantll 712	. . . . . . 7 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(𝐺‘𝑡)))
97	61	adantlr 713	. . . . . . 7 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(𝐹‘𝑡)))
98	94, 95, 96, 97	addge0d 11792	. . . . . 6 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))
99		elxrge0 13436	. . . . . 6 ⊢ (((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ (0[,]+∞) ↔ (((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ ℝ^* ∧ 0 ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))))
100	93, 98, 99	sylanbrc 583	. . . . 5 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ (0[,]+∞))
101	100	fmpttd 7116	. . . 4 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))):ℝ⟶(0[,]+∞))
102	101	3adant2 1131	. . 3 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))):ℝ⟶(0[,]+∞))
103		abs2dif2 15282	. . . . . . . 8 ⊢ (((𝐺‘𝑡) ∈ ℂ ∧ (𝐹‘𝑡) ∈ ℂ) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))
104	2, 5, 103	syl2anr 597	. . . . . . 7 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) ∧ (𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ)) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))
105	104	anandirs 677	. . . . . 6 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))
106	105	ralrimiva 3146	. . . . 5 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) → ∀𝑡 ∈ ℝ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))
107	16	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) → ℝ ∈ V)
108		eqidd 2733	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))) = (𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))))
109		eqidd 2733	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))) = (𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))))
110	107, 9, 92, 108, 109	ofrfval2 7693	. . . . 5 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) → ((𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))) ∘_r ≤ (𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))) ↔ ∀𝑡 ∈ ℝ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))))
111	106, 110	mpbird 256	. . . 4 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))) ∘_r ≤ (𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))))
112	111	3adant2 1131	. . 3 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))) ∘_r ≤ (𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))))
113		itg2le 25264	. . 3 ⊢ (((𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))) ∘_r ≤ (𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))) → (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) ≤ (∫₂‘(𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))))
114	15, 102, 112, 113	syl3anc 1371	. 2 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) ≤ (∫₂‘(𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))))
115		itg2lecl 25263	. 2 ⊢ (((𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))):ℝ⟶(0[,]+∞) ∧ (∫₂‘(𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))) ∈ ℝ ∧ (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) ≤ (∫₂‘(𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))))) → (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) ∈ ℝ)
116	15, 89, 114, 115	syl3anc 1371	1 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ifcif 4528 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ∘_f cof 7670 ∘_r cofr 7671 ℂcc 11110 ℝcr 11111 0cc0 11112 + caddc 11115 +∞cpnf 11247 ℝ^*cxr 11249 ≤ cle 11251 − cmin 11446 [,)cico 13328 [,]cicc 13329 abscabs 15183 MblFncmbf 25138 ∫₁citg1 25139 ∫₂citg2 25140 𝐿¹cibl 25141

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-z 12561 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-fl 13759 df-seq 13969 df-exp 14030 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-sum 15635 df-rest 17370 df-topgen 17391 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-top 22403 df-topon 22420 df-bases 22456 df-cmp 22898 df-ovol 24988 df-vol 24989 df-mbf 25143 df-itg1 25144 df-itg2 25145 df-ibl 25146 df-0p 25194

This theorem is referenced by: ftc1anclem5 36657 ftc1anclem6 36658

W3C validator