ftc1anclem4 - Mathbox for Brendan Leahy

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Theorem ftc1anclem4 36564

Description: Lemma for ftc1anc 36569. (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 7084	. . . . . . . . . 10 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → (𝐺‘𝑡) ∈ ℝ)
2	1	recnd 11242	. . . . . . . . 9 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → (𝐺‘𝑡) ∈ ℂ)
3		i1ff 25193	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹:ℝ⟶ℝ)
4	3	ffvelcdmda 7087	. . . . . . . . . 10 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) → (𝐹‘𝑡) ∈ ℝ)
5	4	recnd 11242	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) → (𝐹‘𝑡) ∈ ℂ)
6		subcl 11459	. . . . . . . . 9 ⊢ (((𝐺‘𝑡) ∈ ℂ ∧ (𝐹‘𝑡) ∈ ℂ) → ((𝐺‘𝑡) − (𝐹‘𝑡)) ∈ ℂ)
7	2, 5, 6	syl2anr 598	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) ∧ (𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ)) → ((𝐺‘𝑡) − (𝐹‘𝑡)) ∈ ℂ)
8	7	anandirs 678	. . . . . . 7 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → ((𝐺‘𝑡) − (𝐹‘𝑡)) ∈ ℂ)
9	8	abscld 15383	. . . . . 6 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ ℝ)
10	9	rexrd 11264	. . . . 5 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ ℝ^*)
11	8	absge0d 15391	. . . . 5 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))
12		elxrge0 13434	. . . . 5 ⊢ ((abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ (0[,]+∞) ↔ ((abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ ℝ^* ∧ 0 ≤ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))))
13	10, 11, 12	sylanbrc 584	. . . 4 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ (0[,]+∞))
14	13	fmpttd 7115	. . 3 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))):ℝ⟶(0[,]+∞))
15	14	3adant2 1132	. 2 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))):ℝ⟶(0[,]+∞))
16		reex 11201	. . . . . . 7 ⊢ ℝ ∈ V
17	16	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → ℝ ∈ V)
18		fvexd 6907	. . . . . 6 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘(𝐺‘𝑡)) ∈ V)
19		fvexd 6907	. . . . . 6 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘(𝐹‘𝑡)) ∈ V)
20		eqidd 2734	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) = (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))))
21		eqidd 2734	. . . . . 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 7690	. . . . 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 6896	. . . 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 6961	. . . . . . . . 9 ⊢ (𝐺:ℝ⟶ℝ → 𝐺 = (𝑡 ∈ ℝ ↦ (𝐺‘𝑡)))
26		absf 15284	. . . . . . . . . . 11 ⊢ abs:ℂ⟶ℝ
27	26	a1i 11	. . . . . . . . . 10 ⊢ (𝐺:ℝ⟶ℝ → abs:ℂ⟶ℝ)
28	27	feqmptd 6961	. . . . . . . . 9 ⊢ (𝐺:ℝ⟶ℝ → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)))
29		fveq2 6892	. . . . . . . . 9 ⊢ (𝑥 = (𝐺‘𝑡) → (abs‘𝑥) = (abs‘(𝐺‘𝑡)))
30	2, 25, 28, 29	fmptco 7127	. . . . . . . 8 ⊢ (𝐺:ℝ⟶ℝ → (abs ∘ 𝐺) = (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))))
31	30	adantl 483	. . . . . . 7 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (abs ∘ 𝐺) = (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))))
32		iblmbf 25285	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝐿¹ → 𝐺 ∈ MblFn)
33		ftc1anclem1 36561	. . . . . . . . 9 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝐺 ∈ MblFn) → (abs ∘ 𝐺) ∈ MblFn)
34	32, 33	sylan2 594	. . . . . . . 8 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝐺 ∈ 𝐿¹) → (abs ∘ 𝐺) ∈ MblFn)
35	34	ancoms 460	. . . . . . 7 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (abs ∘ 𝐺) ∈ MblFn)
36	31, 35	eqeltrrd 2835	. . . . . 6 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) ∈ MblFn)
37	36	3adant1 1131	. . . . 5 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) ∈ MblFn)
38	2	abscld 15383	. . . . . . . 8 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → (abs‘(𝐺‘𝑡)) ∈ ℝ)
39	2	absge0d 15391	. . . . . . . 8 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(𝐺‘𝑡)))
40		elrege0 13431	. . . . . . . 8 ⊢ ((abs‘(𝐺‘𝑡)) ∈ (0[,)+∞) ↔ ((abs‘(𝐺‘𝑡)) ∈ ℝ ∧ 0 ≤ (abs‘(𝐺‘𝑡))))
41	38, 39, 40	sylanbrc 584	. . . . . . 7 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → (abs‘(𝐺‘𝑡)) ∈ (0[,)+∞))
42	41	fmpttd 7115	. . . . . 6 ⊢ (𝐺:ℝ⟶ℝ → (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))):ℝ⟶(0[,)+∞))
43	42	3ad2ant3 1136	. . . . 5 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))):ℝ⟶(0[,)+∞))
44		iftrue 4535	. . . . . . . . 9 ⊢ (𝑡 ∈ ℝ → if(𝑡 ∈ ℝ, (abs‘(𝐺‘𝑡)), 0) = (abs‘(𝐺‘𝑡)))
45	44	mpteq2ia 5252	. . . . . . . 8 ⊢ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐺‘𝑡)), 0)) = (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡)))
46	45	fveq2i 6895	. . . . . . 7 ⊢ (∫₂‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐺‘𝑡)), 0))) = (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))))
47	1	adantll 713	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (𝐺‘𝑡) ∈ ℝ)
48		simpr 486	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → 𝐺:ℝ⟶ℝ)
49	48	feqmptd 6961	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → 𝐺 = (𝑡 ∈ ℝ ↦ (𝐺‘𝑡)))
50		simpl 484	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → 𝐺 ∈ 𝐿¹)
51	49, 50	eqeltrrd 2835	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (𝐺‘𝑡)) ∈ 𝐿¹)
52	47, 51, 36	iblabsnc 36552	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) ∈ 𝐿¹)
53	38	adantll 713	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘(𝐺‘𝑡)) ∈ ℝ)
54	39	adantll 713	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(𝐺‘𝑡)))
55	53, 54	iblpos 25310	. . . . . . . . 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 497	. . . . . . 7 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐺‘𝑡)), 0))) ∈ ℝ)
58	46, 57	eqeltrrid 2839	. . . . . 6 ⊢ ((𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡)))) ∈ ℝ)
59	58	3adant1 1131	. . . . 5 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡)))) ∈ ℝ)
60	5	abscld 15383	. . . . . . . 8 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) → (abs‘(𝐹‘𝑡)) ∈ ℝ)
61	5	absge0d 15391	. . . . . . . 8 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(𝐹‘𝑡)))
62		elrege0 13431	. . . . . . . 8 ⊢ ((abs‘(𝐹‘𝑡)) ∈ (0[,)+∞) ↔ ((abs‘(𝐹‘𝑡)) ∈ ℝ ∧ 0 ≤ (abs‘(𝐹‘𝑡))))
63	60, 61, 62	sylanbrc 584	. . . . . . 7 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) → (abs‘(𝐹‘𝑡)) ∈ (0[,)+∞))
64	63	fmpttd 7115	. . . . . 6 ⊢ (𝐹 ∈ dom ∫₁ → (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))):ℝ⟶(0[,)+∞))
65	64	3ad2ant1 1134	. . . . 5 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))):ℝ⟶(0[,)+∞))
66		iftrue 4535	. . . . . . . . 9 ⊢ (𝑡 ∈ ℝ → if(𝑡 ∈ ℝ, (abs‘(𝐹‘𝑡)), 0) = (abs‘(𝐹‘𝑡)))
67	66	mpteq2ia 5252	. . . . . . . 8 ⊢ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐹‘𝑡)), 0)) = (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡)))
68	67	fveq2i 6895	. . . . . . 7 ⊢ (∫₂‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐹‘𝑡)), 0))) = (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))))
69	3	feqmptd 6961	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹 = (𝑡 ∈ ℝ ↦ (𝐹‘𝑡)))
70		i1fibl 25325	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹 ∈ 𝐿¹)
71	69, 70	eqeltrrd 2835	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫₁ → (𝑡 ∈ ℝ ↦ (𝐹‘𝑡)) ∈ 𝐿¹)
72	26	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ dom ∫₁ → abs:ℂ⟶ℝ)
73	72	feqmptd 6961	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ dom ∫₁ → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)))
74		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐹‘𝑡) → (abs‘𝑥) = (abs‘(𝐹‘𝑡)))
75	5, 69, 73, 74	fmptco 7127	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫₁ → (abs ∘ 𝐹) = (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))))
76		i1fmbf 25192	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹 ∈ MblFn)
77		ftc1anclem1 36561	. . . . . . . . . . . 12 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝐹 ∈ MblFn) → (abs ∘ 𝐹) ∈ MblFn)
78	3, 76, 77	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫₁ → (abs ∘ 𝐹) ∈ MblFn)
79	75, 78	eqeltrrd 2835	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫₁ → (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))) ∈ MblFn)
80	4, 71, 79	iblabsnc 36552	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫₁ → (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))) ∈ 𝐿¹)
81	60, 61	iblpos 25310	. . . . . . . . 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 497	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫₁ → (∫₂‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐹‘𝑡)), 0))) ∈ ℝ)
84	68, 83	eqeltrrid 2839	. . . . . 6 ⊢ (𝐹 ∈ dom ∫₁ → (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡)))) ∈ ℝ)
85	84	3ad2ant1 1134	. . . . 5 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡)))) ∈ ℝ)
86	37, 43, 59, 65, 85	itg2addnc 36542	. . . 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 2775	. . 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 11243	. . 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 2834	. 2 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (∫₂‘(𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))) ∈ ℝ)
90		readdcl 11193	. . . . . . . . 9 ⊢ (((abs‘(𝐺‘𝑡)) ∈ ℝ ∧ (abs‘(𝐹‘𝑡)) ∈ ℝ) → ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ ℝ)
91	38, 60, 90	syl2anr 598	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) ∧ (𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ)) → ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ ℝ)
92	91	anandirs 678	. . . . . . 7 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ ℝ)
93	92	rexrd 11264	. . . . . 6 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ ℝ^*)
94	38	adantll 713	. . . . . . 7 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘(𝐺‘𝑡)) ∈ ℝ)
95	60	adantlr 714	. . . . . . 7 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘(𝐹‘𝑡)) ∈ ℝ)
96	39	adantll 713	. . . . . . 7 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(𝐺‘𝑡)))
97	61	adantlr 714	. . . . . . 7 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(𝐹‘𝑡)))
98	94, 95, 96, 97	addge0d 11790	. . . . . 6 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))
99		elxrge0 13434	. . . . . 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 584	. . . . 5 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ (0[,]+∞))
101	100	fmpttd 7115	. . . 4 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))):ℝ⟶(0[,]+∞))
102	101	3adant2 1132	. . 3 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))):ℝ⟶(0[,]+∞))
103		abs2dif2 15280	. . . . . . . 8 ⊢ (((𝐺‘𝑡) ∈ ℂ ∧ (𝐹‘𝑡) ∈ ℂ) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))
104	2, 5, 103	syl2anr 598	. . . . . . 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 678	. . . . . 6 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))
106	105	ralrimiva 3147	. . . . 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 2734	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))) = (𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))))
109		eqidd 2734	. . . . . 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 7691	. . . . 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 257	. . . 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 1132	. . 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 25257	. . 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 1372	. 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 25256	. 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 1372	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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ifcif 4529 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5677 ∘ ccom 5681 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ∘_f cof 7668 ∘_r cofr 7669 ℂcc 11108 ℝcr 11109 0cc0 11110 + caddc 11113 +∞cpnf 11245 ℝ^*cxr 11247 ≤ cle 11249 − cmin 11444 [,)cico 13326 [,]cicc 13327 abscabs 15181 MblFncmbf 25131 ∫₁citg1 25132 ∫₂citg2 25133 𝐿¹cibl 25134

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-ofr 7671 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-rest 17368 df-topgen 17389 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-top 22396 df-topon 22413 df-bases 22449 df-cmp 22891 df-ovol 24981 df-vol 24982 df-mbf 25136 df-itg1 25137 df-itg2 25138 df-ibl 25139 df-0p 25187

This theorem is referenced by: ftc1anclem5 36565 ftc1anclem6 36566

W3C validator