ftc1anclem4 - Mathbox for Brendan Leahy

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Theorem ftc1anclem4 36552

Description: Lemma for ftc1anc 36557. (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 7080	. . . . . . . . . 10 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → (𝐺‘𝑡) ∈ ℝ)
2	1	recnd 11238	. . . . . . . . 9 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → (𝐺‘𝑡) ∈ ℂ)
3		i1ff 25184	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹:ℝ⟶ℝ)
4	3	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) → (𝐹‘𝑡) ∈ ℝ)
5	4	recnd 11238	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) → (𝐹‘𝑡) ∈ ℂ)
6		subcl 11455	. . . . . . . . 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 15379	. . . . . 6 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ ℝ)
10	9	rexrd 11260	. . . . 5 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ ℝ^*)
11	8	absge0d 15387	. . . . 5 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))
12		elxrge0 13430	. . . . 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 7111	. . 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 11197	. . . . . . 7 ⊢ ℝ ∈ V
17	16	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) → ℝ ∈ V)
18		fvexd 6903	. . . . . 6 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺 ∈ 𝐿¹ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (abs‘(𝐺‘𝑡)) ∈ V)
19		fvexd 6903	. . . . . 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 7686	. . . . 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 6892	. . . 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 6957	. . . . . . . . 9 ⊢ (𝐺:ℝ⟶ℝ → 𝐺 = (𝑡 ∈ ℝ ↦ (𝐺‘𝑡)))
26		absf 15280	. . . . . . . . . . 11 ⊢ abs:ℂ⟶ℝ
27	26	a1i 11	. . . . . . . . . 10 ⊢ (𝐺:ℝ⟶ℝ → abs:ℂ⟶ℝ)
28	27	feqmptd 6957	. . . . . . . . 9 ⊢ (𝐺:ℝ⟶ℝ → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)))
29		fveq2 6888	. . . . . . . . 9 ⊢ (𝑥 = (𝐺‘𝑡) → (abs‘𝑥) = (abs‘(𝐺‘𝑡)))
30	2, 25, 28, 29	fmptco 7123	. . . . . . . 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 25276	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝐿¹ → 𝐺 ∈ MblFn)
33		ftc1anclem1 36549	. . . . . . . . 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 15379	. . . . . . . 8 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → (abs‘(𝐺‘𝑡)) ∈ ℝ)
39	2	absge0d 15387	. . . . . . . 8 ⊢ ((𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(𝐺‘𝑡)))
40		elrege0 13427	. . . . . . . 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 7111	. . . . . 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 4533	. . . . . . . . 9 ⊢ (𝑡 ∈ ℝ → if(𝑡 ∈ ℝ, (abs‘(𝐺‘𝑡)), 0) = (abs‘(𝐺‘𝑡)))
45	44	mpteq2ia 5250	. . . . . . . 8 ⊢ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐺‘𝑡)), 0)) = (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡)))
46	45	fveq2i 6891	. . . . . . 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 6957	. . . . . . . . . . 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 36540	. . . . . . . . 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 25301	. . . . . . . . 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 15379	. . . . . . . 8 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) → (abs‘(𝐹‘𝑡)) ∈ ℝ)
61	5	absge0d 15387	. . . . . . . 8 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(𝐹‘𝑡)))
62		elrege0 13427	. . . . . . . 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 7111	. . . . . 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 4533	. . . . . . . . 9 ⊢ (𝑡 ∈ ℝ → if(𝑡 ∈ ℝ, (abs‘(𝐹‘𝑡)), 0) = (abs‘(𝐹‘𝑡)))
67	66	mpteq2ia 5250	. . . . . . . 8 ⊢ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐹‘𝑡)), 0)) = (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡)))
68	67	fveq2i 6891	. . . . . . 7 ⊢ (∫₂‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐹‘𝑡)), 0))) = (∫₂‘(𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))))
69	3	feqmptd 6957	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹 = (𝑡 ∈ ℝ ↦ (𝐹‘𝑡)))
70		i1fibl 25316	. . . . . . . . . . 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 6957	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ dom ∫₁ → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)))
74		fveq2 6888	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐹‘𝑡) → (abs‘𝑥) = (abs‘(𝐹‘𝑡)))
75	5, 69, 73, 74	fmptco 7123	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫₁ → (abs ∘ 𝐹) = (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))))
76		i1fmbf 25183	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹 ∈ MblFn)
77		ftc1anclem1 36549	. . . . . . . . . . . 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 36540	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫₁ → (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))) ∈ 𝐿¹)
81	60, 61	iblpos 25301	. . . . . . . . 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 36530	. . . 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 11239	. . 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 11189	. . . . . . . . 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 11260	. . . . . 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 11786	. . . . . 6 ⊢ (((𝐹 ∈ dom ∫₁ ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))
99		elxrge0 13430	. . . . . 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 7111	. . . 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 15276	. . . . . . . 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 7687	. . . . 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 25248	. . 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 25247	. 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 4527 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ∘ ccom 5679 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ∘_f cof 7664 ∘_r cofr 7665 ℂcc 11104 ℝcr 11105 0cc0 11106 + caddc 11109 +∞cpnf 11241 ℝ^*cxr 11243 ≤ cle 11245 − cmin 11440 [,)cico 13322 [,]cicc 13323 abscabs 15177 MblFncmbf 25122 ∫₁citg1 25123 ∫₂citg2 25124 𝐿¹cibl 25125

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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-rest 17364 df-topgen 17385 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-top 22387 df-topon 22404 df-bases 22440 df-cmp 22882 df-ovol 24972 df-vol 24973 df-mbf 25127 df-itg1 25128 df-itg2 25129 df-ibl 25130 df-0p 25178

This theorem is referenced by: ftc1anclem5 36553 ftc1anclem6 36554

W3C validator