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Theorem itgle 25948

Description: Monotonicity of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)

**Hypotheses**
Ref	Expression
itgle.1	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
itgle.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹)
itgle.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
itgle.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)
itgle.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)

**Assertion**
Ref	Expression
itgle	⊢ (𝜑 → ∫𝐴𝐵 d𝑥 ≤ ∫𝐴𝐶 d𝑥)

Distinct variable groups: 𝑥,𝐴 𝜑,𝑥 Allowed substitution hints: 𝐵(𝑥) 𝐶(𝑥)

**Proof of Theorem itgle**
Step	Hyp	Ref	Expression
1		itgle.1	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
2		itgle.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
3	2	iblrelem 25929	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ)))
4	1, 3	mpbid 235	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ))
5	4	simp2d 1159	. . 3 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ)
6		itgle.2	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹)
7		itgle.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)
8	7	iblrelem 25929	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ∈ ℝ)))
9	6, 8	mpbid 235	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ∈ ℝ))
10	9	simp3d 1160	. . 3 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ∈ ℝ)
11	9	simp2d 1159	. . 3 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)
12	4	simp3d 1160	. . 3 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ)
13	2	ad2ant2r 759	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ)
14	13	rexrd 11258	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ^*)
15		simprr 784	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐵)
16		elxrge0 13483	. . . . . . 7 ⊢ (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵))
17	14, 15, 16	sylanbrc 594	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ (0[,]+∞))
18		0e0iccpnf 13485	. . . . . . 7 ⊢ 0 ∈ (0[,]+∞)
19	18	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ∈ (0[,]+∞))
20	17, 19	ifclda 4522	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,]+∞))
21	20	fmpttd 7110	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,]+∞))
22	7	ad2ant2r 759	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ)
23	22	rexrd 11258	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ^*)
24		simprr 784	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 0 ≤ 𝐶)
25		elxrge0 13483	. . . . . . 7 ⊢ (𝐶 ∈ (0[,]+∞) ↔ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶))
26	23, 24, 25	sylanbrc 594	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ (0[,]+∞))
27	18	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 0 ∈ (0[,]+∞))
28	26, 27	ifclda 4522	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,]+∞))
29	28	fmpttd 7110	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,]+∞))
30		0re 11209	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
31		max1 13210	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
32	30, 7, 31	sylancr 598	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
33		ifcl 4532	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
34	7, 30, 33	sylancl 597	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
35		itgle.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)
36		max2 13212	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
37	30, 7, 36	sylancr 598	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
38	2, 7, 34, 35, 37	letrd 11366	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0))
39		maxle 13216	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ) → (if(0 ≤ 𝐵, 𝐵, 0) ≤ if(0 ≤ 𝐶, 𝐶, 0) ↔ (0 ≤ if(0 ≤ 𝐶, 𝐶, 0) ∧ 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0))))
40	30, 2, 34, 39	mp3an2i 1493	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) ≤ if(0 ≤ 𝐶, 𝐶, 0) ↔ (0 ≤ if(0 ≤ 𝐶, 𝐶, 0) ∧ 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0))))
41	32, 38, 40	mpbir2and 725	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ≤ if(0 ≤ 𝐶, 𝐶, 0))
42		iftrue 4492	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0))
43	42	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0))
44		iftrue 4492	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0))
45	44	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0))
46	41, 43, 45	3brtr4d 5142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
47	46	ex 417	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)))
48		0le0 12341	. . . . . . . . . 10 ⊢ 0 ≤ 0
49	48	a1i 11	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
50		iffalse 4495	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = 0)
51		iffalse 4495	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = 0)
52	49, 50, 51	3brtr4d 5142	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
53	47, 52	pm2.61d1 182	. . . . . . 7 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
54		ifan 4540	. . . . . . 7 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0)
55		ifan 4540	. . . . . . 7 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)
56	53, 54, 55	3brtr4g 5144	. . . . . 6 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
57	56	ralrimivw 3159	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
58		reex 11190	. . . . . . 7 ⊢ ℝ ∈ V
59	58	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
60		eqidd 2762	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)))
61		eqidd 2762	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
62	59, 20, 28, 60, 61	ofrfval2 7695	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) ↔ ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
63	57, 62	mpbird 260	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
64		itg2le 25877	. . . 4 ⊢ (((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))))
65	21, 29, 63, 64	syl3anc 1396	. . 3 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))))
66	7	renegcld 11640	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ∈ ℝ)
67	66	ad2ant2r 759	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ)
68	67	rexrd 11258	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ^*)
69		simprr 784	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶)) → 0 ≤ -𝐶)
70		elxrge0 13483	. . . . . . 7 ⊢ (-𝐶 ∈ (0[,]+∞) ↔ (-𝐶 ∈ ℝ^* ∧ 0 ≤ -𝐶))
71	68, 69, 70	sylanbrc 594	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ (0[,]+∞))
72	18	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶)) → 0 ∈ (0[,]+∞))
73	71, 72	ifclda 4522	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ∈ (0[,]+∞))
74	73	fmpttd 7110	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)):ℝ⟶(0[,]+∞))
75	2	renegcld 11640	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
76	75	ad2ant2r 759	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ)
77	76	rexrd 11258	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ^*)
78		simprr 784	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → 0 ≤ -𝐵)
79		elxrge0 13483	. . . . . . 7 ⊢ (-𝐵 ∈ (0[,]+∞) ↔ (-𝐵 ∈ ℝ^* ∧ 0 ≤ -𝐵))
80	77, 78, 79	sylanbrc 594	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ (0[,]+∞))
81	18	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → 0 ∈ (0[,]+∞))
82	80, 81	ifclda 4522	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) ∈ (0[,]+∞))
83	82	fmpttd 7110	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)):ℝ⟶(0[,]+∞))
84		max1 13210	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
85	30, 75, 84	sylancr 598	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
86		ifcl 4532	. . . . . . . . . . . . 13 ⊢ ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
87	75, 30, 86	sylancl 597	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
88	2, 7	lenegd 11792	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ≤ 𝐶 ↔ -𝐶 ≤ -𝐵))
89	35, 88	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ≤ -𝐵)
90		max2 13212	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
91	30, 75, 90	sylancr 598	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
92	66, 75, 87, 89, 91	letrd 11366	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0))
93		maxle 13216	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ -𝐶 ∈ ℝ ∧ if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ) → (if(0 ≤ -𝐶, -𝐶, 0) ≤ if(0 ≤ -𝐵, -𝐵, 0) ↔ (0 ≤ if(0 ≤ -𝐵, -𝐵, 0) ∧ -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0))))
94	30, 66, 87, 93	mp3an2i 1493	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) ≤ if(0 ≤ -𝐵, -𝐵, 0) ↔ (0 ≤ if(0 ≤ -𝐵, -𝐵, 0) ∧ -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0))))
95	85, 92, 94	mpbir2and 725	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ≤ if(0 ≤ -𝐵, -𝐵, 0))
96		iftrue 4492	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0))
97	96	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0))
98		iftrue 4492	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0))
99	98	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0))
100	95, 97, 99	3brtr4d 5142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
101	100	ex 417	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)))
102		iffalse 4495	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = 0)
103		iffalse 4495	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = 0)
104	49, 102, 103	3brtr4d 5142	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
105	101, 104	pm2.61d1 182	. . . . . . 7 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
106		ifan 4540	. . . . . . 7 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0)
107		ifan 4540	. . . . . . 7 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)
108	105, 106, 107	3brtr4g 5144	. . . . . 6 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
109	108	ralrimivw 3159	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
110		eqidd 2762	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)))
111		eqidd 2762	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))
112	59, 73, 82, 110, 111	ofrfval2 7695	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) ↔ ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))
113	109, 112	mpbird 260	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))
114		itg2le 25877	. . . 4 ⊢ (((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))))
115	74, 83, 113, 114	syl3anc 1396	. . 3 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))))
116	5, 10, 11, 12, 65, 115	le2subd 11833	. 2 ⊢ (𝜑 → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) − (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))) ≤ ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) − (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)))))
117	2, 1	itgrevallem1 25933	. 2 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) − (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))))
118	7, 6	itgrevallem1 25933	. 2 ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) − (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)))))
119	116, 117, 118	3brtr4d 5142	1 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 ≤ ∫𝐴𝐶 d𝑥)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 ∀wral 3077 Vcvv 3453 ifcif 4486 class class class wbr 5108 ↦ cmpt 5191 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ∘_r cofr 7673 ℝcr 11098 0cc0 11099 +∞cpnf 11239 ℝ^*cxr 11241 ≤ cle 11243 − cmin 11440 -cneg 11441 [,]cicc 13374 MblFncmbf 25752 ∫₂citg2 25754 𝐿¹cibl 25755 ∫citg 25756

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9609 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-disj 5076 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-ofr 7675 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-map 8825 df-pm 8826 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9401 df-inf 9402 df-oi 9471 df-dju 9886 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-n0 12504 df-z 12591 df-uz 12862 df-q 12972 df-rp 13016 df-xadd 13137 df-ioo 13375 df-ico 13377 df-icc 13378 df-fz 13535 df-fzo 13682 df-fl 13824 df-mod 13902 df-seq 14037 df-exp 14097 df-hash 14366 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-clim 15538 df-sum 15737 df-xmet 21494 df-met 21495 df-ovol 25602 df-vol 25603 df-mbf 25757 df-itg1 25758 df-itg2 25759 df-ibl 25760 df-itg 25761 df-0p 25808

This theorem is referenced by: itgge0 25949 itgless 25955 itgabs 25973 itgulm 26547 itgabsnc 38284 intlewftc 42774 wallispilem1 46727 fourierdlem47 46815 fourierdlem87 46855 etransclem23 46919

W3C validator