itgle - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > itgle		Structured version Visualization version GIF version

Theorem itgle 25777

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 25758	. . . . 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 232	. . . 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 1144	. . 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 25758	. . . . 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 232	. . . 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 1145	. . 3 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ∈ ℝ)
11	9	simp2d 1144	. . 3 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)
12	4	simp3d 1145	. . 3 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ)
13	2	ad2ant2r 748	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ)
14	13	rexrd 11195	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ^*)
15		simprr 773	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐵)
16		elxrge0 13410	. . . . . . 7 ⊢ (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵))
17	14, 15, 16	sylanbrc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ (0[,]+∞))
18		0e0iccpnf 13412	. . . . . . 7 ⊢ 0 ∈ (0[,]+∞)
19	18	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ∈ (0[,]+∞))
20	17, 19	ifclda 4503	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,]+∞))
21	20	fmpttd 7068	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,]+∞))
22	7	ad2ant2r 748	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ)
23	22	rexrd 11195	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ^*)
24		simprr 773	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 0 ≤ 𝐶)
25		elxrge0 13410	. . . . . . 7 ⊢ (𝐶 ∈ (0[,]+∞) ↔ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶))
26	23, 24, 25	sylanbrc 584	. . . . . 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 4503	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,]+∞))
29	28	fmpttd 7068	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,]+∞))
30		0re 11146	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
31		max1 13137	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
32	30, 7, 31	sylancr 588	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
33		ifcl 4513	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
34	7, 30, 33	sylancl 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
35		itgle.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)
36		max2 13139	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
37	30, 7, 36	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
38	2, 7, 34, 35, 37	letrd 11303	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0))
39		maxle 13143	. . . . . . . . . . . 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 1469	. . . . . . . . . . 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 714	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ≤ if(0 ≤ 𝐶, 𝐶, 0))
42		iftrue 4473	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0))
43	42	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0))
44		iftrue 4473	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0))
45	44	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0))
46	41, 43, 45	3brtr4d 5118	. . . . . . . . 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 412	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)))
48		0le0 12282	. . . . . . . . . 10 ⊢ 0 ≤ 0
49	48	a1i 11	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
50		iffalse 4476	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = 0)
51		iffalse 4476	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = 0)
52	49, 50, 51	3brtr4d 5118	. . . . . . . 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 180	. . . . . . 7 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
54		ifan 4521	. . . . . . 7 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0)
55		ifan 4521	. . . . . . 7 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)
56	53, 54, 55	3brtr4g 5120	. . . . . 6 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
57	56	ralrimivw 3134	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
58		reex 11129	. . . . . . 7 ⊢ ℝ ∈ V
59	58	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
60		eqidd 2738	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)))
61		eqidd 2738	. . . . . 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 7652	. . . . 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 257	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
64		itg2le 25706	. . . 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 1374	. . 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 11577	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ∈ ℝ)
67	66	ad2ant2r 748	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ)
68	67	rexrd 11195	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ^*)
69		simprr 773	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶)) → 0 ≤ -𝐶)
70		elxrge0 13410	. . . . . . 7 ⊢ (-𝐶 ∈ (0[,]+∞) ↔ (-𝐶 ∈ ℝ^* ∧ 0 ≤ -𝐶))
71	68, 69, 70	sylanbrc 584	. . . . . 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 4503	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ∈ (0[,]+∞))
74	73	fmpttd 7068	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)):ℝ⟶(0[,]+∞))
75	2	renegcld 11577	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
76	75	ad2ant2r 748	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ)
77	76	rexrd 11195	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ^*)
78		simprr 773	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → 0 ≤ -𝐵)
79		elxrge0 13410	. . . . . . 7 ⊢ (-𝐵 ∈ (0[,]+∞) ↔ (-𝐵 ∈ ℝ^* ∧ 0 ≤ -𝐵))
80	77, 78, 79	sylanbrc 584	. . . . . 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 4503	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) ∈ (0[,]+∞))
83	82	fmpttd 7068	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)):ℝ⟶(0[,]+∞))
84		max1 13137	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
85	30, 75, 84	sylancr 588	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
86		ifcl 4513	. . . . . . . . . . . . 13 ⊢ ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
87	75, 30, 86	sylancl 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
88	2, 7	lenegd 11729	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ≤ 𝐶 ↔ -𝐶 ≤ -𝐵))
89	35, 88	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ≤ -𝐵)
90		max2 13139	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
91	30, 75, 90	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
92	66, 75, 87, 89, 91	letrd 11303	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0))
93		maxle 13143	. . . . . . . . . . . 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 1469	. . . . . . . . . . 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 714	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ≤ if(0 ≤ -𝐵, -𝐵, 0))
96		iftrue 4473	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0))
97	96	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0))
98		iftrue 4473	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0))
99	98	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0))
100	95, 97, 99	3brtr4d 5118	. . . . . . . . 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 412	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)))
102		iffalse 4476	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = 0)
103		iffalse 4476	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = 0)
104	49, 102, 103	3brtr4d 5118	. . . . . . . 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 180	. . . . . . 7 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
106		ifan 4521	. . . . . . 7 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0)
107		ifan 4521	. . . . . . 7 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)
108	105, 106, 107	3brtr4g 5120	. . . . . 6 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
109	108	ralrimivw 3134	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
110		eqidd 2738	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)))
111		eqidd 2738	. . . . . 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 7652	. . . . 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 257	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))
114		itg2le 25706	. . . 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 1374	. . 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 11770	. 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 25762	. 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 25762	. 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 5118	1 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 ≤ ∫𝐴𝐶 d𝑥)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3430 ifcif 4467 class class class wbr 5086 ↦ cmpt 5167 ⟶wf 6495 ‘cfv 6499 (class class class)co 7367 ∘_r cofr 7630 ℝcr 11037 0cc0 11038 +∞cpnf 11176 ℝ^*cxr 11178 ≤ cle 11180 − cmin 11377 -cneg 11378 [,]cicc 13301 MblFncmbf 25581 ∫₂citg2 25583 𝐿¹cibl 25584 ∫citg 25585

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-ofr 7632 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-xadd 13064 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-sum 15649 df-xmet 21345 df-met 21346 df-ovol 25431 df-vol 25432 df-mbf 25586 df-itg1 25587 df-itg2 25588 df-ibl 25589 df-itg 25590 df-0p 25637

This theorem is referenced by: itgge0 25778 itgless 25784 itgabs 25802 itgulm 26373 itgabsnc 38010 intlewftc 42500 wallispilem1 46493 fourierdlem47 46581 fourierdlem87 46621 etransclem23 46685

W3C validator