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 25845

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 25826	. . . . 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 25826	. . . . 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 747	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ)
14	13	rexrd 11311	. . . . . . 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 13497	. . . . . . 7 ⊢ (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵))
17	14, 15, 16	sylanbrc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ (0[,]+∞))
18		0e0iccpnf 13499	. . . . . . 7 ⊢ 0 ∈ (0[,]+∞)
19	18	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ∈ (0[,]+∞))
20	17, 19	ifclda 4561	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,]+∞))
21	20	fmpttd 7135	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,]+∞))
22	7	ad2ant2r 747	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ)
23	22	rexrd 11311	. . . . . . 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 13497	. . . . . . 7 ⊢ (𝐶 ∈ (0[,]+∞) ↔ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶))
26	23, 24, 25	sylanbrc 583	. . . . . 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 4561	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,]+∞))
29	28	fmpttd 7135	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,]+∞))
30		0re 11263	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
31		max1 13227	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
32	30, 7, 31	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
33		ifcl 4571	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
34	7, 30, 33	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
35		itgle.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)
36		max2 13229	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
37	30, 7, 36	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
38	2, 7, 34, 35, 37	letrd 11418	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0))
39		maxle 13233	. . . . . . . . . . . 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 1468	. . . . . . . . . . 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 713	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ≤ if(0 ≤ 𝐶, 𝐶, 0))
42		iftrue 4531	. . . . . . . . . . 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 4531	. . . . . . . . . . 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 5175	. . . . . . . . 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 12367	. . . . . . . . . 10 ⊢ 0 ≤ 0
49	48	a1i 11	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
50		iffalse 4534	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = 0)
51		iffalse 4534	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = 0)
52	49, 50, 51	3brtr4d 5175	. . . . . . . 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 4579	. . . . . . 7 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0)
55		ifan 4579	. . . . . . 7 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)
56	53, 54, 55	3brtr4g 5177	. . . . . 6 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
57	56	ralrimivw 3150	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
58		reex 11246	. . . . . . 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 7718	. . . . 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 25774	. . . 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 1373	. . 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 11690	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ∈ ℝ)
67	66	ad2ant2r 747	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ)
68	67	rexrd 11311	. . . . . . 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 13497	. . . . . . 7 ⊢ (-𝐶 ∈ (0[,]+∞) ↔ (-𝐶 ∈ ℝ^* ∧ 0 ≤ -𝐶))
71	68, 69, 70	sylanbrc 583	. . . . . 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 4561	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ∈ (0[,]+∞))
74	73	fmpttd 7135	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)):ℝ⟶(0[,]+∞))
75	2	renegcld 11690	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
76	75	ad2ant2r 747	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ)
77	76	rexrd 11311	. . . . . . 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 13497	. . . . . . 7 ⊢ (-𝐵 ∈ (0[,]+∞) ↔ (-𝐵 ∈ ℝ^* ∧ 0 ≤ -𝐵))
80	77, 78, 79	sylanbrc 583	. . . . . 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 4561	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) ∈ (0[,]+∞))
83	82	fmpttd 7135	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)):ℝ⟶(0[,]+∞))
84		max1 13227	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
85	30, 75, 84	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
86		ifcl 4571	. . . . . . . . . . . . 13 ⊢ ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
87	75, 30, 86	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
88	2, 7	lenegd 11842	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ≤ 𝐶 ↔ -𝐶 ≤ -𝐵))
89	35, 88	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ≤ -𝐵)
90		max2 13229	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
91	30, 75, 90	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
92	66, 75, 87, 89, 91	letrd 11418	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0))
93		maxle 13233	. . . . . . . . . . . 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 1468	. . . . . . . . . . 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 713	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ≤ if(0 ≤ -𝐵, -𝐵, 0))
96		iftrue 4531	. . . . . . . . . . 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 4531	. . . . . . . . . . 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 5175	. . . . . . . . 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 4534	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = 0)
103		iffalse 4534	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = 0)
104	49, 102, 103	3brtr4d 5175	. . . . . . . 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 4579	. . . . . . 7 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0)
107		ifan 4579	. . . . . . 7 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)
108	105, 106, 107	3brtr4g 5177	. . . . . 6 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
109	108	ralrimivw 3150	. . . . 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 7718	. . . . 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 25774	. . . 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 1373	. . 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 11883	. 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 25830	. 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 25830	. 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 5175	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 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘_r cofr 7696 ℝcr 11154 0cc0 11155 +∞cpnf 11292 ℝ^*cxr 11294 ≤ cle 11296 − cmin 11492 -cneg 11493 [,]cicc 13390 MblFncmbf 25649 ∫₂citg2 25651 𝐿¹cibl 25652 ∫citg 25653

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xadd 13155 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-xmet 21357 df-met 21358 df-ovol 25499 df-vol 25500 df-mbf 25654 df-itg1 25655 df-itg2 25656 df-ibl 25657 df-itg 25658 df-0p 25705

This theorem is referenced by: itgge0 25846 itgless 25852 itgabs 25870 itgulm 26451 itgabsnc 37696 intlewftc 42062 wallispilem1 46080 fourierdlem47 46168 fourierdlem87 46208 etransclem23 46272

W3C validator