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

Theorem itgle 24013
Description: Monotonicity of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)
Hypotheses
Ref Expression
itgle.1 (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)
itgle.2 (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)
itgle.3 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
itgle.4 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)
itgle.5 ((𝜑𝑥𝐴) → 𝐵𝐶)
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
StepHypRef Expression
1 itgle.1 . . . . 5 (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)
2 itgle.3 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
32iblrelem 23994 . . . . 5 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ)))
41, 3mpbid 224 . . . 4 (𝜑 → ((𝑥𝐴𝐵) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ))
54simp2d 1134 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ)
6 itgle.2 . . . . 5 (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)
7 itgle.4 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)
87iblrelem 23994 . . . . 5 (𝜑 → ((𝑥𝐴𝐶) ∈ 𝐿1 ↔ ((𝑥𝐴𝐶) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ∈ ℝ)))
96, 8mpbid 224 . . . 4 (𝜑 → ((𝑥𝐴𝐶) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ∈ ℝ))
109simp3d 1135 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ∈ ℝ)
119simp2d 1134 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)
124simp3d 1135 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ)
132ad2ant2r 737 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ)
1413rexrd 10426 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ*)
15 simprr 763 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐵)
16 elxrge0 12595 . . . . . . 7 (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵))
1714, 15, 16sylanbrc 578 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ (0[,]+∞))
18 0e0iccpnf 12597 . . . . . . 7 0 ∈ (0[,]+∞)
1918a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 0 ∈ (0[,]+∞))
2017, 19ifclda 4341 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,]+∞))
2120fmpttd 6649 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,]+∞))
227ad2ant2r 737 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ)
2322rexrd 10426 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ*)
24 simprr 763 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 0 ≤ 𝐶)
25 elxrge0 12595 . . . . . . 7 (𝐶 ∈ (0[,]+∞) ↔ (𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶))
2623, 24, 25sylanbrc 578 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ (0[,]+∞))
2718a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 0 ∈ (0[,]+∞))
2826, 27ifclda 4341 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,]+∞))
2928fmpttd 6649 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,]+∞))
30 0re 10378 . . . . . . . . . . . 12 0 ∈ ℝ
31 max1 12328 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
3230, 7, 31sylancr 581 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
33 ifcl 4351 . . . . . . . . . . . . 13 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
347, 30, 33sylancl 580 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
35 itgle.5 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵𝐶)
36 max2 12330 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
3730, 7, 36sylancr 581 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
382, 7, 34, 35, 37letrd 10533 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0))
3930a1i 11 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 0 ∈ ℝ)
40 maxle 12334 . . . . . . . . . . . 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))))
4139, 2, 34, 40syl3anc 1439 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) ≤ if(0 ≤ 𝐶, 𝐶, 0) ↔ (0 ≤ if(0 ≤ 𝐶, 𝐶, 0) ∧ 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0))))
4232, 38, 41mpbir2and 703 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ≤ if(0 ≤ 𝐶, 𝐶, 0))
43 iftrue 4313 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0))
4443adantl 475 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0))
45 iftrue 4313 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0))
4645adantl 475 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0))
4742, 44, 463brtr4d 4918 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
4847ex 403 . . . . . . . 8 (𝜑 → (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)))
49 0le0 11483 . . . . . . . . . 10 0 ≤ 0
5049a1i 11 . . . . . . . . 9 𝑥𝐴 → 0 ≤ 0)
51 iffalse 4316 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = 0)
52 iffalse 4316 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = 0)
5350, 51, 523brtr4d 4918 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
5448, 53pm2.61d1 173 . . . . . . 7 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
55 ifan 4358 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0)
56 ifan 4358 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)
5754, 55, 563brtr4g 4920 . . . . . 6 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
5857ralrimivw 3149 . . . . 5 (𝜑 → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
59 reex 10363 . . . . . . 7 ℝ ∈ V
6059a1i 11 . . . . . 6 (𝜑 → ℝ ∈ V)
61 eqidd 2779 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)))
62 eqidd 2779 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
6360, 20, 28, 61, 62ofrfval2 7192 . . . . 5 (𝜑 → ((𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) ↔ ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
6458, 63mpbird 249 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
65 itg2le 23943 . . . 4 (((𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))))
6621, 29, 64, 65syl3anc 1439 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))))
677renegcld 10802 . . . . . . . . 9 ((𝜑𝑥𝐴) → -𝐶 ∈ ℝ)
6867ad2ant2r 737 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ)
6968rexrd 10426 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ*)
70 simprr 763 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → 0 ≤ -𝐶)
71 elxrge0 12595 . . . . . . 7 (-𝐶 ∈ (0[,]+∞) ↔ (-𝐶 ∈ ℝ* ∧ 0 ≤ -𝐶))
7269, 70, 71sylanbrc 578 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ (0[,]+∞))
7318a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → 0 ∈ (0[,]+∞))
7472, 73ifclda 4341 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ∈ (0[,]+∞))
7574fmpttd 6649 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)):ℝ⟶(0[,]+∞))
762renegcld 10802 . . . . . . . . 9 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
7776ad2ant2r 737 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ)
7877rexrd 10426 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ*)
79 simprr 763 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → 0 ≤ -𝐵)
80 elxrge0 12595 . . . . . . 7 (-𝐵 ∈ (0[,]+∞) ↔ (-𝐵 ∈ ℝ* ∧ 0 ≤ -𝐵))
8178, 79, 80sylanbrc 578 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ (0[,]+∞))
8218a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → 0 ∈ (0[,]+∞))
8381, 82ifclda 4341 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) ∈ (0[,]+∞))
8483fmpttd 6649 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)):ℝ⟶(0[,]+∞))
85 max1 12328 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
8630, 76, 85sylancr 581 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
87 ifcl 4351 . . . . . . . . . . . . 13 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
8876, 30, 87sylancl 580 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
892, 7lenegd 10954 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝐵𝐶 ↔ -𝐶 ≤ -𝐵))
9035, 89mpbid 224 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -𝐶 ≤ -𝐵)
91 max2 12330 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
9230, 76, 91sylancr 581 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
9367, 76, 88, 90, 92letrd 10533 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0))
94 maxle 12334 . . . . . . . . . . . 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))))
9539, 67, 88, 94syl3anc 1439 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) ≤ if(0 ≤ -𝐵, -𝐵, 0) ↔ (0 ≤ if(0 ≤ -𝐵, -𝐵, 0) ∧ -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0))))
9686, 93, 95mpbir2and 703 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ≤ if(0 ≤ -𝐵, -𝐵, 0))
97 iftrue 4313 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0))
9897adantl 475 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0))
99 iftrue 4313 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0))
10099adantl 475 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0))
10196, 98, 1003brtr4d 4918 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
102101ex 403 . . . . . . . 8 (𝜑 → (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)))
103 iffalse 4316 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = 0)
104 iffalse 4316 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = 0)
10550, 103, 1043brtr4d 4918 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
106102, 105pm2.61d1 173 . . . . . . 7 (𝜑 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
107 ifan 4358 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) = if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0)
108 ifan 4358 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) = if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)
109106, 107, 1083brtr4g 4920 . . . . . 6 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
110109ralrimivw 3149 . . . . 5 (𝜑 → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
111 eqidd 2779 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)))
112 eqidd 2779 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))
11360, 74, 83, 111, 112ofrfval2 7192 . . . . 5 (𝜑 → ((𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) ↔ ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))
114110, 113mpbird 249 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))
115 itg2le 23943 . . . 4 (((𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))))
11675, 84, 114, 115syl3anc 1439 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))))
1175, 10, 11, 12, 66, 116le2subd 10995 . 2 (𝜑 → ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) − (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))) ≤ ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) − (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)))))
1182, 1itgrevallem1 23998 . 2 (𝜑 → ∫𝐴𝐵 d𝑥 = ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) − (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))))
1197, 6itgrevallem1 23998 . 2 (𝜑 → ∫𝐴𝐶 d𝑥 = ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) − (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)))))
120117, 118, 1193brtr4d 4918 1 (𝜑 → ∫𝐴𝐵 d𝑥 ≤ ∫𝐴𝐶 d𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090  Vcvv 3398  ifcif 4307   class class class wbr 4886  cmpt 4965  wf 6131  cfv 6135  (class class class)co 6922  𝑟 cofr 7173  cr 10271  0cc0 10272  +∞cpnf 10408  *cxr 10410  cle 10412  cmin 10606  -cneg 10607  [,]cicc 12490  MblFncmbf 23818  2citg2 23820  𝐿1cibl 23821  citg 23822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350  ax-addf 10351
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-disj 4855  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-ofr 7175  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-n0 11643  df-z 11729  df-uz 11993  df-q 12096  df-rp 12138  df-xadd 12258  df-ioo 12491  df-ico 12493  df-icc 12494  df-fz 12644  df-fzo 12785  df-fl 12912  df-mod 12988  df-seq 13120  df-exp 13179  df-hash 13436  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-clim 14627  df-sum 14825  df-xmet 20135  df-met 20136  df-ovol 23668  df-vol 23669  df-mbf 23823  df-itg1 23824  df-itg2 23825  df-ibl 23826  df-itg 23827  df-0p 23874
This theorem is referenced by:  itgge0  24014  itgless  24020  itgabs  24038  itgulm  24599  itgabsnc  34104  wallispilem1  41209  fourierdlem47  41297  fourierdlem87  41337  etransclem23  41401
  Copyright terms: Public domain W3C validator