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

Theorem itgle 25865
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 25846 . . . . 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 232 . . . 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 1143 . . 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 25846 . . . . 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 232 . . . 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 1144 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ∈ ℝ)
119simp2d 1143 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)
124simp3d 1144 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ)
132ad2ant2r 746 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ)
1413rexrd 11340 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ*)
15 simprr 772 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐵)
16 elxrge0 13517 . . . . . . 7 (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵))
1714, 15, 16sylanbrc 582 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ (0[,]+∞))
18 0e0iccpnf 13519 . . . . . . 7 0 ∈ (0[,]+∞)
1918a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 0 ∈ (0[,]+∞))
2017, 19ifclda 4583 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,]+∞))
2120fmpttd 7149 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,]+∞))
227ad2ant2r 746 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ)
2322rexrd 11340 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ*)
24 simprr 772 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 0 ≤ 𝐶)
25 elxrge0 13517 . . . . . . 7 (𝐶 ∈ (0[,]+∞) ↔ (𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶))
2623, 24, 25sylanbrc 582 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ (0[,]+∞))
2718a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 0 ∈ (0[,]+∞))
2826, 27ifclda 4583 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,]+∞))
2928fmpttd 7149 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,]+∞))
30 0re 11292 . . . . . . . . . . . 12 0 ∈ ℝ
31 max1 13247 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
3230, 7, 31sylancr 586 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
33 ifcl 4593 . . . . . . . . . . . . 13 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
347, 30, 33sylancl 585 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
35 itgle.5 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵𝐶)
36 max2 13249 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
3730, 7, 36sylancr 586 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
382, 7, 34, 35, 37letrd 11447 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0))
39 maxle 13253 . . . . . . . . . . . 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))))
4030, 2, 34, 39mp3an2i 1466 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) ≤ if(0 ≤ 𝐶, 𝐶, 0) ↔ (0 ≤ if(0 ≤ 𝐶, 𝐶, 0) ∧ 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0))))
4132, 38, 40mpbir2and 712 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ≤ if(0 ≤ 𝐶, 𝐶, 0))
42 iftrue 4554 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0))
4342adantl 481 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0))
44 iftrue 4554 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0))
4544adantl 481 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0))
4641, 43, 453brtr4d 5198 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
4746ex 412 . . . . . . . 8 (𝜑 → (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)))
48 0le0 12394 . . . . . . . . . 10 0 ≤ 0
4948a1i 11 . . . . . . . . 9 𝑥𝐴 → 0 ≤ 0)
50 iffalse 4557 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = 0)
51 iffalse 4557 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = 0)
5249, 50, 513brtr4d 5198 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
5347, 52pm2.61d1 180 . . . . . . 7 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
54 ifan 4601 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0)
55 ifan 4601 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)
5653, 54, 553brtr4g 5200 . . . . . 6 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
5756ralrimivw 3156 . . . . 5 (𝜑 → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
58 reex 11275 . . . . . . 7 ℝ ∈ V
5958a1i 11 . . . . . 6 (𝜑 → ℝ ∈ V)
60 eqidd 2741 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)))
61 eqidd 2741 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
6259, 20, 28, 60, 61ofrfval2 7735 . . . . 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)))
6357, 62mpbird 257 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘r ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
64 itg2le 25794 . . . 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))))
6521, 29, 63, 64syl3anc 1371 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))))
667renegcld 11717 . . . . . . . . 9 ((𝜑𝑥𝐴) → -𝐶 ∈ ℝ)
6766ad2ant2r 746 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ)
6867rexrd 11340 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ*)
69 simprr 772 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → 0 ≤ -𝐶)
70 elxrge0 13517 . . . . . . 7 (-𝐶 ∈ (0[,]+∞) ↔ (-𝐶 ∈ ℝ* ∧ 0 ≤ -𝐶))
7168, 69, 70sylanbrc 582 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ (0[,]+∞))
7218a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → 0 ∈ (0[,]+∞))
7371, 72ifclda 4583 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ∈ (0[,]+∞))
7473fmpttd 7149 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)):ℝ⟶(0[,]+∞))
752renegcld 11717 . . . . . . . . 9 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
7675ad2ant2r 746 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ)
7776rexrd 11340 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ*)
78 simprr 772 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → 0 ≤ -𝐵)
79 elxrge0 13517 . . . . . . 7 (-𝐵 ∈ (0[,]+∞) ↔ (-𝐵 ∈ ℝ* ∧ 0 ≤ -𝐵))
8077, 78, 79sylanbrc 582 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ (0[,]+∞))
8118a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → 0 ∈ (0[,]+∞))
8280, 81ifclda 4583 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) ∈ (0[,]+∞))
8382fmpttd 7149 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)):ℝ⟶(0[,]+∞))
84 max1 13247 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
8530, 75, 84sylancr 586 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
86 ifcl 4593 . . . . . . . . . . . . 13 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
8775, 30, 86sylancl 585 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
882, 7lenegd 11869 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝐵𝐶 ↔ -𝐶 ≤ -𝐵))
8935, 88mpbid 232 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -𝐶 ≤ -𝐵)
90 max2 13249 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
9130, 75, 90sylancr 586 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
9266, 75, 87, 89, 91letrd 11447 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0))
93 maxle 13253 . . . . . . . . . . . 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))))
9430, 66, 87, 93mp3an2i 1466 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) ≤ if(0 ≤ -𝐵, -𝐵, 0) ↔ (0 ≤ if(0 ≤ -𝐵, -𝐵, 0) ∧ -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0))))
9585, 92, 94mpbir2and 712 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ≤ if(0 ≤ -𝐵, -𝐵, 0))
96 iftrue 4554 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0))
9796adantl 481 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0))
98 iftrue 4554 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0))
9998adantl 481 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0))
10095, 97, 993brtr4d 5198 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
101100ex 412 . . . . . . . 8 (𝜑 → (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)))
102 iffalse 4557 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = 0)
103 iffalse 4557 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = 0)
10449, 102, 1033brtr4d 5198 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
105101, 104pm2.61d1 180 . . . . . . 7 (𝜑 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
106 ifan 4601 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) = if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0)
107 ifan 4601 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) = if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)
108105, 106, 1073brtr4g 5200 . . . . . 6 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
109108ralrimivw 3156 . . . . 5 (𝜑 → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
110 eqidd 2741 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)))
111 eqidd 2741 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))
11259, 73, 82, 110, 111ofrfval2 7735 . . . . 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)))
113109, 112mpbird 257 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) ∘r ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))
114 itg2le 25794 . . . 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))))
11574, 83, 113, 114syl3anc 1371 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))))
1165, 10, 11, 12, 65, 115le2subd 11910 . 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)))))
1172, 1itgrevallem1 25850 . 2 (𝜑 → ∫𝐴𝐵 d𝑥 = ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) − (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))))
1187, 6itgrevallem1 25850 . 2 (𝜑 → ∫𝐴𝐶 d𝑥 = ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) − (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)))))
119116, 117, 1183brtr4d 5198 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 1537  wcel 2108  wral 3067  Vcvv 3488  ifcif 4548   class class class wbr 5166  cmpt 5249  wf 6569  cfv 6573  (class class class)co 7448  r cofr 7713  cr 11183  0cc0 11184  +∞cpnf 11321  *cxr 11323  cle 11325  cmin 11520  -cneg 11521  [,]cicc 13410  MblFncmbf 25668  2citg2 25670  𝐿1cibl 25671  citg 25672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262  ax-addf 11263
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-disj 5134  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-ofr 7715  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-oi 9579  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-xadd 13176  df-ioo 13411  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-fl 13843  df-mod 13921  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-xmet 21380  df-met 21381  df-ovol 25518  df-vol 25519  df-mbf 25673  df-itg1 25674  df-itg2 25675  df-ibl 25676  df-itg 25677  df-0p 25724
This theorem is referenced by:  itgge0  25866  itgless  25872  itgabs  25890  itgulm  26469  itgabsnc  37649  intlewftc  42018  wallispilem1  45986  fourierdlem47  46074  fourierdlem87  46114  etransclem23  46178
  Copyright terms: Public domain W3C validator