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Theorem itgle 24417
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 24398 . . . . 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 235 . . . 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 1140 . . 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 24398 . . . . 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 235 . . . 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 1141 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ∈ ℝ)
119simp2d 1140 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)
124simp3d 1141 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ)
132ad2ant2r 746 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ)
1413rexrd 10684 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ*)
15 simprr 772 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐵)
16 elxrge0 12839 . . . . . . 7 (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵))
1714, 15, 16sylanbrc 586 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ (0[,]+∞))
18 0e0iccpnf 12841 . . . . . . 7 0 ∈ (0[,]+∞)
1918a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 0 ∈ (0[,]+∞))
2017, 19ifclda 4462 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,]+∞))
2120fmpttd 6860 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,]+∞))
227ad2ant2r 746 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ)
2322rexrd 10684 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ*)
24 simprr 772 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 0 ≤ 𝐶)
25 elxrge0 12839 . . . . . . 7 (𝐶 ∈ (0[,]+∞) ↔ (𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶))
2623, 24, 25sylanbrc 586 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ (0[,]+∞))
2718a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 0 ∈ (0[,]+∞))
2826, 27ifclda 4462 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,]+∞))
2928fmpttd 6860 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,]+∞))
30 0re 10636 . . . . . . . . . . . 12 0 ∈ ℝ
31 max1 12570 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
3230, 7, 31sylancr 590 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
33 ifcl 4472 . . . . . . . . . . . . 13 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
347, 30, 33sylancl 589 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
35 itgle.5 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵𝐶)
36 max2 12572 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
3730, 7, 36sylancr 590 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
382, 7, 34, 35, 37letrd 10790 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0))
39 maxle 12576 . . . . . . . . . . . 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 1463 . . . . . . . . . . 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 4434 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0))
4342adantl 485 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0))
44 iftrue 4434 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0))
4544adantl 485 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0))
4641, 43, 453brtr4d 5065 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
4746ex 416 . . . . . . . 8 (𝜑 → (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)))
48 0le0 11730 . . . . . . . . . 10 0 ≤ 0
4948a1i 11 . . . . . . . . 9 𝑥𝐴 → 0 ≤ 0)
50 iffalse 4437 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = 0)
51 iffalse 4437 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = 0)
5249, 50, 513brtr4d 5065 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
5347, 52pm2.61d1 183 . . . . . . 7 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
54 ifan 4479 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0)
55 ifan 4479 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)
5653, 54, 553brtr4g 5067 . . . . . 6 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
5756ralrimivw 3153 . . . . 5 (𝜑 → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
58 reex 10621 . . . . . . 7 ℝ ∈ V
5958a1i 11 . . . . . 6 (𝜑 → ℝ ∈ V)
60 eqidd 2802 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)))
61 eqidd 2802 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
6259, 20, 28, 60, 61ofrfval2 7411 . . . . 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 260 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘r ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
64 itg2le 24347 . . . 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 1368 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))))
667renegcld 11060 . . . . . . . . 9 ((𝜑𝑥𝐴) → -𝐶 ∈ ℝ)
6766ad2ant2r 746 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ)
6867rexrd 10684 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ*)
69 simprr 772 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → 0 ≤ -𝐶)
70 elxrge0 12839 . . . . . . 7 (-𝐶 ∈ (0[,]+∞) ↔ (-𝐶 ∈ ℝ* ∧ 0 ≤ -𝐶))
7168, 69, 70sylanbrc 586 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ (0[,]+∞))
7218a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → 0 ∈ (0[,]+∞))
7371, 72ifclda 4462 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ∈ (0[,]+∞))
7473fmpttd 6860 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)):ℝ⟶(0[,]+∞))
752renegcld 11060 . . . . . . . . 9 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
7675ad2ant2r 746 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ)
7776rexrd 10684 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ*)
78 simprr 772 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → 0 ≤ -𝐵)
79 elxrge0 12839 . . . . . . 7 (-𝐵 ∈ (0[,]+∞) ↔ (-𝐵 ∈ ℝ* ∧ 0 ≤ -𝐵))
8077, 78, 79sylanbrc 586 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ (0[,]+∞))
8118a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → 0 ∈ (0[,]+∞))
8280, 81ifclda 4462 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) ∈ (0[,]+∞))
8382fmpttd 6860 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)):ℝ⟶(0[,]+∞))
84 max1 12570 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
8530, 75, 84sylancr 590 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
86 ifcl 4472 . . . . . . . . . . . . 13 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
8775, 30, 86sylancl 589 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
882, 7lenegd 11212 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝐵𝐶 ↔ -𝐶 ≤ -𝐵))
8935, 88mpbid 235 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -𝐶 ≤ -𝐵)
90 max2 12572 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
9130, 75, 90sylancr 590 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
9266, 75, 87, 89, 91letrd 10790 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0))
93 maxle 12576 . . . . . . . . . . . 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 1463 . . . . . . . . . . 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 4434 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0))
9796adantl 485 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0))
98 iftrue 4434 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0))
9998adantl 485 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0))
10095, 97, 993brtr4d 5065 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
101100ex 416 . . . . . . . 8 (𝜑 → (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)))
102 iffalse 4437 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = 0)
103 iffalse 4437 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = 0)
10449, 102, 1033brtr4d 5065 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
105101, 104pm2.61d1 183 . . . . . . 7 (𝜑 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
106 ifan 4479 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) = if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0)
107 ifan 4479 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) = if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)
108105, 106, 1073brtr4g 5067 . . . . . 6 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
109108ralrimivw 3153 . . . . 5 (𝜑 → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
110 eqidd 2802 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)))
111 eqidd 2802 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))
11259, 73, 82, 110, 111ofrfval2 7411 . . . . 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 260 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) ∘r ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))
114 itg2le 24347 . . . 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 1368 . . 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 11253 . 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 24402 . 2 (𝜑 → ∫𝐴𝐵 d𝑥 = ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) − (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))))
1187, 6itgrevallem1 24402 . 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 5065 1 (𝜑 → ∫𝐴𝐵 d𝑥 ≤ ∫𝐴𝐶 d𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  Vcvv 3444  ifcif 4428   class class class wbr 5033  cmpt 5113  wf 6324  cfv 6328  (class class class)co 7139  r cofr 7392  cr 10529  0cc0 10530  +∞cpnf 10665  *cxr 10667  cle 10669  cmin 10863  -cneg 10864  [,]cicc 12733  MblFncmbf 24222  2citg2 24224  𝐿1cibl 24225  citg 24226
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-addf 10609
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-disj 4999  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-ofr 7394  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-xadd 12500  df-ioo 12734  df-ico 12736  df-icc 12737  df-fz 12890  df-fzo 13033  df-fl 13161  df-mod 13237  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-clim 14841  df-sum 15039  df-xmet 20088  df-met 20089  df-ovol 24072  df-vol 24073  df-mbf 24227  df-itg1 24228  df-itg2 24229  df-ibl 24230  df-itg 24231  df-0p 24278
This theorem is referenced by:  itgge0  24418  itgless  24424  itgabs  24442  itgulm  25007  itgabsnc  35125  wallispilem1  42700  fourierdlem47  42788  fourierdlem87  42828  etransclem23  42892
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