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Theorem itgle 23621
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 23602 . . . . 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 222 . . . 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 1094 . . 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 23602 . . . . 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 222 . . . 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 1095 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ∈ ℝ)
119simp2d 1094 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)
124simp3d 1095 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ)
132ad2ant2r 798 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ)
1413rexrd 10127 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ*)
15 simprr 811 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐵)
16 elxrge0 12319 . . . . . . 7 (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵))
1714, 15, 16sylanbrc 699 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ (0[,]+∞))
18 0e0iccpnf 12321 . . . . . . 7 0 ∈ (0[,]+∞)
1918a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 0 ∈ (0[,]+∞))
2017, 19ifclda 4153 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,]+∞))
21 eqid 2651 . . . . 5 (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
2220, 21fmptd 6425 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,]+∞))
237ad2ant2r 798 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ)
2423rexrd 10127 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ*)
25 simprr 811 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 0 ≤ 𝐶)
26 elxrge0 12319 . . . . . . 7 (𝐶 ∈ (0[,]+∞) ↔ (𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶))
2724, 25, 26sylanbrc 699 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ (0[,]+∞))
2818a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 0 ∈ (0[,]+∞))
2927, 28ifclda 4153 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,]+∞))
30 eqid 2651 . . . . 5 (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
3129, 30fmptd 6425 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,]+∞))
32 0re 10078 . . . . . . . . . . . 12 0 ∈ ℝ
33 max1 12054 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
3432, 7, 33sylancr 696 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
35 ifcl 4163 . . . . . . . . . . . . 13 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
367, 32, 35sylancl 695 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
37 itgle.5 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵𝐶)
38 max2 12056 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
3932, 7, 38sylancr 696 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
402, 7, 36, 37, 39letrd 10232 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0))
4132a1i 11 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 0 ∈ ℝ)
42 maxle 12060 . . . . . . . . . . . 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))))
4341, 2, 36, 42syl3anc 1366 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) ≤ if(0 ≤ 𝐶, 𝐶, 0) ↔ (0 ≤ if(0 ≤ 𝐶, 𝐶, 0) ∧ 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0))))
4434, 40, 43mpbir2and 977 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ≤ if(0 ≤ 𝐶, 𝐶, 0))
45 iftrue 4125 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0))
4645adantl 481 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0))
47 iftrue 4125 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0))
4847adantl 481 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0))
4944, 46, 483brtr4d 4717 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
5049ex 449 . . . . . . . 8 (𝜑 → (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)))
51 0le0 11148 . . . . . . . . . 10 0 ≤ 0
5251a1i 11 . . . . . . . . 9 𝑥𝐴 → 0 ≤ 0)
53 iffalse 4128 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = 0)
54 iffalse 4128 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = 0)
5552, 53, 543brtr4d 4717 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
5650, 55pm2.61d1 171 . . . . . . 7 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
57 ifan 4167 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0)
58 ifan 4167 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)
5956, 57, 583brtr4g 4719 . . . . . 6 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
6059ralrimivw 2996 . . . . 5 (𝜑 → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
61 reex 10065 . . . . . . 7 ℝ ∈ V
6261a1i 11 . . . . . 6 (𝜑 → ℝ ∈ V)
63 eqidd 2652 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)))
64 eqidd 2652 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
6562, 20, 29, 63, 64ofrfval2 6957 . . . . 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)))
6660, 65mpbird 247 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
67 itg2le 23551 . . . 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))))
6822, 31, 66, 67syl3anc 1366 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))))
697renegcld 10495 . . . . . . . . 9 ((𝜑𝑥𝐴) → -𝐶 ∈ ℝ)
7069ad2ant2r 798 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ)
7170rexrd 10127 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ*)
72 simprr 811 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → 0 ≤ -𝐶)
73 elxrge0 12319 . . . . . . 7 (-𝐶 ∈ (0[,]+∞) ↔ (-𝐶 ∈ ℝ* ∧ 0 ≤ -𝐶))
7471, 72, 73sylanbrc 699 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ (0[,]+∞))
7518a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → 0 ∈ (0[,]+∞))
7674, 75ifclda 4153 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ∈ (0[,]+∞))
77 eqid 2651 . . . . 5 (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))
7876, 77fmptd 6425 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)):ℝ⟶(0[,]+∞))
792renegcld 10495 . . . . . . . . 9 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
8079ad2ant2r 798 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ)
8180rexrd 10127 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ*)
82 simprr 811 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → 0 ≤ -𝐵)
83 elxrge0 12319 . . . . . . 7 (-𝐵 ∈ (0[,]+∞) ↔ (-𝐵 ∈ ℝ* ∧ 0 ≤ -𝐵))
8481, 82, 83sylanbrc 699 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ (0[,]+∞))
8518a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → 0 ∈ (0[,]+∞))
8684, 85ifclda 4153 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) ∈ (0[,]+∞))
87 eqid 2651 . . . . 5 (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
8886, 87fmptd 6425 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)):ℝ⟶(0[,]+∞))
89 max1 12054 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
9032, 79, 89sylancr 696 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
91 ifcl 4163 . . . . . . . . . . . . 13 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
9279, 32, 91sylancl 695 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
932, 7lenegd 10644 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝐵𝐶 ↔ -𝐶 ≤ -𝐵))
9437, 93mpbid 222 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -𝐶 ≤ -𝐵)
95 max2 12056 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
9632, 79, 95sylancr 696 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
9769, 79, 92, 94, 96letrd 10232 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0))
98 maxle 12060 . . . . . . . . . . . 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))))
9941, 69, 92, 98syl3anc 1366 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) ≤ if(0 ≤ -𝐵, -𝐵, 0) ↔ (0 ≤ if(0 ≤ -𝐵, -𝐵, 0) ∧ -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0))))
10090, 97, 99mpbir2and 977 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ≤ if(0 ≤ -𝐵, -𝐵, 0))
101 iftrue 4125 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0))
102101adantl 481 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0))
103 iftrue 4125 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0))
104103adantl 481 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0))
105100, 102, 1043brtr4d 4717 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
106105ex 449 . . . . . . . 8 (𝜑 → (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)))
107 iffalse 4128 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = 0)
108 iffalse 4128 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = 0)
10952, 107, 1083brtr4d 4717 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
110106, 109pm2.61d1 171 . . . . . . 7 (𝜑 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
111 ifan 4167 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) = if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0)
112 ifan 4167 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) = if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)
113110, 111, 1123brtr4g 4719 . . . . . 6 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
114113ralrimivw 2996 . . . . 5 (𝜑 → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
115 eqidd 2652 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)))
116 eqidd 2652 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))
11762, 76, 86, 115, 116ofrfval2 6957 . . . . 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)))
118114, 117mpbird 247 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))
119 itg2le 23551 . . . 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))))
12078, 88, 118, 119syl3anc 1366 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))))
1215, 10, 11, 12, 68, 120le2subd 10685 . 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)))))
1222, 1itgrevallem1 23606 . 2 (𝜑 → ∫𝐴𝐵 d𝑥 = ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) − (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))))
1237, 6itgrevallem1 23606 . 2 (𝜑 → ∫𝐴𝐶 d𝑥 = ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) − (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)))))
124121, 122, 1233brtr4d 4717 1 (𝜑 → ∫𝐴𝐵 d𝑥 ≤ ∫𝐴𝐶 d𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  ifcif 4119   class class class wbr 4685  cmpt 4762  wf 5922  cfv 5926  (class class class)co 6690  𝑟 cofr 6938  cr 9973  0cc0 9974  +∞cpnf 10109  *cxr 10111  cle 10113  cmin 10304  -cneg 10305  [,]cicc 12216  MblFncmbf 23428  2citg2 23430  𝐿1cibl 23431  citg 23432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xadd 11985  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-xmet 19787  df-met 19788  df-ovol 23279  df-vol 23280  df-mbf 23433  df-itg1 23434  df-itg2 23435  df-ibl 23436  df-itg 23437  df-0p 23482
This theorem is referenced by:  itgge0  23622  itgless  23628  itgabs  23646  itgulm  24207  itgabsnc  33609  wallispilem1  40600  fourierdlem47  40688  fourierdlem87  40728  etransclem23  40792
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