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Theorem itgle 23784
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 23765 . . . . 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 223 . . . 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 1166 . . 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 23765 . . . . 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 223 . . . 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 1167 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ∈ ℝ)
119simp2d 1166 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)
124simp3d 1167 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ)
132ad2ant2r 744 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ)
1413rexrd 10368 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ*)
15 simprr 780 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐵)
16 elxrge0 12495 . . . . . . 7 (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵))
1714, 15, 16sylanbrc 574 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ (0[,]+∞))
18 0e0iccpnf 12497 . . . . . . 7 0 ∈ (0[,]+∞)
1918a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ 𝐵)) → 0 ∈ (0[,]+∞))
2017, 19ifclda 4307 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,]+∞))
2120fmpttd 6601 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,]+∞))
227ad2ant2r 744 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ)
2322rexrd 10368 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ*)
24 simprr 780 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 0 ≤ 𝐶)
25 elxrge0 12495 . . . . . . 7 (𝐶 ∈ (0[,]+∞) ↔ (𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶))
2623, 24, 25sylanbrc 574 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ (0[,]+∞))
2718a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 0 ∈ (0[,]+∞))
2826, 27ifclda 4307 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,]+∞))
2928fmpttd 6601 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,]+∞))
30 0re 10321 . . . . . . . . . . . 12 0 ∈ ℝ
31 max1 12228 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
3230, 7, 31sylancr 577 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
33 ifcl 4317 . . . . . . . . . . . . 13 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
347, 30, 33sylancl 576 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
35 itgle.5 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵𝐶)
36 max2 12230 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
3730, 7, 36sylancr 577 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
382, 7, 34, 35, 37letrd 10473 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0))
3930a1i 11 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 0 ∈ ℝ)
40 maxle 12234 . . . . . . . . . . . 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 1483 . . . . . . . . . . 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 695 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ≤ if(0 ≤ 𝐶, 𝐶, 0))
43 iftrue 4279 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0))
4443adantl 469 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0))
45 iftrue 4279 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0))
4645adantl 469 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0))
4742, 44, 463brtr4d 4869 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
4847ex 399 . . . . . . . 8 (𝜑 → (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)))
49 0le0 11387 . . . . . . . . . 10 0 ≤ 0
5049a1i 11 . . . . . . . . 9 𝑥𝐴 → 0 ≤ 0)
51 iffalse 4282 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = 0)
52 iffalse 4282 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = 0)
5350, 51, 523brtr4d 4869 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
5448, 53pm2.61d1 172 . . . . . . 7 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))
55 ifan 4324 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0)
56 ifan 4324 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)
5754, 55, 563brtr4g 4871 . . . . . 6 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
5857ralrimivw 3151 . . . . 5 (𝜑 → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
59 reex 10306 . . . . . . 7 ℝ ∈ V
6059a1i 11 . . . . . 6 (𝜑 → ℝ ∈ V)
61 eqidd 2803 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)))
62 eqidd 2803 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
6360, 20, 28, 61, 62ofrfval2 7139 . . . . 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 248 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
65 itg2le 23714 . . . 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 1483 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))))
677renegcld 10736 . . . . . . . . 9 ((𝜑𝑥𝐴) → -𝐶 ∈ ℝ)
6867ad2ant2r 744 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ)
6968rexrd 10368 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ*)
70 simprr 780 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → 0 ≤ -𝐶)
71 elxrge0 12495 . . . . . . 7 (-𝐶 ∈ (0[,]+∞) ↔ (-𝐶 ∈ ℝ* ∧ 0 ≤ -𝐶))
7269, 70, 71sylanbrc 574 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ (0[,]+∞))
7318a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ -𝐶)) → 0 ∈ (0[,]+∞))
7472, 73ifclda 4307 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ∈ (0[,]+∞))
7574fmpttd 6601 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)):ℝ⟶(0[,]+∞))
762renegcld 10736 . . . . . . . . 9 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
7776ad2ant2r 744 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ)
7877rexrd 10368 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ*)
79 simprr 780 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → 0 ≤ -𝐵)
80 elxrge0 12495 . . . . . . 7 (-𝐵 ∈ (0[,]+∞) ↔ (-𝐵 ∈ ℝ* ∧ 0 ≤ -𝐵))
8178, 79, 80sylanbrc 574 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ (0[,]+∞))
8218a1i 11 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ¬ (𝑥𝐴 ∧ 0 ≤ -𝐵)) → 0 ∈ (0[,]+∞))
8381, 82ifclda 4307 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) ∈ (0[,]+∞))
8483fmpttd 6601 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)):ℝ⟶(0[,]+∞))
85 max1 12228 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
8630, 76, 85sylancr 577 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
87 ifcl 4317 . . . . . . . . . . . . 13 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
8876, 30, 87sylancl 576 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
892, 7lenegd 10885 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝐵𝐶 ↔ -𝐶 ≤ -𝐵))
9035, 89mpbid 223 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -𝐶 ≤ -𝐵)
91 max2 12230 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
9230, 76, 91sylancr 577 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0))
9367, 76, 88, 90, 92letrd 10473 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0))
94 maxle 12234 . . . . . . . . . . . 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 1483 . . . . . . . . . . 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 695 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ≤ if(0 ≤ -𝐵, -𝐵, 0))
97 iftrue 4279 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0))
9897adantl 469 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0))
99 iftrue 4279 . . . . . . . . . . 11 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0))
10099adantl 469 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0))
10196, 98, 1003brtr4d 4869 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
102101ex 399 . . . . . . . 8 (𝜑 → (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)))
103 iffalse 4282 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = 0)
104 iffalse 4282 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = 0)
10550, 103, 1043brtr4d 4869 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
106102, 105pm2.61d1 172 . . . . . . 7 (𝜑 → if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))
107 ifan 4324 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) = if(𝑥𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0)
108 ifan 4324 . . . . . . 7 if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) = if(𝑥𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)
109106, 107, 1083brtr4g 4871 . . . . . 6 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
110109ralrimivw 3151 . . . . 5 (𝜑 → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))
111 eqidd 2803 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)))
112 eqidd 2803 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))
11360, 74, 83, 111, 112ofrfval2 7139 . . . . 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 248 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))
115 itg2le 23714 . . . 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 1483 . . 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 10926 . 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 23769 . 2 (𝜑 → ∫𝐴𝐵 d𝑥 = ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) − (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))))
1197, 6itgrevallem1 23769 . 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 4869 1 (𝜑 → ∫𝐴𝐵 d𝑥 ≤ ∫𝐴𝐶 d𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2155  wral 3092  Vcvv 3387  ifcif 4273   class class class wbr 4837  cmpt 4916  wf 6091  cfv 6095  (class class class)co 6868  𝑟 cofr 7120  cr 10214  0cc0 10215  +∞cpnf 10350  *cxr 10352  cle 10354  cmin 10545  -cneg 10546  [,]cicc 12390  MblFncmbf 23589  2citg2 23591  𝐿1cibl 23592  citg 23593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173  ax-inf2 8779  ax-cnex 10271  ax-resscn 10272  ax-1cn 10273  ax-icn 10274  ax-addcl 10275  ax-addrcl 10276  ax-mulcl 10277  ax-mulrcl 10278  ax-mulcom 10279  ax-addass 10280  ax-mulass 10281  ax-distr 10282  ax-i2m1 10283  ax-1ne0 10284  ax-1rid 10285  ax-rnegex 10286  ax-rrecex 10287  ax-cnre 10288  ax-pre-lttri 10289  ax-pre-lttrn 10290  ax-pre-ltadd 10291  ax-pre-mulgt0 10292  ax-pre-sup 10293  ax-addf 10294
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-nel 3078  df-ral 3097  df-rex 3098  df-reu 3099  df-rmo 3100  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-int 4663  df-iun 4707  df-disj 4806  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-se 5265  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-isom 6104  df-riota 6829  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-of 7121  df-ofr 7122  df-om 7290  df-1st 7392  df-2nd 7393  df-wrecs 7636  df-recs 7698  df-rdg 7736  df-1o 7790  df-2o 7791  df-oadd 7794  df-er 7973  df-map 8088  df-pm 8089  df-en 8187  df-dom 8188  df-sdom 8189  df-fin 8190  df-sup 8581  df-inf 8582  df-oi 8648  df-card 9042  df-cda 9269  df-pnf 10355  df-mnf 10356  df-xr 10357  df-ltxr 10358  df-le 10359  df-sub 10547  df-neg 10548  df-div 10964  df-nn 11300  df-2 11358  df-3 11359  df-4 11360  df-n0 11554  df-z 11638  df-uz 11899  df-q 12002  df-rp 12041  df-xadd 12157  df-ioo 12391  df-ico 12393  df-icc 12394  df-fz 12544  df-fzo 12684  df-fl 12811  df-mod 12887  df-seq 13019  df-exp 13078  df-hash 13332  df-cj 14056  df-re 14057  df-im 14058  df-sqrt 14192  df-abs 14193  df-clim 14436  df-sum 14634  df-xmet 19941  df-met 19942  df-ovol 23439  df-vol 23440  df-mbf 23594  df-itg1 23595  df-itg2 23596  df-ibl 23597  df-itg 23598  df-0p 23645
This theorem is referenced by:  itgge0  23785  itgless  23791  itgabs  23809  itgulm  24370  itgabsnc  33785  wallispilem1  40755  fourierdlem47  40843  fourierdlem87  40883  etransclem23  40947
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