Mathbox for Brendan Leahy < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ibladdnclem Structured version   Visualization version   GIF version

 Description: Lemma for ibladdnc 33099; cf ibladdlem 23492, whose fifth hypothesis is rendered unnecessary by the weakened hypotheses of itg2addnc 33096. (Contributed by Brendan Leahy, 31-Oct-2017.)
Hypotheses
Ref Expression
ibladdnclem.1 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
ibladdnclem.2 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)
ibladdnclem.3 ((𝜑𝑥𝐴) → 𝐷 = (𝐵 + 𝐶))
ibladdnclem.4 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
ibladdnclem.6 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ)
ibladdnclem.7 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)
Assertion
Ref Expression
ibladdnclem (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ∈ ℝ)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)

StepHypRef Expression
1 ifan 4106 . . . 4 if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) = if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0)
2 ibladdnclem.3 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐷 = (𝐵 + 𝐶))
3 ibladdnclem.1 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
4 ibladdnclem.2 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)
53, 4readdcld 10013 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐵 + 𝐶) ∈ ℝ)
62, 5eqeltrd 2698 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐷 ∈ ℝ)
7 0re 9984 . . . . . . . . 9 0 ∈ ℝ
8 ifcl 4102 . . . . . . . . 9 ((𝐷 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ)
96, 7, 8sylancl 693 . . . . . . . 8 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ)
109rexrd 10033 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ*)
11 max1 11959 . . . . . . . 8 ((0 ∈ ℝ ∧ 𝐷 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐷, 𝐷, 0))
127, 6, 11sylancr 694 . . . . . . 7 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐷, 𝐷, 0))
13 elxrge0 12223 . . . . . . 7 (if(0 ≤ 𝐷, 𝐷, 0) ∈ (0[,]+∞) ↔ (if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ* ∧ 0 ≤ if(0 ≤ 𝐷, 𝐷, 0)))
1410, 12, 13sylanbrc 697 . . . . . 6 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ (0[,]+∞))
15 0e0iccpnf 12225 . . . . . . 7 0 ∈ (0[,]+∞)
1615a1i 11 . . . . . 6 ((𝜑 ∧ ¬ 𝑥𝐴) → 0 ∈ (0[,]+∞))
1714, 16ifclda 4092 . . . . 5 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ∈ (0[,]+∞))
1817adantr 481 . . . 4 ((𝜑𝑥 ∈ ℝ) → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ∈ (0[,]+∞))
191, 18syl5eqel 2702 . . 3 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ∈ (0[,]+∞))
20 eqid 2621 . . 3 (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))
2119, 20fmptd 6340 . 2 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)):ℝ⟶(0[,]+∞))
22 reex 9971 . . . . . . . 8 ℝ ∈ V
2322a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
24 ifan 4106 . . . . . . . . 9 if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0)
25 ifcl 4102 . . . . . . . . . . 11 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
263, 7, 25sylancl 693 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
277a1i 11 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑥𝐴) → 0 ∈ ℝ)
2826, 27ifclda 4092 . . . . . . . . 9 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ∈ ℝ)
2924, 28syl5eqel 2702 . . . . . . . 8 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
3029adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
31 ifan 4106 . . . . . . . . 9 if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)
32 ifcl 4102 . . . . . . . . . . 11 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
334, 7, 32sylancl 693 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
3433, 27ifclda 4092 . . . . . . . . 9 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) ∈ ℝ)
3531, 34syl5eqel 2702 . . . . . . . 8 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
3635adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
37 eqidd 2622 . . . . . . 7 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)))
38 eqidd 2622 . . . . . . 7 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
3923, 30, 36, 37, 38offval2 6867 . . . . . 6 (𝜑 → ((𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘𝑓 + (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) = (𝑥 ∈ ℝ ↦ (if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))))
40 iftrue 4064 . . . . . . . . 9 (𝑥𝐴 → if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
41 ibar 525 . . . . . . . . . . 11 (𝑥𝐴 → (0 ≤ 𝐵 ↔ (𝑥𝐴 ∧ 0 ≤ 𝐵)))
4241ifbid 4080 . . . . . . . . . 10 (𝑥𝐴 → if(0 ≤ 𝐵, 𝐵, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
43 ibar 525 . . . . . . . . . . 11 (𝑥𝐴 → (0 ≤ 𝐶 ↔ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
4443ifbid 4080 . . . . . . . . . 10 (𝑥𝐴 → if(0 ≤ 𝐶, 𝐶, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
4542, 44oveq12d 6622 . . . . . . . . 9 (𝑥𝐴 → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
4640, 45eqtr2d 2656 . . . . . . . 8 (𝑥𝐴 → (if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
47 00id 10155 . . . . . . . . 9 (0 + 0) = 0
48 simpl 473 . . . . . . . . . . . 12 ((𝑥𝐴 ∧ 0 ≤ 𝐵) → 𝑥𝐴)
4948con3i 150 . . . . . . . . . . 11 𝑥𝐴 → ¬ (𝑥𝐴 ∧ 0 ≤ 𝐵))
5049iffalsed 4069 . . . . . . . . . 10 𝑥𝐴 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = 0)
51 simpl 473 . . . . . . . . . . . 12 ((𝑥𝐴 ∧ 0 ≤ 𝐶) → 𝑥𝐴)
5251con3i 150 . . . . . . . . . . 11 𝑥𝐴 → ¬ (𝑥𝐴 ∧ 0 ≤ 𝐶))
5352iffalsed 4069 . . . . . . . . . 10 𝑥𝐴 → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = 0)
5450, 53oveq12d 6622 . . . . . . . . 9 𝑥𝐴 → (if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (0 + 0))
55 iffalse 4067 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) = 0)
5647, 54, 553eqtr4a 2681 . . . . . . . 8 𝑥𝐴 → (if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
5746, 56pm2.61i 176 . . . . . . 7 (if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)
5857mpteq2i 4701 . . . . . 6 (𝑥 ∈ ℝ ↦ (if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
5939, 58syl6eq 2671 . . . . 5 (𝜑 → ((𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘𝑓 + (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)))
6059fveq2d 6152 . . . 4 (𝜑 → (∫2‘((𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘𝑓 + (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))) = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))))
61 ibladdnclem.4 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
6261, 3mbfdm2 23311 . . . . . . 7 (𝜑𝐴 ∈ dom vol)
63 mblss 23206 . . . . . . 7 (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
6462, 63syl 17 . . . . . 6 (𝜑𝐴 ⊆ ℝ)
65 rembl 23215 . . . . . . 7 ℝ ∈ dom vol
6665a1i 11 . . . . . 6 (𝜑 → ℝ ∈ dom vol)
6729adantr 481 . . . . . 6 ((𝜑𝑥𝐴) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
68 eldifn 3711 . . . . . . . . 9 (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥𝐴)
6968adantl 482 . . . . . . . 8 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ 𝑥𝐴)
7069intnanrd 962 . . . . . . 7 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ (𝑥𝐴 ∧ 0 ≤ 𝐵))
7170iffalsed 4069 . . . . . 6 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = 0)
7242mpteq2ia 4700 . . . . . . 7 (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑥𝐴 ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
733, 61mbfpos 23324 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
7472, 73syl5eqelr 2703 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∈ MblFn)
7564, 66, 67, 71, 74mbfss 23319 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∈ MblFn)
76 max1 11959 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
777, 3, 76sylancr 694 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
78 elrege0 12220 . . . . . . . . . 10 (if(0 ≤ 𝐵, 𝐵, 0) ∈ (0[,)+∞) ↔ (if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝐵, 𝐵, 0)))
7926, 77, 78sylanbrc 697 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ (0[,)+∞))
80 0e0icopnf 12224 . . . . . . . . . 10 0 ∈ (0[,)+∞)
8180a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑥𝐴) → 0 ∈ (0[,)+∞))
8279, 81ifclda 4092 . . . . . . . 8 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ∈ (0[,)+∞))
8324, 82syl5eqel 2702 . . . . . . 7 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,)+∞))
8483adantr 481 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,)+∞))
85 eqid 2621 . . . . . 6 (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
8684, 85fmptd 6340 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,)+∞))
87 ibladdnclem.6 . . . . 5 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ)
88 max1 11959 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
897, 4, 88sylancr 694 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
90 elrege0 12220 . . . . . . . . . 10 (if(0 ≤ 𝐶, 𝐶, 0) ∈ (0[,)+∞) ↔ (if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝐶, 𝐶, 0)))
9133, 89, 90sylanbrc 697 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ (0[,)+∞))
9291, 81ifclda 4092 . . . . . . . 8 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) ∈ (0[,)+∞))
9331, 92syl5eqel 2702 . . . . . . 7 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,)+∞))
9493adantr 481 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,)+∞))
95 eqid 2621 . . . . . 6 (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
9694, 95fmptd 6340 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,)+∞))
97 ibladdnclem.7 . . . . 5 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)
9875, 86, 87, 96, 97itg2addnc 33096 . . . 4 (𝜑 → (∫2‘((𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘𝑓 + (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)))))
9960, 98eqtr3d 2657 . . 3 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))) = ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) + (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))))
10087, 97readdcld 10013 . . 3 (𝜑 → ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) + (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))) ∈ ℝ)
10199, 100eqeltrd 2698 . 2 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))) ∈ ℝ)
10226, 33readdcld 10013 . . . . . . . 8 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ)
103102rexrd 10033 . . . . . . 7 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ*)
10426, 33, 77, 89addge0d 10547 . . . . . . 7 ((𝜑𝑥𝐴) → 0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
105 elxrge0 12223 . . . . . . 7 ((if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ (0[,]+∞) ↔ ((if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ* ∧ 0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
106103, 104, 105sylanbrc 697 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ (0[,]+∞))
107106, 16ifclda 4092 . . . . 5 (𝜑 → if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) ∈ (0[,]+∞))
108107adantr 481 . . . 4 ((𝜑𝑥 ∈ ℝ) → if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) ∈ (0[,]+∞))
109 eqid 2621 . . . 4 (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
110108, 109fmptd 6340 . . 3 (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)):ℝ⟶(0[,]+∞))
111 max2 11961 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(0 ≤ 𝐵, 𝐵, 0))
1127, 3, 111sylancr 694 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵 ≤ if(0 ≤ 𝐵, 𝐵, 0))
113 max2 11961 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
1147, 4, 113sylancr 694 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
1153, 4, 26, 33, 112, 114le2addd 10590 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐵 + 𝐶) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
1162, 115eqbrtrd 4635 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐷 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
117 breq1 4616 . . . . . . . . . . 11 (𝐷 = if(0 ≤ 𝐷, 𝐷, 0) → (𝐷 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ↔ if(0 ≤ 𝐷, 𝐷, 0) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
118 breq1 4616 . . . . . . . . . . 11 (0 = if(0 ≤ 𝐷, 𝐷, 0) → (0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ↔ if(0 ≤ 𝐷, 𝐷, 0) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
119117, 118ifboth 4096 . . . . . . . . . 10 ((𝐷 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∧ 0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) → if(0 ≤ 𝐷, 𝐷, 0) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
120116, 104, 119syl2anc 692 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
121 iftrue 4064 . . . . . . . . . 10 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = if(0 ≤ 𝐷, 𝐷, 0))
122121adantl 482 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = if(0 ≤ 𝐷, 𝐷, 0))
12340adantl 482 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
124120, 122, 1233brtr4d 4645 . . . . . . . 8 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
125124ex 450 . . . . . . 7 (𝜑 → (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)))
126 0le0 11054 . . . . . . . . 9 0 ≤ 0
127126a1i 11 . . . . . . . 8 𝑥𝐴 → 0 ≤ 0)
128 iffalse 4067 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = 0)
129127, 128, 553brtr4d 4645 . . . . . . 7 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
130125, 129pm2.61d1 171 . . . . . 6 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
1311, 130syl5eqbr 4648 . . . . 5 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
132131ralrimivw 2961 . . . 4 (𝜑 → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
133 eqidd 2622 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)))
134 eqidd 2622 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)))
13523, 19, 108, 133, 134ofrfval2 6868 . . . 4 (𝜑 → ((𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)) ↔ ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)))
136132, 135mpbird 247 . . 3 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)))
137 itg2le 23412 . . 3 (((𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))))
13821, 110, 136, 137syl3anc 1323 . 2 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))))
139 itg2lecl 23411 . 2 (((𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)):ℝ⟶(0[,]+∞) ∧ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)))) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ∈ ℝ)
14021, 101, 138, 139syl3anc 1323 1 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ∈ ℝ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  Vcvv 3186   ∖ cdif 3552   ⊆ wss 3555  ifcif 4058   class class class wbr 4613   ↦ cmpt 4673  dom cdm 5074  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604   ∘𝑓 cof 6848   ∘𝑟 cofr 6849  ℝcr 9879  0cc0 9880   + caddc 9883  +∞cpnf 10015  ℝ*cxr 10017   ≤ cle 10019  [,)cico 12119  [,]cicc 12120  volcvol 23139  MblFncmbf 23289  ∫2citg2 23291 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-disj 4584  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-ofr 6851  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-sum 14351  df-rest 16004  df-topgen 16025  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-top 20621  df-bases 20622  df-topon 20623  df-cmp 21100  df-ovol 23140  df-vol 23141  df-mbf 23294  df-itg1 23295  df-itg2 23296 This theorem is referenced by:  ibladdnc  33099
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