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

Theorem ibladdlem 23631
Description: Lemma for ibladd 23632. (Contributed by Mario Carneiro, 17-Aug-2014.)
Hypotheses
Ref Expression
ibladd.1 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
ibladd.2 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)
ibladd.3 ((𝜑𝑥𝐴) → 𝐷 = (𝐵 + 𝐶))
ibladd.4 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
ibladd.5 (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)
ibladd.6 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ)
ibladd.7 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)
Assertion
Ref Expression
ibladdlem (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)

Proof of Theorem ibladdlem
StepHypRef Expression
1 ifan 4167 . . . 4 if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) = if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0)
2 ibladd.3 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐷 = (𝐵 + 𝐶))
3 ibladd.1 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
4 ibladd.2 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)
53, 4readdcld 10107 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐵 + 𝐶) ∈ ℝ)
62, 5eqeltrd 2730 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐷 ∈ ℝ)
7 0re 10078 . . . . . . . . 9 0 ∈ ℝ
8 ifcl 4163 . . . . . . . . 9 ((𝐷 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ)
96, 7, 8sylancl 695 . . . . . . . 8 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ)
109rexrd 10127 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ*)
11 max1 12054 . . . . . . . 8 ((0 ∈ ℝ ∧ 𝐷 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐷, 𝐷, 0))
127, 6, 11sylancr 696 . . . . . . 7 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐷, 𝐷, 0))
13 elxrge0 12319 . . . . . . 7 (if(0 ≤ 𝐷, 𝐷, 0) ∈ (0[,]+∞) ↔ (if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ* ∧ 0 ≤ if(0 ≤ 𝐷, 𝐷, 0)))
1410, 12, 13sylanbrc 699 . . . . . 6 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ (0[,]+∞))
15 0e0iccpnf 12321 . . . . . . 7 0 ∈ (0[,]+∞)
1615a1i 11 . . . . . 6 ((𝜑 ∧ ¬ 𝑥𝐴) → 0 ∈ (0[,]+∞))
1714, 16ifclda 4153 . . . . 5 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ∈ (0[,]+∞))
1817adantr 480 . . . 4 ((𝜑𝑥 ∈ ℝ) → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ∈ (0[,]+∞))
191, 18syl5eqel 2734 . . 3 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ∈ (0[,]+∞))
20 eqid 2651 . . 3 (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))
2119, 20fmptd 6425 . 2 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)):ℝ⟶(0[,]+∞))
22 reex 10065 . . . . . . . 8 ℝ ∈ V
2322a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
24 ifan 4167 . . . . . . . . 9 if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0)
25 ifcl 4163 . . . . . . . . . . 11 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
263, 7, 25sylancl 695 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
277a1i 11 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑥𝐴) → 0 ∈ ℝ)
2826, 27ifclda 4153 . . . . . . . . 9 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ∈ ℝ)
2924, 28syl5eqel 2734 . . . . . . . 8 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
3029adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
31 ifan 4167 . . . . . . . . 9 if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)
32 ifcl 4163 . . . . . . . . . . 11 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
334, 7, 32sylancl 695 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
3433, 27ifclda 4153 . . . . . . . . 9 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) ∈ ℝ)
3531, 34syl5eqel 2734 . . . . . . . 8 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
3635adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
37 eqidd 2652 . . . . . . 7 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)))
38 eqidd 2652 . . . . . . 7 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
3923, 30, 36, 37, 38offval2 6956 . . . . . 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 4125 . . . . . . . . 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 524 . . . . . . . . . . 11 (𝑥𝐴 → (0 ≤ 𝐵 ↔ (𝑥𝐴 ∧ 0 ≤ 𝐵)))
4241ifbid 4141 . . . . . . . . . 10 (𝑥𝐴 → if(0 ≤ 𝐵, 𝐵, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
43 ibar 524 . . . . . . . . . . 11 (𝑥𝐴 → (0 ≤ 𝐶 ↔ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
4443ifbid 4141 . . . . . . . . . 10 (𝑥𝐴 → if(0 ≤ 𝐶, 𝐶, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
4542, 44oveq12d 6708 . . . . . . . . 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 2686 . . . . . . . 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 10249 . . . . . . . . 9 (0 + 0) = 0
48 simpl 472 . . . . . . . . . . . 12 ((𝑥𝐴 ∧ 0 ≤ 𝐵) → 𝑥𝐴)
4948con3i 150 . . . . . . . . . . 11 𝑥𝐴 → ¬ (𝑥𝐴 ∧ 0 ≤ 𝐵))
5049iffalsed 4130 . . . . . . . . . 10 𝑥𝐴 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = 0)
51 simpl 472 . . . . . . . . . . . 12 ((𝑥𝐴 ∧ 0 ≤ 𝐶) → 𝑥𝐴)
5251con3i 150 . . . . . . . . . . 11 𝑥𝐴 → ¬ (𝑥𝐴 ∧ 0 ≤ 𝐶))
5352iffalsed 4130 . . . . . . . . . 10 𝑥𝐴 → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = 0)
5450, 53oveq12d 6708 . . . . . . . . 9 𝑥𝐴 → (if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (0 + 0))
55 iffalse 4128 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) = 0)
5647, 54, 553eqtr4a 2711 . . . . . . . 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 4774 . . . . . 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 2701 . . . . 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 6233 . . . 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 ibladd.4 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
6261, 3mbfdm2 23450 . . . . . . 7 (𝜑𝐴 ∈ dom vol)
63 mblss 23345 . . . . . . 7 (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
6462, 63syl 17 . . . . . 6 (𝜑𝐴 ⊆ ℝ)
65 rembl 23354 . . . . . . 7 ℝ ∈ dom vol
6665a1i 11 . . . . . 6 (𝜑 → ℝ ∈ dom vol)
6729adantr 480 . . . . . 6 ((𝜑𝑥𝐴) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
68 eldifn 3766 . . . . . . . . 9 (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥𝐴)
6968adantl 481 . . . . . . . 8 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ 𝑥𝐴)
7069intnanrd 983 . . . . . . 7 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ (𝑥𝐴 ∧ 0 ≤ 𝐵))
7170iffalsed 4130 . . . . . 6 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = 0)
7242mpteq2ia 4773 . . . . . . 7 (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑥𝐴 ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
733, 61mbfpos 23463 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
7472, 73syl5eqelr 2735 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∈ MblFn)
7564, 66, 67, 71, 74mbfss 23458 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∈ MblFn)
76 max1 12054 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
777, 3, 76sylancr 696 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
78 elrege0 12316 . . . . . . . . . 10 (if(0 ≤ 𝐵, 𝐵, 0) ∈ (0[,)+∞) ↔ (if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝐵, 𝐵, 0)))
7926, 77, 78sylanbrc 699 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ (0[,)+∞))
80 0e0icopnf 12320 . . . . . . . . . 10 0 ∈ (0[,)+∞)
8180a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑥𝐴) → 0 ∈ (0[,)+∞))
8279, 81ifclda 4153 . . . . . . . 8 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ∈ (0[,)+∞))
8324, 82syl5eqel 2734 . . . . . . 7 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,)+∞))
8483adantr 480 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,)+∞))
85 eqid 2651 . . . . . 6 (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
8684, 85fmptd 6425 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,)+∞))
87 ibladd.6 . . . . 5 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ)
8835adantr 480 . . . . . 6 ((𝜑𝑥𝐴) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
8969, 53syl 17 . . . . . 6 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = 0)
9044mpteq2ia 4773 . . . . . . 7 (𝑥𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) = (𝑥𝐴 ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
91 ibladd.5 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)
924, 91mbfpos 23463 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ MblFn)
9390, 92syl5eqelr 2735 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) ∈ MblFn)
9464, 66, 88, 89, 93mbfss 23458 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) ∈ MblFn)
95 max1 12054 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
967, 4, 95sylancr 696 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
97 elrege0 12316 . . . . . . . . . 10 (if(0 ≤ 𝐶, 𝐶, 0) ∈ (0[,)+∞) ↔ (if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝐶, 𝐶, 0)))
9833, 96, 97sylanbrc 699 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ (0[,)+∞))
9998, 81ifclda 4153 . . . . . . . 8 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) ∈ (0[,)+∞))
10031, 99syl5eqel 2734 . . . . . . 7 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,)+∞))
101100adantr 480 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,)+∞))
102 eqid 2651 . . . . . 6 (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
103101, 102fmptd 6425 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,)+∞))
104 ibladd.7 . . . . 5 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)
10575, 86, 87, 94, 103, 104itg2add 23571 . . . 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)))))
10660, 105eqtr3d 2687 . . 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)))))
10787, 104readdcld 10107 . . 3 (𝜑 → ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) + (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))) ∈ ℝ)
108106, 107eqeltrd 2730 . 2 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))) ∈ ℝ)
10926, 33readdcld 10107 . . . . . . . 8 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ)
110109rexrd 10127 . . . . . . 7 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ*)
11126, 33, 77, 96addge0d 10641 . . . . . . 7 ((𝜑𝑥𝐴) → 0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
112 elxrge0 12319 . . . . . . 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))))
113110, 111, 112sylanbrc 699 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ (0[,]+∞))
114113, 16ifclda 4153 . . . . 5 (𝜑 → if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) ∈ (0[,]+∞))
115114adantr 480 . . . 4 ((𝜑𝑥 ∈ ℝ) → if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) ∈ (0[,]+∞))
116 eqid 2651 . . . 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))
117115, 116fmptd 6425 . . 3 (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)):ℝ⟶(0[,]+∞))
118 max2 12056 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(0 ≤ 𝐵, 𝐵, 0))
1197, 3, 118sylancr 696 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵 ≤ if(0 ≤ 𝐵, 𝐵, 0))
120 max2 12056 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
1217, 4, 120sylancr 696 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
1223, 4, 26, 33, 119, 121le2addd 10684 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐵 + 𝐶) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
1232, 122eqbrtrd 4707 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐷 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
124 breq1 4688 . . . . . . . . . . 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))))
125 breq1 4688 . . . . . . . . . . 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))))
126124, 125ifboth 4157 . . . . . . . . . 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)))
127123, 111, 126syl2anc 694 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
128 iftrue 4125 . . . . . . . . . 10 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = if(0 ≤ 𝐷, 𝐷, 0))
129128adantl 481 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = if(0 ≤ 𝐷, 𝐷, 0))
13040adantl 481 . . . . . . . . 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)))
131127, 129, 1303brtr4d 4717 . . . . . . . 8 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
132131ex 449 . . . . . . 7 (𝜑 → (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)))
133 0le0 11148 . . . . . . . . 9 0 ≤ 0
134133a1i 11 . . . . . . . 8 𝑥𝐴 → 0 ≤ 0)
135 iffalse 4128 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = 0)
136134, 135, 553brtr4d 4717 . . . . . . 7 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
137132, 136pm2.61d1 171 . . . . . 6 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
1381, 137syl5eqbr 4720 . . . . 5 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
139138ralrimivw 2996 . . . 4 (𝜑 → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
140 eqidd 2652 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)))
141 eqidd 2652 . . . . 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)))
14223, 19, 115, 140, 141ofrfval2 6957 . . . 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)))
143139, 142mpbird 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)))
144 itg2le 23551 . . 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))))
14521, 117, 143, 144syl3anc 1366 . 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))))
146 itg2lecl 23550 . 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)
14721, 108, 145, 146syl3anc 1366 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 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cdif 3604  wss 3607  ifcif 4119   class class class wbr 4685  cmpt 4762  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690  𝑓 cof 6937  𝑟 cofr 6938  cr 9973  0cc0 9974   + caddc 9977  +∞cpnf 10109  *cxr 10111  cle 10113  [,)cico 12215  [,]cicc 12216  volcvol 23278  MblFncmbf 23428  2citg2 23430
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-cc 9295  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-omul 7610  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  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-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  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-rlim 14264  df-sum 14461  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280  df-mbf 23433  df-itg1 23434  df-itg2 23435  df-0p 23482
This theorem is referenced by:  ibladd  23632
  Copyright terms: Public domain W3C validator