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 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)

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  ℝcr 9973  0cc0 9974   + caddc 9977  +∞cpnf 10109  ℝ*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
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