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 Description: Lemma for ibladd 24431. (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 4476 . . . 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 10661 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐵 + 𝐶) ∈ ℝ)
62, 5eqeltrd 2890 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐷 ∈ ℝ)
7 0re 10634 . . . . . . . . 9 0 ∈ ℝ
8 ifcl 4469 . . . . . . . . 9 ((𝐷 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ)
96, 7, 8sylancl 589 . . . . . . . 8 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ)
109rexrd 10682 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ*)
11 max1 12568 . . . . . . . 8 ((0 ∈ ℝ ∧ 𝐷 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐷, 𝐷, 0))
127, 6, 11sylancr 590 . . . . . . 7 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐷, 𝐷, 0))
13 elxrge0 12837 . . . . . . 7 (if(0 ≤ 𝐷, 𝐷, 0) ∈ (0[,]+∞) ↔ (if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ* ∧ 0 ≤ if(0 ≤ 𝐷, 𝐷, 0)))
1410, 12, 13sylanbrc 586 . . . . . 6 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ (0[,]+∞))
15 0e0iccpnf 12839 . . . . . . 7 0 ∈ (0[,]+∞)
1615a1i 11 . . . . . 6 ((𝜑 ∧ ¬ 𝑥𝐴) → 0 ∈ (0[,]+∞))
1714, 16ifclda 4459 . . . . 5 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ∈ (0[,]+∞))
1817adantr 484 . . . 4 ((𝜑𝑥 ∈ ℝ) → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ∈ (0[,]+∞))
191, 18eqeltrid 2894 . . 3 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ∈ (0[,]+∞))
2019fmpttd 6856 . 2 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)):ℝ⟶(0[,]+∞))
21 reex 10619 . . . . . . . 8 ℝ ∈ V
2221a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
23 ifan 4476 . . . . . . . . 9 if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0)
24 ifcl 4469 . . . . . . . . . . 11 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
253, 7, 24sylancl 589 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
267a1i 11 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑥𝐴) → 0 ∈ ℝ)
2725, 26ifclda 4459 . . . . . . . . 9 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ∈ ℝ)
2823, 27eqeltrid 2894 . . . . . . . 8 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
2928adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
30 ifan 4476 . . . . . . . . 9 if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)
31 ifcl 4469 . . . . . . . . . . 11 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
324, 7, 31sylancl 589 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
3332, 26ifclda 4459 . . . . . . . . 9 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) ∈ ℝ)
3430, 33eqeltrid 2894 . . . . . . . 8 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
3534adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
36 eqidd 2799 . . . . . . 7 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)))
37 eqidd 2799 . . . . . . 7 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
3822, 29, 35, 36, 37offval2 7408 . . . . . 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))))
39 iftrue 4431 . . . . . . . . 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)))
40 ibar 532 . . . . . . . . . . 11 (𝑥𝐴 → (0 ≤ 𝐵 ↔ (𝑥𝐴 ∧ 0 ≤ 𝐵)))
4140ifbid 4447 . . . . . . . . . 10 (𝑥𝐴 → if(0 ≤ 𝐵, 𝐵, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
42 ibar 532 . . . . . . . . . . 11 (𝑥𝐴 → (0 ≤ 𝐶 ↔ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
4342ifbid 4447 . . . . . . . . . 10 (𝑥𝐴 → if(0 ≤ 𝐶, 𝐶, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
4441, 43oveq12d 7153 . . . . . . . . 9 (𝑥𝐴 → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
4539, 44eqtr2d 2834 . . . . . . . 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))
46 00id 10806 . . . . . . . . 9 (0 + 0) = 0
47 simpl 486 . . . . . . . . . . . 12 ((𝑥𝐴 ∧ 0 ≤ 𝐵) → 𝑥𝐴)
4847con3i 157 . . . . . . . . . . 11 𝑥𝐴 → ¬ (𝑥𝐴 ∧ 0 ≤ 𝐵))
4948iffalsed 4436 . . . . . . . . . 10 𝑥𝐴 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = 0)
50 simpl 486 . . . . . . . . . . . 12 ((𝑥𝐴 ∧ 0 ≤ 𝐶) → 𝑥𝐴)
5150con3i 157 . . . . . . . . . . 11 𝑥𝐴 → ¬ (𝑥𝐴 ∧ 0 ≤ 𝐶))
5251iffalsed 4436 . . . . . . . . . 10 𝑥𝐴 → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = 0)
5349, 52oveq12d 7153 . . . . . . . . 9 𝑥𝐴 → (if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (0 + 0))
54 iffalse 4434 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) = 0)
5546, 53, 543eqtr4a 2859 . . . . . . . 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))
5645, 55pm2.61i 185 . . . . . . 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)
5756mpteq2i 5122 . . . . . 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))
5838, 57eqtrdi 2849 . . . . 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)))
5958fveq2d 6649 . . . 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))))
60 ibladd.4 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
6160, 3mbfdm2 24248 . . . . . . 7 (𝜑𝐴 ∈ dom vol)
62 mblss 24142 . . . . . . 7 (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
6361, 62syl 17 . . . . . 6 (𝜑𝐴 ⊆ ℝ)
64 rembl 24151 . . . . . . 7 ℝ ∈ dom vol
6564a1i 11 . . . . . 6 (𝜑 → ℝ ∈ dom vol)
6628adantr 484 . . . . . 6 ((𝜑𝑥𝐴) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
67 eldifn 4055 . . . . . . . . 9 (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥𝐴)
6867adantl 485 . . . . . . . 8 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ 𝑥𝐴)
6968intnanrd 493 . . . . . . 7 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ (𝑥𝐴 ∧ 0 ≤ 𝐵))
7069iffalsed 4436 . . . . . 6 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = 0)
7141mpteq2ia 5121 . . . . . . 7 (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑥𝐴 ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
723, 60mbfpos 24262 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
7371, 72eqeltrrid 2895 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∈ MblFn)
7463, 65, 66, 70, 73mbfss 24257 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∈ MblFn)
75 max1 12568 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
767, 3, 75sylancr 590 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
77 elrege0 12834 . . . . . . . . . 10 (if(0 ≤ 𝐵, 𝐵, 0) ∈ (0[,)+∞) ↔ (if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝐵, 𝐵, 0)))
7825, 76, 77sylanbrc 586 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ (0[,)+∞))
79 0e0icopnf 12838 . . . . . . . . . 10 0 ∈ (0[,)+∞)
8079a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑥𝐴) → 0 ∈ (0[,)+∞))
8178, 80ifclda 4459 . . . . . . . 8 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ∈ (0[,)+∞))
8223, 81eqeltrid 2894 . . . . . . 7 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,)+∞))
8382adantr 484 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,)+∞))
8483fmpttd 6856 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,)+∞))
85 ibladd.6 . . . . 5 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ)
8634adantr 484 . . . . . 6 ((𝜑𝑥𝐴) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
8768, 52syl 17 . . . . . 6 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = 0)
8843mpteq2ia 5121 . . . . . . 7 (𝑥𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) = (𝑥𝐴 ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
89 ibladd.5 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)
904, 89mbfpos 24262 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ MblFn)
9188, 90eqeltrrid 2895 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) ∈ MblFn)
9263, 65, 86, 87, 91mbfss 24257 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) ∈ MblFn)
93 max1 12568 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
947, 4, 93sylancr 590 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
95 elrege0 12834 . . . . . . . . . 10 (if(0 ≤ 𝐶, 𝐶, 0) ∈ (0[,)+∞) ↔ (if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝐶, 𝐶, 0)))
9632, 94, 95sylanbrc 586 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ (0[,)+∞))
9796, 80ifclda 4459 . . . . . . . 8 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) ∈ (0[,)+∞))
9830, 97eqeltrid 2894 . . . . . . 7 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,)+∞))
9998adantr 484 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,)+∞))
10099fmpttd 6856 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,)+∞))
101 ibladd.7 . . . . 5 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)
10274, 84, 85, 92, 100, 101itg2add 24370 . . . 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)))))
10359, 102eqtr3d 2835 . . 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)))))
10485, 101readdcld 10661 . . 3 (𝜑 → ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) + (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))) ∈ ℝ)
105103, 104eqeltrd 2890 . 2 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))) ∈ ℝ)
10625, 32readdcld 10661 . . . . . . . 8 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ)
107106rexrd 10682 . . . . . . 7 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ*)
10825, 32, 76, 94addge0d 11207 . . . . . . 7 ((𝜑𝑥𝐴) → 0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
109 elxrge0 12837 . . . . . . 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))))
110107, 108, 109sylanbrc 586 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ (0[,]+∞))
111110, 16ifclda 4459 . . . . 5 (𝜑 → if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) ∈ (0[,]+∞))
112111adantr 484 . . . 4 ((𝜑𝑥 ∈ ℝ) → if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) ∈ (0[,]+∞))
113112fmpttd 6856 . . 3 (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)):ℝ⟶(0[,]+∞))
114 max2 12570 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(0 ≤ 𝐵, 𝐵, 0))
1157, 3, 114sylancr 590 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵 ≤ if(0 ≤ 𝐵, 𝐵, 0))
116 max2 12570 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
1177, 4, 116sylancr 590 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
1183, 4, 25, 32, 115, 117le2addd 11250 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐵 + 𝐶) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
1192, 118eqbrtrd 5052 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐷 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
120 breq1 5033 . . . . . . . . . . 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))))
121 breq1 5033 . . . . . . . . . . 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))))
122120, 121ifboth 4463 . . . . . . . . . 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)))
123119, 108, 122syl2anc 587 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
124 iftrue 4431 . . . . . . . . . 10 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = if(0 ≤ 𝐷, 𝐷, 0))
125124adantl 485 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = if(0 ≤ 𝐷, 𝐷, 0))
12639adantl 485 . . . . . . . . 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)))
127123, 125, 1263brtr4d 5062 . . . . . . . 8 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
128127ex 416 . . . . . . 7 (𝜑 → (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)))
129 0le0 11728 . . . . . . . . 9 0 ≤ 0
130129a1i 11 . . . . . . . 8 𝑥𝐴 → 0 ≤ 0)
131 iffalse 4434 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = 0)
132130, 131, 543brtr4d 5062 . . . . . . 7 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
133128, 132pm2.61d1 183 . . . . . 6 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
1341, 133eqbrtrid 5065 . . . . 5 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
135134ralrimivw 3150 . . . 4 (𝜑 → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
136 eqidd 2799 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)))
137 eqidd 2799 . . . . 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)))
13822, 19, 112, 136, 137ofrfval2 7409 . . . 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)))
139135, 138mpbird 260 . . 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)))
140 itg2le 24350 . . 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))))
14120, 113, 139, 140syl3anc 1368 . 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))))
142 itg2lecl 24349 . 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)
14320, 105, 141, 142syl3anc 1368 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 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  ifcif 4425   class class class wbr 5030   ↦ cmpt 5110  dom cdm 5519  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∘f cof 7388   ∘r cofr 7389  ℝcr 10527  0cc0 10528   + caddc 10531  +∞cpnf 10663  ℝ*cxr 10665   ≤ cle 10667  [,)cico 12730  [,]cicc 12731  volcvol 24074  MblFncmbf 24225  ∫2citg2 24227 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-inf2 9090  ax-cc 9848  ax-cnex 10584  ax-resscn 10585  ax-1cn 10586  ax-icn 10587  ax-addcl 10588  ax-addrcl 10589  ax-mulcl 10590  ax-mulrcl 10591  ax-mulcom 10592  ax-addass 10593  ax-mulass 10594  ax-distr 10595  ax-i2m1 10596  ax-1ne0 10597  ax-1rid 10598  ax-rnegex 10599  ax-rrecex 10600  ax-cnre 10601  ax-pre-lttri 10602  ax-pre-lttrn 10603  ax-pre-ltadd 10604  ax-pre-mulgt0 10605  ax-pre-sup 10606  ax-addf 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-disj 4996  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7390  df-ofr 7391  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-2o 8088  df-oadd 8091  df-omul 8092  df-er 8274  df-map 8393  df-pm 8394  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-fi 8861  df-sup 8892  df-inf 8893  df-oi 8960  df-dju 9316  df-card 9354  df-acn 9357  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-sub 10863  df-neg 10864  df-div 11289  df-nn 11628  df-2 11690  df-3 11691  df-n0 11888  df-z 11972  df-uz 12234  df-q 12339  df-rp 12380  df-xneg 12497  df-xadd 12498  df-xmul 12499  df-ioo 12732  df-ioc 12733  df-ico 12734  df-icc 12735  df-fz 12888  df-fzo 13031  df-fl 13159  df-seq 13367  df-exp 13428  df-hash 13689  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-clim 14839  df-rlim 14840  df-sum 15037  df-rest 16690  df-topgen 16711  df-psmet 20086  df-xmet 20087  df-met 20088  df-bl 20089  df-mopn 20090  df-top 21506  df-topon 21523  df-bases 21558  df-cmp 21999  df-ovol 24075  df-vol 24076  df-mbf 24230  df-itg1 24231  df-itg2 24232  df-0p 24281 This theorem is referenced by:  ibladd  24431
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