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Theorem ibladdlem 25669
Description: Lemma for ibladd 25670. (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 4581 . . . 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 11250 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐵 + 𝐶) ∈ ℝ)
62, 5eqeltrd 2832 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐷 ∈ ℝ)
7 0re 11223 . . . . . . . . 9 0 ∈ ℝ
8 ifcl 4573 . . . . . . . . 9 ((𝐷 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ)
96, 7, 8sylancl 585 . . . . . . . 8 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ)
109rexrd 11271 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ*)
11 max1 13171 . . . . . . . 8 ((0 ∈ ℝ ∧ 𝐷 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐷, 𝐷, 0))
127, 6, 11sylancr 586 . . . . . . 7 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐷, 𝐷, 0))
13 elxrge0 13441 . . . . . . 7 (if(0 ≤ 𝐷, 𝐷, 0) ∈ (0[,]+∞) ↔ (if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ* ∧ 0 ≤ if(0 ≤ 𝐷, 𝐷, 0)))
1410, 12, 13sylanbrc 582 . . . . . 6 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ (0[,]+∞))
15 0e0iccpnf 13443 . . . . . . 7 0 ∈ (0[,]+∞)
1615a1i 11 . . . . . 6 ((𝜑 ∧ ¬ 𝑥𝐴) → 0 ∈ (0[,]+∞))
1714, 16ifclda 4563 . . . . 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, 18eqeltrid 2836 . . 3 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ∈ (0[,]+∞))
2019fmpttd 7116 . 2 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)):ℝ⟶(0[,]+∞))
21 reex 11207 . . . . . . . 8 ℝ ∈ V
2221a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
23 ifan 4581 . . . . . . . . 9 if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0)
24 ifcl 4573 . . . . . . . . . . 11 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
253, 7, 24sylancl 585 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
267a1i 11 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑥𝐴) → 0 ∈ ℝ)
2725, 26ifclda 4563 . . . . . . . . 9 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ∈ ℝ)
2823, 27eqeltrid 2836 . . . . . . . 8 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
2928adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
30 ifan 4581 . . . . . . . . 9 if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)
31 ifcl 4573 . . . . . . . . . . 11 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
324, 7, 31sylancl 585 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
3332, 26ifclda 4563 . . . . . . . . 9 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) ∈ ℝ)
3430, 33eqeltrid 2836 . . . . . . . 8 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
3534adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
36 eqidd 2732 . . . . . . 7 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)))
37 eqidd 2732 . . . . . . 7 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
3822, 29, 35, 36, 37offval2 7694 . . . . . 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 4534 . . . . . . . . 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 528 . . . . . . . . . . 11 (𝑥𝐴 → (0 ≤ 𝐵 ↔ (𝑥𝐴 ∧ 0 ≤ 𝐵)))
4140ifbid 4551 . . . . . . . . . 10 (𝑥𝐴 → if(0 ≤ 𝐵, 𝐵, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
42 ibar 528 . . . . . . . . . . 11 (𝑥𝐴 → (0 ≤ 𝐶 ↔ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
4342ifbid 4551 . . . . . . . . . 10 (𝑥𝐴 → if(0 ≤ 𝐶, 𝐶, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
4441, 43oveq12d 7430 . . . . . . . . 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 2772 . . . . . . . 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 11396 . . . . . . . . 9 (0 + 0) = 0
47 simpl 482 . . . . . . . . . . . 12 ((𝑥𝐴 ∧ 0 ≤ 𝐵) → 𝑥𝐴)
4847con3i 154 . . . . . . . . . . 11 𝑥𝐴 → ¬ (𝑥𝐴 ∧ 0 ≤ 𝐵))
4948iffalsed 4539 . . . . . . . . . 10 𝑥𝐴 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = 0)
50 simpl 482 . . . . . . . . . . . 12 ((𝑥𝐴 ∧ 0 ≤ 𝐶) → 𝑥𝐴)
5150con3i 154 . . . . . . . . . . 11 𝑥𝐴 → ¬ (𝑥𝐴 ∧ 0 ≤ 𝐶))
5251iffalsed 4539 . . . . . . . . . 10 𝑥𝐴 → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = 0)
5349, 52oveq12d 7430 . . . . . . . . 9 𝑥𝐴 → (if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (0 + 0))
54 iffalse 4537 . . . . . . . . 9 𝑥𝐴 → if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) = 0)
5546, 53, 543eqtr4a 2797 . . . . . . . 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 182 . . . . . . 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 5253 . . . . . 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 2787 . . . . 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 6895 . . . 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 25486 . . . . . . 7 (𝜑𝐴 ∈ dom vol)
62 mblss 25380 . . . . . . 7 (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
6361, 62syl 17 . . . . . 6 (𝜑𝐴 ⊆ ℝ)
64 rembl 25389 . . . . . . 7 ℝ ∈ dom vol
6564a1i 11 . . . . . 6 (𝜑 → ℝ ∈ dom vol)
6628adantr 480 . . . . . 6 ((𝜑𝑥𝐴) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
67 eldifn 4127 . . . . . . . . 9 (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥𝐴)
6867adantl 481 . . . . . . . 8 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ 𝑥𝐴)
6968intnanrd 489 . . . . . . 7 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ (𝑥𝐴 ∧ 0 ≤ 𝐵))
7069iffalsed 4539 . . . . . 6 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = 0)
7141mpteq2ia 5251 . . . . . . 7 (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑥𝐴 ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
723, 60mbfpos 25500 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
7371, 72eqeltrrid 2837 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∈ MblFn)
7463, 65, 66, 70, 73mbfss 25495 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∈ MblFn)
75 max1 13171 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
767, 3, 75sylancr 586 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
77 elrege0 13438 . . . . . . . . . 10 (if(0 ≤ 𝐵, 𝐵, 0) ∈ (0[,)+∞) ↔ (if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝐵, 𝐵, 0)))
7825, 76, 77sylanbrc 582 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ (0[,)+∞))
79 0e0icopnf 13442 . . . . . . . . . 10 0 ∈ (0[,)+∞)
8079a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑥𝐴) → 0 ∈ (0[,)+∞))
8178, 80ifclda 4563 . . . . . . . 8 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ∈ (0[,)+∞))
8223, 81eqeltrid 2836 . . . . . . 7 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,)+∞))
8382adantr 480 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,)+∞))
8483fmpttd 7116 . . . . 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 480 . . . . . 6 ((𝜑𝑥𝐴) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
8768, 52syl 17 . . . . . 6 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = 0)
8843mpteq2ia 5251 . . . . . . 7 (𝑥𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) = (𝑥𝐴 ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
89 ibladd.5 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)
904, 89mbfpos 25500 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ MblFn)
9188, 90eqeltrrid 2837 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) ∈ MblFn)
9263, 65, 86, 87, 91mbfss 25495 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) ∈ MblFn)
93 max1 13171 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
947, 4, 93sylancr 586 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
95 elrege0 13438 . . . . . . . . . 10 (if(0 ≤ 𝐶, 𝐶, 0) ∈ (0[,)+∞) ↔ (if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝐶, 𝐶, 0)))
9632, 94, 95sylanbrc 582 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ (0[,)+∞))
9796, 80ifclda 4563 . . . . . . . 8 (𝜑 → if(𝑥𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) ∈ (0[,)+∞))
9830, 97eqeltrid 2836 . . . . . . 7 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,)+∞))
9998adantr 480 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,)+∞))
10099fmpttd 7116 . . . . 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 25609 . . . 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 2773 . . 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 11250 . . 3 (𝜑 → ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) + (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))) ∈ ℝ)
105103, 104eqeltrd 2832 . 2 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))) ∈ ℝ)
10625, 32readdcld 11250 . . . . . . . 8 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ)
107106rexrd 11271 . . . . . . 7 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ*)
10825, 32, 76, 94addge0d 11797 . . . . . . 7 ((𝜑𝑥𝐴) → 0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
109 elxrge0 13441 . . . . . . 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 582 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ (0[,]+∞))
111110, 16ifclda 4563 . . . . 5 (𝜑 → if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) ∈ (0[,]+∞))
112111adantr 480 . . . 4 ((𝜑𝑥 ∈ ℝ) → if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) ∈ (0[,]+∞))
113112fmpttd 7116 . . 3 (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)):ℝ⟶(0[,]+∞))
114 max2 13173 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(0 ≤ 𝐵, 𝐵, 0))
1157, 3, 114sylancr 586 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵 ≤ if(0 ≤ 𝐵, 𝐵, 0))
116 max2 13173 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
1177, 4, 116sylancr 586 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
1183, 4, 25, 32, 115, 117le2addd 11840 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐵 + 𝐶) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
1192, 118eqbrtrd 5170 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐷 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
120 breq1 5151 . . . . . . . . . . 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 5151 . . . . . . . . . . 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 4567 . . . . . . . . . 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 583 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
124 iftrue 4534 . . . . . . . . . 10 (𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = if(0 ≤ 𝐷, 𝐷, 0))
125124adantl 481 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = if(0 ≤ 𝐷, 𝐷, 0))
12639adantl 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)))
127123, 125, 1263brtr4d 5180 . . . . . . . 8 ((𝜑𝑥𝐴) → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
128127ex 412 . . . . . . 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 12320 . . . . . . . . 9 0 ≤ 0
130129a1i 11 . . . . . . . 8 𝑥𝐴 → 0 ≤ 0)
131 iffalse 4537 . . . . . . . 8 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = 0)
132130, 131, 543brtr4d 5180 . . . . . . 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 180 . . . . . 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 5183 . . . . 5 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ≤ if(𝑥𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
135134ralrimivw 3149 . . . 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 2732 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)))
137 eqidd 2732 . . . . 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 7695 . . . 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 257 . . 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 25589 . . 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 1370 . 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 25588 . 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 1370 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 395   = wceq 1540  wcel 2105  wral 3060  Vcvv 3473  cdif 3945  wss 3948  ifcif 4528   class class class wbr 5148  cmpt 5231  dom cdm 5676  wf 6539  cfv 6543  (class class class)co 7412  f cof 7672  r cofr 7673  cr 11115  0cc0 11116   + caddc 11119  +∞cpnf 11252  *cxr 11254  cle 11256  [,)cico 13333  [,]cicc 13334  volcvol 25312  MblFncmbf 25463  2citg2 25465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642  ax-cc 10436  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194  ax-addf 11195
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-ofr 7675  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-oadd 8476  df-omul 8477  df-er 8709  df-map 8828  df-pm 8829  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fi 9412  df-sup 9443  df-inf 9444  df-oi 9511  df-dju 9902  df-card 9940  df-acn 9943  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-z 12566  df-uz 12830  df-q 12940  df-rp 12982  df-xneg 13099  df-xadd 13100  df-xmul 13101  df-ioo 13335  df-ioc 13336  df-ico 13337  df-icc 13338  df-fz 13492  df-fzo 13635  df-fl 13764  df-seq 13974  df-exp 14035  df-hash 14298  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15439  df-rlim 15440  df-sum 15640  df-rest 17375  df-topgen 17396  df-psmet 21225  df-xmet 21226  df-met 21227  df-bl 21228  df-mopn 21229  df-top 22716  df-topon 22733  df-bases 22769  df-cmp 23211  df-ovol 25313  df-vol 25314  df-mbf 25468  df-itg1 25469  df-itg2 25470  df-0p 25519
This theorem is referenced by:  ibladd  25670
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