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Theorem itgaddnclem2 34012
Description: Lemma for itgaddnc 34013; cf. itgaddlem2 23989. (Contributed by Brendan Leahy, 10-Nov-2017.) (Revised by Brendan Leahy, 3-Apr-2018.)
Hypotheses
Ref Expression
ibladdnc.1 ((𝜑𝑥𝐴) → 𝐵𝑉)
ibladdnc.2 (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)
ibladdnc.3 ((𝜑𝑥𝐴) → 𝐶𝑉)
ibladdnc.4 (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)
ibladdnc.m (𝜑 → (𝑥𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)
itgaddnclem.1 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
itgaddnclem.2 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)
Assertion
Ref Expression
itgaddnclem2 (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem itgaddnclem2
StepHypRef Expression
1 itgaddnclem.1 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
2 max0sub 12315 . . . . . . . . . 10 (𝐵 ∈ ℝ → (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) = 𝐵)
31, 2syl 17 . . . . . . . . 9 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) = 𝐵)
4 itgaddnclem.2 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)
5 max0sub 12315 . . . . . . . . . 10 (𝐶 ∈ ℝ → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
64, 5syl 17 . . . . . . . . 9 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
73, 6oveq12d 6923 . . . . . . . 8 ((𝜑𝑥𝐴) → ((if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) + (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0))) = (𝐵 + 𝐶))
8 0re 10358 . . . . . . . . . . 11 0 ∈ ℝ
9 ifcl 4350 . . . . . . . . . . 11 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
101, 8, 9sylancl 582 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
1110recnd 10385 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
12 ifcl 4350 . . . . . . . . . . 11 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
134, 8, 12sylancl 582 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
1413recnd 10385 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
151renegcld 10781 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
16 ifcl 4350 . . . . . . . . . . 11 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
1715, 8, 16sylancl 582 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
1817recnd 10385 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℂ)
194renegcld 10781 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → -𝐶 ∈ ℝ)
20 ifcl 4350 . . . . . . . . . . 11 ((-𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
2119, 8, 20sylancl 582 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
2221recnd 10385 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
2311, 14, 18, 22addsub4d 10760 . . . . . . . 8 ((𝜑𝑥𝐴) → ((if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) − (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) = ((if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) + (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0))))
241, 4readdcld 10386 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐵 + 𝐶) ∈ ℝ)
25 max0sub 12315 . . . . . . . . 9 ((𝐵 + 𝐶) ∈ ℝ → (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) − if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) = (𝐵 + 𝐶))
2624, 25syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) − if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) = (𝐵 + 𝐶))
277, 23, 263eqtr4rd 2872 . . . . . . 7 ((𝜑𝑥𝐴) → (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) − if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) = ((if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) − (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))))
2824renegcld 10781 . . . . . . . . . 10 ((𝜑𝑥𝐴) → -(𝐵 + 𝐶) ∈ ℝ)
29 ifcl 4350 . . . . . . . . . 10 ((-(𝐵 + 𝐶) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) ∈ ℝ)
3028, 8, 29sylancl 582 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) ∈ ℝ)
3130recnd 10385 . . . . . . . 8 ((𝜑𝑥𝐴) → if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) ∈ ℂ)
3210, 13readdcld 10386 . . . . . . . . 9 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ)
3332recnd 10385 . . . . . . . 8 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℂ)
34 ifcl 4350 . . . . . . . . . 10 (((𝐵 + 𝐶) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) ∈ ℝ)
3524, 8, 34sylancl 582 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) ∈ ℝ)
3635recnd 10385 . . . . . . . 8 ((𝜑𝑥𝐴) → if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) ∈ ℂ)
3717, 21readdcld 10386 . . . . . . . . 9 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) ∈ ℝ)
3837recnd 10385 . . . . . . . 8 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) ∈ ℂ)
3931, 33, 36, 38addsubeq4d 10764 . . . . . . 7 ((𝜑𝑥𝐴) → ((if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) ↔ (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) − if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) = ((if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) − (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)))))
4027, 39mpbird 249 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))))
4140itgeq2dv 23947 . . . . 5 (𝜑 → ∫𝐴(if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) d𝑥 = ∫𝐴(if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) d𝑥)
42 ibladdnc.2 . . . . . . . . 9 (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)
43 ibladdnc.4 . . . . . . . . 9 (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)
44 ibladdnc.m . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)
451, 42, 4, 43, 44ibladdnc 34010 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ (𝐵 + 𝐶)) ∈ 𝐿1)
4624iblre 23959 . . . . . . . 8 (𝜑 → ((𝑥𝐴 ↦ (𝐵 + 𝐶)) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) ∈ 𝐿1)))
4745, 46mpbid 224 . . . . . . 7 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) ∈ 𝐿1))
4847simprd 491 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) ∈ 𝐿1)
491iblre 23959 . . . . . . . . 9 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)))
5042, 49mpbid 224 . . . . . . . 8 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1))
5150simpld 490 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1)
524iblre 23959 . . . . . . . . 9 (𝜑 → ((𝑥𝐴𝐶) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)) ∈ 𝐿1)))
5343, 52mpbid 224 . . . . . . . 8 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)) ∈ 𝐿1))
5453simpld 490 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ 𝐿1)
55 iblmbf 23933 . . . . . . . . 9 ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn)
5642, 55syl 17 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
57 iblmbf 23933 . . . . . . . . 9 ((𝑥𝐴𝐶) ∈ 𝐿1 → (𝑥𝐴𝐶) ∈ MblFn)
5843, 57syl 17 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)
5956, 1, 58, 4, 44mbfposadd 34000 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ∈ MblFn)
6010, 51, 13, 54, 59ibladdnc 34010 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ∈ 𝐿1)
61 max1 12304 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
628, 1, 61sylancr 583 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
63 max1 12304 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
648, 4, 63sylancr 583 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
6510, 13, 62, 64addge0d 10928 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
6665iftrued 4314 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
6766oveq2d 6921 . . . . . . . 8 ((𝜑𝑥𝐴) → (if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + if(0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)) = (if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
6867mpteq2dva 4967 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + if(0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))) = (𝑥𝐴 ↦ (if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))))
6924, 44mbfneg 23816 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ -(𝐵 + 𝐶)) ∈ MblFn)
701recnd 10385 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
714recnd 10385 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)
7270, 71negdid 10726 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -(𝐵 + 𝐶) = (-𝐵 + -𝐶))
7372oveq1d 6920 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (-(𝐵 + 𝐶) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = ((-𝐵 + -𝐶) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
7415recnd 10385 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -𝐵 ∈ ℂ)
7519recnd 10385 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -𝐶 ∈ ℂ)
7674, 75, 11, 14add4d 10583 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((-𝐵 + -𝐶) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = ((-𝐵 + if(0 ≤ 𝐵, 𝐵, 0)) + (-𝐶 + if(0 ≤ 𝐶, 𝐶, 0))))
77 negeq 10593 . . . . . . . . . . . . . . . 16 (𝐵 = 0 → -𝐵 = -0)
78 neg0 10648 . . . . . . . . . . . . . . . 16 -0 = 0
7977, 78syl6eq 2877 . . . . . . . . . . . . . . 15 (𝐵 = 0 → -𝐵 = 0)
80 0le0 11459 . . . . . . . . . . . . . . . . 17 0 ≤ 0
8180, 79syl5breqr 4911 . . . . . . . . . . . . . . . 16 (𝐵 = 0 → 0 ≤ -𝐵)
8281iftrued 4314 . . . . . . . . . . . . . . 15 (𝐵 = 0 → if(0 ≤ -𝐵, -𝐵, 0) = -𝐵)
83 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝐵 = 0 → 𝐵 = 0)
8480, 83syl5breqr 4911 . . . . . . . . . . . . . . . . . . 19 (𝐵 = 0 → 0 ≤ 𝐵)
8584iftrued 4314 . . . . . . . . . . . . . . . . . 18 (𝐵 = 0 → if(0 ≤ 𝐵, 𝐵, 0) = 𝐵)
8685, 83eqtrd 2861 . . . . . . . . . . . . . . . . 17 (𝐵 = 0 → if(0 ≤ 𝐵, 𝐵, 0) = 0)
8779, 86oveq12d 6923 . . . . . . . . . . . . . . . 16 (𝐵 = 0 → (-𝐵 + if(0 ≤ 𝐵, 𝐵, 0)) = (0 + 0))
88 00id 10530 . . . . . . . . . . . . . . . 16 (0 + 0) = 0
8987, 88syl6eq 2877 . . . . . . . . . . . . . . 15 (𝐵 = 0 → (-𝐵 + if(0 ≤ 𝐵, 𝐵, 0)) = 0)
9079, 82, 893eqtr4rd 2872 . . . . . . . . . . . . . 14 (𝐵 = 0 → (-𝐵 + if(0 ≤ 𝐵, 𝐵, 0)) = if(0 ≤ -𝐵, -𝐵, 0))
9190adantl 475 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝐵 = 0) → (-𝐵 + if(0 ≤ 𝐵, 𝐵, 0)) = if(0 ≤ -𝐵, -𝐵, 0))
92 ovif2 6998 . . . . . . . . . . . . . 14 (-𝐵 + if(0 ≤ 𝐵, 𝐵, 0)) = if(0 ≤ 𝐵, (-𝐵 + 𝐵), (-𝐵 + 0))
9370negne0bd 10706 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (𝐵 ≠ 0 ↔ -𝐵 ≠ 0))
9493biimpa 470 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → -𝐵 ≠ 0)
951le0neg2d 10924 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥𝐴) → (0 ≤ 𝐵 ↔ -𝐵 ≤ 0))
96 leloe 10443 . . . . . . . . . . . . . . . . . . . . 21 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝐵 ≤ 0 ↔ (-𝐵 < 0 ∨ -𝐵 = 0)))
9715, 8, 96sylancl 582 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥𝐴) → (-𝐵 ≤ 0 ↔ (-𝐵 < 0 ∨ -𝐵 = 0)))
9895, 97bitrd 271 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (0 ≤ 𝐵 ↔ (-𝐵 < 0 ∨ -𝐵 = 0)))
99 df-ne 3000 . . . . . . . . . . . . . . . . . . . . 21 (-𝐵 ≠ 0 ↔ ¬ -𝐵 = 0)
100 biorf 967 . . . . . . . . . . . . . . . . . . . . 21 (¬ -𝐵 = 0 → (-𝐵 < 0 ↔ (-𝐵 = 0 ∨ -𝐵 < 0)))
10199, 100sylbi 209 . . . . . . . . . . . . . . . . . . . 20 (-𝐵 ≠ 0 → (-𝐵 < 0 ↔ (-𝐵 = 0 ∨ -𝐵 < 0)))
102 orcom 903 . . . . . . . . . . . . . . . . . . . 20 ((-𝐵 = 0 ∨ -𝐵 < 0) ↔ (-𝐵 < 0 ∨ -𝐵 = 0))
103101, 102syl6rbb 280 . . . . . . . . . . . . . . . . . . 19 (-𝐵 ≠ 0 → ((-𝐵 < 0 ∨ -𝐵 = 0) ↔ -𝐵 < 0))
10498, 103sylan9bb 507 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ -𝐵 ≠ 0) → (0 ≤ 𝐵 ↔ -𝐵 < 0))
10594, 104syldan 587 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (0 ≤ 𝐵 ↔ -𝐵 < 0))
106 ltnle 10436 . . . . . . . . . . . . . . . . . . 19 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝐵 < 0 ↔ ¬ 0 ≤ -𝐵))
10715, 8, 106sylancl 582 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (-𝐵 < 0 ↔ ¬ 0 ≤ -𝐵))
108107adantr 474 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (-𝐵 < 0 ↔ ¬ 0 ≤ -𝐵))
109105, 108bitrd 271 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (0 ≤ 𝐵 ↔ ¬ 0 ≤ -𝐵))
11074, 70addcomd 10557 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (-𝐵 + 𝐵) = (𝐵 + -𝐵))
11170negidd 10703 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (𝐵 + -𝐵) = 0)
112110, 111eqtrd 2861 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (-𝐵 + 𝐵) = 0)
113112adantr 474 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (-𝐵 + 𝐵) = 0)
11474addid1d 10555 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (-𝐵 + 0) = -𝐵)
115114adantr 474 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (-𝐵 + 0) = -𝐵)
116109, 113, 115ifbieq12d 4333 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → if(0 ≤ 𝐵, (-𝐵 + 𝐵), (-𝐵 + 0)) = if(¬ 0 ≤ -𝐵, 0, -𝐵))
117 ifnot 4356 . . . . . . . . . . . . . . 15 if(¬ 0 ≤ -𝐵, 0, -𝐵) = if(0 ≤ -𝐵, -𝐵, 0)
118116, 117syl6eq 2877 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → if(0 ≤ 𝐵, (-𝐵 + 𝐵), (-𝐵 + 0)) = if(0 ≤ -𝐵, -𝐵, 0))
11992, 118syl5eq 2873 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (-𝐵 + if(0 ≤ 𝐵, 𝐵, 0)) = if(0 ≤ -𝐵, -𝐵, 0))
12091, 119pm2.61dane 3086 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (-𝐵 + if(0 ≤ 𝐵, 𝐵, 0)) = if(0 ≤ -𝐵, -𝐵, 0))
121 negeq 10593 . . . . . . . . . . . . . . . 16 (𝐶 = 0 → -𝐶 = -0)
122121, 78syl6eq 2877 . . . . . . . . . . . . . . 15 (𝐶 = 0 → -𝐶 = 0)
12380, 122syl5breqr 4911 . . . . . . . . . . . . . . . 16 (𝐶 = 0 → 0 ≤ -𝐶)
124123iftrued 4314 . . . . . . . . . . . . . . 15 (𝐶 = 0 → if(0 ≤ -𝐶, -𝐶, 0) = -𝐶)
125 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝐶 = 0 → 𝐶 = 0)
12680, 125syl5breqr 4911 . . . . . . . . . . . . . . . . . . 19 (𝐶 = 0 → 0 ≤ 𝐶)
127126iftrued 4314 . . . . . . . . . . . . . . . . . 18 (𝐶 = 0 → if(0 ≤ 𝐶, 𝐶, 0) = 𝐶)
128127, 125eqtrd 2861 . . . . . . . . . . . . . . . . 17 (𝐶 = 0 → if(0 ≤ 𝐶, 𝐶, 0) = 0)
129122, 128oveq12d 6923 . . . . . . . . . . . . . . . 16 (𝐶 = 0 → (-𝐶 + if(0 ≤ 𝐶, 𝐶, 0)) = (0 + 0))
130129, 88syl6eq 2877 . . . . . . . . . . . . . . 15 (𝐶 = 0 → (-𝐶 + if(0 ≤ 𝐶, 𝐶, 0)) = 0)
131122, 124, 1303eqtr4rd 2872 . . . . . . . . . . . . . 14 (𝐶 = 0 → (-𝐶 + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ -𝐶, -𝐶, 0))
132131adantl 475 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝐶 = 0) → (-𝐶 + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ -𝐶, -𝐶, 0))
133 ovif2 6998 . . . . . . . . . . . . . 14 (-𝐶 + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐶, (-𝐶 + 𝐶), (-𝐶 + 0))
13471negne0bd 10706 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (𝐶 ≠ 0 ↔ -𝐶 ≠ 0))
135134biimpa 470 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → -𝐶 ≠ 0)
1364le0neg2d 10924 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥𝐴) → (0 ≤ 𝐶 ↔ -𝐶 ≤ 0))
137 leloe 10443 . . . . . . . . . . . . . . . . . . . . 21 ((-𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝐶 ≤ 0 ↔ (-𝐶 < 0 ∨ -𝐶 = 0)))
13819, 8, 137sylancl 582 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥𝐴) → (-𝐶 ≤ 0 ↔ (-𝐶 < 0 ∨ -𝐶 = 0)))
139136, 138bitrd 271 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (0 ≤ 𝐶 ↔ (-𝐶 < 0 ∨ -𝐶 = 0)))
140 df-ne 3000 . . . . . . . . . . . . . . . . . . . . 21 (-𝐶 ≠ 0 ↔ ¬ -𝐶 = 0)
141 biorf 967 . . . . . . . . . . . . . . . . . . . . 21 (¬ -𝐶 = 0 → (-𝐶 < 0 ↔ (-𝐶 = 0 ∨ -𝐶 < 0)))
142140, 141sylbi 209 . . . . . . . . . . . . . . . . . . . 20 (-𝐶 ≠ 0 → (-𝐶 < 0 ↔ (-𝐶 = 0 ∨ -𝐶 < 0)))
143 orcom 903 . . . . . . . . . . . . . . . . . . . 20 ((-𝐶 = 0 ∨ -𝐶 < 0) ↔ (-𝐶 < 0 ∨ -𝐶 = 0))
144142, 143syl6rbb 280 . . . . . . . . . . . . . . . . . . 19 (-𝐶 ≠ 0 → ((-𝐶 < 0 ∨ -𝐶 = 0) ↔ -𝐶 < 0))
145139, 144sylan9bb 507 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ -𝐶 ≠ 0) → (0 ≤ 𝐶 ↔ -𝐶 < 0))
146135, 145syldan 587 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (0 ≤ 𝐶 ↔ -𝐶 < 0))
147 ltnle 10436 . . . . . . . . . . . . . . . . . . 19 ((-𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝐶 < 0 ↔ ¬ 0 ≤ -𝐶))
14819, 8, 147sylancl 582 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (-𝐶 < 0 ↔ ¬ 0 ≤ -𝐶))
149148adantr 474 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (-𝐶 < 0 ↔ ¬ 0 ≤ -𝐶))
150146, 149bitrd 271 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (0 ≤ 𝐶 ↔ ¬ 0 ≤ -𝐶))
15175, 71addcomd 10557 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (-𝐶 + 𝐶) = (𝐶 + -𝐶))
15271negidd 10703 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (𝐶 + -𝐶) = 0)
153151, 152eqtrd 2861 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (-𝐶 + 𝐶) = 0)
154153adantr 474 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (-𝐶 + 𝐶) = 0)
15575addid1d 10555 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (-𝐶 + 0) = -𝐶)
156155adantr 474 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (-𝐶 + 0) = -𝐶)
157150, 154, 156ifbieq12d 4333 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → if(0 ≤ 𝐶, (-𝐶 + 𝐶), (-𝐶 + 0)) = if(¬ 0 ≤ -𝐶, 0, -𝐶))
158 ifnot 4356 . . . . . . . . . . . . . . 15 if(¬ 0 ≤ -𝐶, 0, -𝐶) = if(0 ≤ -𝐶, -𝐶, 0)
159157, 158syl6eq 2877 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → if(0 ≤ 𝐶, (-𝐶 + 𝐶), (-𝐶 + 0)) = if(0 ≤ -𝐶, -𝐶, 0))
160133, 159syl5eq 2873 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (-𝐶 + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ -𝐶, -𝐶, 0))
161132, 160pm2.61dane 3086 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (-𝐶 + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ -𝐶, -𝐶, 0))
162120, 161oveq12d 6923 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((-𝐵 + if(0 ≤ 𝐵, 𝐵, 0)) + (-𝐶 + if(0 ≤ 𝐶, 𝐶, 0))) = (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)))
16373, 76, 1623eqtrd 2865 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (-(𝐵 + 𝐶) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)))
164163mpteq2dva 4967 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ (-(𝐵 + 𝐶) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))) = (𝑥𝐴 ↦ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))))
1651, 56mbfneg 23816 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 ↦ -𝐵) ∈ MblFn)
1664, 58mbfneg 23816 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 ↦ -𝐶) ∈ MblFn)
16772mpteq2dva 4967 . . . . . . . . . . 11 (𝜑 → (𝑥𝐴 ↦ -(𝐵 + 𝐶)) = (𝑥𝐴 ↦ (-𝐵 + -𝐶)))
168167, 69eqeltrrd 2907 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 ↦ (-𝐵 + -𝐶)) ∈ MblFn)
169165, 15, 166, 19, 168mbfposadd 34000 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) ∈ MblFn)
170164, 169eqeltrd 2906 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ (-(𝐵 + 𝐶) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))) ∈ MblFn)
17169, 28, 59, 32, 170mbfposadd 34000 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + if(0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))) ∈ MblFn)
17268, 171eqeltrrd 2907 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))) ∈ MblFn)
173 max1 12304 . . . . . . 7 ((0 ∈ ℝ ∧ -(𝐵 + 𝐶) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0))
1748, 28, 173sylancr 583 . . . . . 6 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0))
17530, 48, 32, 60, 172, 30, 32, 174, 65itgaddnclem1 34011 . . . . 5 (𝜑 → ∫𝐴(if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) d𝑥 = (∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥))
17647simpld 490 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0)) ∈ 𝐿1)
17750simprd 491 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)
17853simprd 491 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)) ∈ 𝐿1)
17917, 177, 21, 178, 169ibladdnc 34010 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) ∈ 𝐿1)
180 max1 12304 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
1818, 15, 180sylancr 583 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
182 max1 12304 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ -𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
1838, 19, 182sylancr 583 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
18417, 21, 181, 183addge0d 10928 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 0 ≤ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)))
185184iftrued 4314 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)), (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)), 0) = (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)))
186185oveq2d 6921 . . . . . . . 8 ((𝜑𝑥𝐴) → (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + if(0 ≤ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)), (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)), 0)) = (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))))
187186mpteq2dva 4967 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + if(0 ≤ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)), (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)), 0))) = (𝑥𝐴 ↦ (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)))))
18870, 71, 18, 22add4d 10583 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝐵 + 𝐶) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) = ((𝐵 + if(0 ≤ -𝐵, -𝐵, 0)) + (𝐶 + if(0 ≤ -𝐶, -𝐶, 0))))
18982, 79eqtrd 2861 . . . . . . . . . . . . . . . . 17 (𝐵 = 0 → if(0 ≤ -𝐵, -𝐵, 0) = 0)
19083, 189oveq12d 6923 . . . . . . . . . . . . . . . 16 (𝐵 = 0 → (𝐵 + if(0 ≤ -𝐵, -𝐵, 0)) = (0 + 0))
191190, 88syl6eq 2877 . . . . . . . . . . . . . . 15 (𝐵 = 0 → (𝐵 + if(0 ≤ -𝐵, -𝐵, 0)) = 0)
19283, 85, 1913eqtr4rd 2872 . . . . . . . . . . . . . 14 (𝐵 = 0 → (𝐵 + if(0 ≤ -𝐵, -𝐵, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
193192adantl 475 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝐵 = 0) → (𝐵 + if(0 ≤ -𝐵, -𝐵, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
194 ovif2 6998 . . . . . . . . . . . . . 14 (𝐵 + if(0 ≤ -𝐵, -𝐵, 0)) = if(0 ≤ -𝐵, (𝐵 + -𝐵), (𝐵 + 0))
1951le0neg1d 10923 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
196 leloe 10443 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐵 ≤ 0 ↔ (𝐵 < 0 ∨ 𝐵 = 0)))
1971, 8, 196sylancl 582 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (𝐵 ≤ 0 ↔ (𝐵 < 0 ∨ 𝐵 = 0)))
198195, 197bitr3d 273 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (0 ≤ -𝐵 ↔ (𝐵 < 0 ∨ 𝐵 = 0)))
199 df-ne 3000 . . . . . . . . . . . . . . . . . . . 20 (𝐵 ≠ 0 ↔ ¬ 𝐵 = 0)
200 biorf 967 . . . . . . . . . . . . . . . . . . . 20 𝐵 = 0 → (𝐵 < 0 ↔ (𝐵 = 0 ∨ 𝐵 < 0)))
201199, 200sylbi 209 . . . . . . . . . . . . . . . . . . 19 (𝐵 ≠ 0 → (𝐵 < 0 ↔ (𝐵 = 0 ∨ 𝐵 < 0)))
202 orcom 903 . . . . . . . . . . . . . . . . . . 19 ((𝐵 = 0 ∨ 𝐵 < 0) ↔ (𝐵 < 0 ∨ 𝐵 = 0))
203201, 202syl6rbb 280 . . . . . . . . . . . . . . . . . 18 (𝐵 ≠ 0 → ((𝐵 < 0 ∨ 𝐵 = 0) ↔ 𝐵 < 0))
204198, 203sylan9bb 507 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (0 ≤ -𝐵𝐵 < 0))
205 ltnle 10436 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐵 < 0 ↔ ¬ 0 ≤ 𝐵))
2061, 8, 205sylancl 582 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (𝐵 < 0 ↔ ¬ 0 ≤ 𝐵))
207206adantr 474 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (𝐵 < 0 ↔ ¬ 0 ≤ 𝐵))
208204, 207bitrd 271 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (0 ≤ -𝐵 ↔ ¬ 0 ≤ 𝐵))
209111adantr 474 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (𝐵 + -𝐵) = 0)
21070addid1d 10555 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (𝐵 + 0) = 𝐵)
211210adantr 474 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (𝐵 + 0) = 𝐵)
212208, 209, 211ifbieq12d 4333 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → if(0 ≤ -𝐵, (𝐵 + -𝐵), (𝐵 + 0)) = if(¬ 0 ≤ 𝐵, 0, 𝐵))
213 ifnot 4356 . . . . . . . . . . . . . . 15 if(¬ 0 ≤ 𝐵, 0, 𝐵) = if(0 ≤ 𝐵, 𝐵, 0)
214212, 213syl6eq 2877 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → if(0 ≤ -𝐵, (𝐵 + -𝐵), (𝐵 + 0)) = if(0 ≤ 𝐵, 𝐵, 0))
215194, 214syl5eq 2873 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (𝐵 + if(0 ≤ -𝐵, -𝐵, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
216193, 215pm2.61dane 3086 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐵 + if(0 ≤ -𝐵, -𝐵, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
217124, 122eqtrd 2861 . . . . . . . . . . . . . . . . 17 (𝐶 = 0 → if(0 ≤ -𝐶, -𝐶, 0) = 0)
218125, 217oveq12d 6923 . . . . . . . . . . . . . . . 16 (𝐶 = 0 → (𝐶 + if(0 ≤ -𝐶, -𝐶, 0)) = (0 + 0))
219218, 88syl6eq 2877 . . . . . . . . . . . . . . 15 (𝐶 = 0 → (𝐶 + if(0 ≤ -𝐶, -𝐶, 0)) = 0)
220125, 127, 2193eqtr4rd 2872 . . . . . . . . . . . . . 14 (𝐶 = 0 → (𝐶 + if(0 ≤ -𝐶, -𝐶, 0)) = if(0 ≤ 𝐶, 𝐶, 0))
221220adantl 475 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝐶 = 0) → (𝐶 + if(0 ≤ -𝐶, -𝐶, 0)) = if(0 ≤ 𝐶, 𝐶, 0))
222 ovif2 6998 . . . . . . . . . . . . . 14 (𝐶 + if(0 ≤ -𝐶, -𝐶, 0)) = if(0 ≤ -𝐶, (𝐶 + -𝐶), (𝐶 + 0))
2234le0neg1d 10923 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (𝐶 ≤ 0 ↔ 0 ≤ -𝐶))
224 leloe 10443 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐶 ≤ 0 ↔ (𝐶 < 0 ∨ 𝐶 = 0)))
2254, 8, 224sylancl 582 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (𝐶 ≤ 0 ↔ (𝐶 < 0 ∨ 𝐶 = 0)))
226223, 225bitr3d 273 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (0 ≤ -𝐶 ↔ (𝐶 < 0 ∨ 𝐶 = 0)))
227 df-ne 3000 . . . . . . . . . . . . . . . . . . . 20 (𝐶 ≠ 0 ↔ ¬ 𝐶 = 0)
228 biorf 967 . . . . . . . . . . . . . . . . . . . 20 𝐶 = 0 → (𝐶 < 0 ↔ (𝐶 = 0 ∨ 𝐶 < 0)))
229227, 228sylbi 209 . . . . . . . . . . . . . . . . . . 19 (𝐶 ≠ 0 → (𝐶 < 0 ↔ (𝐶 = 0 ∨ 𝐶 < 0)))
230 orcom 903 . . . . . . . . . . . . . . . . . . 19 ((𝐶 = 0 ∨ 𝐶 < 0) ↔ (𝐶 < 0 ∨ 𝐶 = 0))
231229, 230syl6rbb 280 . . . . . . . . . . . . . . . . . 18 (𝐶 ≠ 0 → ((𝐶 < 0 ∨ 𝐶 = 0) ↔ 𝐶 < 0))
232226, 231sylan9bb 507 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (0 ≤ -𝐶𝐶 < 0))
233 ltnle 10436 . . . . . . . . . . . . . . . . . . 19 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐶 < 0 ↔ ¬ 0 ≤ 𝐶))
2344, 8, 233sylancl 582 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (𝐶 < 0 ↔ ¬ 0 ≤ 𝐶))
235234adantr 474 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (𝐶 < 0 ↔ ¬ 0 ≤ 𝐶))
236232, 235bitrd 271 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (0 ≤ -𝐶 ↔ ¬ 0 ≤ 𝐶))
237152adantr 474 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (𝐶 + -𝐶) = 0)
23871addid1d 10555 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (𝐶 + 0) = 𝐶)
239238adantr 474 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (𝐶 + 0) = 𝐶)
240236, 237, 239ifbieq12d 4333 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → if(0 ≤ -𝐶, (𝐶 + -𝐶), (𝐶 + 0)) = if(¬ 0 ≤ 𝐶, 0, 𝐶))
241 ifnot 4356 . . . . . . . . . . . . . . 15 if(¬ 0 ≤ 𝐶, 0, 𝐶) = if(0 ≤ 𝐶, 𝐶, 0)
242240, 241syl6eq 2877 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → if(0 ≤ -𝐶, (𝐶 + -𝐶), (𝐶 + 0)) = if(0 ≤ 𝐶, 𝐶, 0))
243222, 242syl5eq 2873 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (𝐶 + if(0 ≤ -𝐶, -𝐶, 0)) = if(0 ≤ 𝐶, 𝐶, 0))
244221, 243pm2.61dane 3086 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐶 + if(0 ≤ -𝐶, -𝐶, 0)) = if(0 ≤ 𝐶, 𝐶, 0))
245216, 244oveq12d 6923 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝐵 + if(0 ≤ -𝐵, -𝐵, 0)) + (𝐶 + if(0 ≤ -𝐶, -𝐶, 0))) = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
246188, 245eqtrd 2861 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝐵 + 𝐶) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
247246mpteq2dva 4967 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ ((𝐵 + 𝐶) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)))) = (𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
248247, 59eqeltrd 2906 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ ((𝐵 + 𝐶) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)))) ∈ MblFn)
24944, 24, 169, 37, 248mbfposadd 34000 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + if(0 ≤ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)), (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)), 0))) ∈ MblFn)
250187, 249eqeltrrd 2907 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)))) ∈ MblFn)
251 max1 12304 . . . . . . 7 ((0 ∈ ℝ ∧ (𝐵 + 𝐶) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0))
2528, 24, 251sylancr 583 . . . . . 6 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0))
25335, 176, 37, 179, 250, 35, 37, 252, 184itgaddnclem1 34011 . . . . 5 (𝜑 → ∫𝐴(if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) d𝑥 = (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥))
25441, 175, 2533eqtr3d 2869 . . . 4 (𝜑 → (∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥) = (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥))
25530, 48itgcl 23949 . . . . 5 (𝜑 → ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥 ∈ ℂ)
25610, 51, 13, 54, 59, 10, 13, 62, 64itgaddnclem1 34011 . . . . . 6 (𝜑 → ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥))
25710, 51itgcl 23949 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 ∈ ℂ)
25813, 54itgcl 23949 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 ∈ ℂ)
259257, 258addcld 10376 . . . . . 6 (𝜑 → (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥) ∈ ℂ)
260256, 259eqeltrd 2906 . . . . 5 (𝜑 → ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 ∈ ℂ)
26135, 176itgcl 23949 . . . . 5 (𝜑 → ∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 ∈ ℂ)
26217, 177, 21, 178, 169, 17, 21, 181, 183itgaddnclem1 34011 . . . . . 6 (𝜑 → ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥 = (∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥))
26317, 177itgcl 23949 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 ∈ ℂ)
26421, 178itgcl 23949 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥 ∈ ℂ)
265263, 264addcld 10376 . . . . . 6 (𝜑 → (∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥) ∈ ℂ)
266262, 265eqeltrd 2906 . . . . 5 (𝜑 → ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥 ∈ ℂ)
267255, 260, 261, 266addsubeq4d 10764 . . . 4 (𝜑 → ((∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥) = (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥) ↔ (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 − ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥) = (∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 − ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥)))
268254, 267mpbid 224 . . 3 (𝜑 → (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 − ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥) = (∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 − ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥))
269256, 262oveq12d 6923 . . 3 (𝜑 → (∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 − ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥) = ((∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥) − (∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥)))
270257, 258, 263, 264addsub4d 10760 . . 3 (𝜑 → ((∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥) − (∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥)) = ((∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥) + (∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥)))
271268, 269, 2703eqtrd 2865 . 2 (𝜑 → (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 − ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥) = ((∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥) + (∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥)))
27224, 45itgreval 23962 . 2 (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 − ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥))
2731, 42itgreval 23962 . . 3 (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥))
2744, 43itgreval 23962 . . 3 (𝜑 → ∫𝐴𝐶 d𝑥 = (∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥))
275273, 274oveq12d 6923 . 2 (𝜑 → (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥) = ((∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥) + (∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥)))
276271, 272, 2753eqtr4d 2871 1 (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 880   = wceq 1658  wcel 2166  wne 2999  ifcif 4306   class class class wbr 4873  cmpt 4952  (class class class)co 6905  cc 10250  cr 10251  0cc0 10252   + caddc 10255   < clt 10391  cle 10392  cmin 10585  -cneg 10586  MblFncmbf 23780  𝐿1cibl 23783  citg 23784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-inf2 8815  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329  ax-pre-sup 10330  ax-addf 10331
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-disj 4842  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-of 7157  df-ofr 7158  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-oadd 7830  df-er 8009  df-map 8124  df-pm 8125  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-fi 8586  df-sup 8617  df-inf 8618  df-oi 8684  df-card 9078  df-cda 9305  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-div 11010  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-n0 11619  df-z 11705  df-uz 11969  df-q 12072  df-rp 12113  df-xneg 12232  df-xadd 12233  df-xmul 12234  df-ioo 12467  df-ico 12469  df-icc 12470  df-fz 12620  df-fzo 12761  df-fl 12888  df-mod 12964  df-seq 13096  df-exp 13155  df-hash 13411  df-cj 14216  df-re 14217  df-im 14218  df-sqrt 14352  df-abs 14353  df-clim 14596  df-sum 14794  df-rest 16436  df-topgen 16457  df-psmet 20098  df-xmet 20099  df-met 20100  df-bl 20101  df-mopn 20102  df-top 21069  df-topon 21086  df-bases 21121  df-cmp 21561  df-ovol 23630  df-vol 23631  df-mbf 23785  df-itg1 23786  df-itg2 23787  df-ibl 23788  df-itg 23789  df-0p 23836
This theorem is referenced by:  itgaddnc  34013
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