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Theorem itgaddnclem2 34837
Description: Lemma for itgaddnc 34838; cf. itgaddlem2 24358. (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 12584 . . . . . . . . . 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 12584 . . . . . . . . . 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 7168 . . . . . . . 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 10637 . . . . . . . . . . 11 0 ∈ ℝ
9 ifcl 4514 . . . . . . . . . . 11 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
101, 8, 9sylancl 586 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
1110recnd 10663 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
12 ifcl 4514 . . . . . . . . . . 11 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
134, 8, 12sylancl 586 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
1413recnd 10663 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
151renegcld 11061 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
16 ifcl 4514 . . . . . . . . . . 11 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
1715, 8, 16sylancl 586 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
1817recnd 10663 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℂ)
194renegcld 11061 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → -𝐶 ∈ ℝ)
20 ifcl 4514 . . . . . . . . . . 11 ((-𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
2119, 8, 20sylancl 586 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
2221recnd 10663 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
2311, 14, 18, 22addsub4d 11038 . . . . . . . 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 10664 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐵 + 𝐶) ∈ ℝ)
25 max0sub 12584 . . . . . . . . 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 11061 . . . . . . . . . 10 ((𝜑𝑥𝐴) → -(𝐵 + 𝐶) ∈ ℝ)
29 ifcl 4514 . . . . . . . . . 10 ((-(𝐵 + 𝐶) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) ∈ ℝ)
3028, 8, 29sylancl 586 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) ∈ ℝ)
3130recnd 10663 . . . . . . . 8 ((𝜑𝑥𝐴) → if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) ∈ ℂ)
3210, 13readdcld 10664 . . . . . . . . 9 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ)
3332recnd 10663 . . . . . . . 8 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℂ)
34 ifcl 4514 . . . . . . . . . 10 (((𝐵 + 𝐶) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) ∈ ℝ)
3524, 8, 34sylancl 586 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) ∈ ℝ)
3635recnd 10663 . . . . . . . 8 ((𝜑𝑥𝐴) → if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) ∈ ℂ)
3717, 21readdcld 10664 . . . . . . . . 9 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) ∈ ℝ)
3837recnd 10663 . . . . . . . 8 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) ∈ ℂ)
3931, 33, 36, 38addsubeq4d 11042 . . . . . . 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 258 . . . . . 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 24316 . . . . 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 34835 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ (𝐵 + 𝐶)) ∈ 𝐿1)
4624iblre 24328 . . . . . . . 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 233 . . . . . . 7 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) ∈ 𝐿1))
4847simprd 496 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) ∈ 𝐿1)
491iblre 24328 . . . . . . . . 9 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)))
5042, 49mpbid 233 . . . . . . . 8 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1))
5150simpld 495 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1)
524iblre 24328 . . . . . . . . 9 (𝜑 → ((𝑥𝐴𝐶) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)) ∈ 𝐿1)))
5343, 52mpbid 233 . . . . . . . 8 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)) ∈ 𝐿1))
5453simpld 495 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ 𝐿1)
55 iblmbf 24302 . . . . . . . . 9 ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn)
5642, 55syl 17 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
57 iblmbf 24302 . . . . . . . . 9 ((𝑥𝐴𝐶) ∈ 𝐿1 → (𝑥𝐴𝐶) ∈ MblFn)
5843, 57syl 17 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)
5956, 1, 58, 4, 44mbfposadd 34825 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ∈ MblFn)
6010, 51, 13, 54, 59ibladdnc 34835 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ∈ 𝐿1)
61 max1 12573 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
628, 1, 61sylancr 587 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
63 max1 12573 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
648, 4, 63sylancr 587 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
6510, 13, 62, 64addge0d 11210 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
6665iftrued 4478 . . . . . . . . 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 7166 . . . . . . . 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 5158 . . . . . . 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 24185 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ -(𝐵 + 𝐶)) ∈ MblFn)
701recnd 10663 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
714recnd 10663 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)
7270, 71negdid 11004 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -(𝐵 + 𝐶) = (-𝐵 + -𝐶))
7372oveq1d 7165 . . . . . . . . . . 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 10663 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -𝐵 ∈ ℂ)
7519recnd 10663 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -𝐶 ∈ ℂ)
7674, 75, 11, 14add4d 10862 . . . . . . . . . . 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 10872 . . . . . . . . . . . . . . . 16 (𝐵 = 0 → -𝐵 = -0)
78 neg0 10926 . . . . . . . . . . . . . . . 16 -0 = 0
7977, 78syl6eq 2877 . . . . . . . . . . . . . . 15 (𝐵 = 0 → -𝐵 = 0)
80 0le0 11732 . . . . . . . . . . . . . . . . 17 0 ≤ 0
8180, 79breqtrrid 5101 . . . . . . . . . . . . . . . 16 (𝐵 = 0 → 0 ≤ -𝐵)
8281iftrued 4478 . . . . . . . . . . . . . . 15 (𝐵 = 0 → if(0 ≤ -𝐵, -𝐵, 0) = -𝐵)
83 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝐵 = 0 → 𝐵 = 0)
8480, 83breqtrrid 5101 . . . . . . . . . . . . . . . . . . 19 (𝐵 = 0 → 0 ≤ 𝐵)
8584iftrued 4478 . . . . . . . . . . . . . . . . . 18 (𝐵 = 0 → if(0 ≤ 𝐵, 𝐵, 0) = 𝐵)
8685, 83eqtrd 2861 . . . . . . . . . . . . . . . . 17 (𝐵 = 0 → if(0 ≤ 𝐵, 𝐵, 0) = 0)
8779, 86oveq12d 7168 . . . . . . . . . . . . . . . 16 (𝐵 = 0 → (-𝐵 + if(0 ≤ 𝐵, 𝐵, 0)) = (0 + 0))
88 00id 10809 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝐵 = 0) → (-𝐵 + if(0 ≤ 𝐵, 𝐵, 0)) = if(0 ≤ -𝐵, -𝐵, 0))
92 ovif2 7246 . . . . . . . . . . . . . 14 (-𝐵 + if(0 ≤ 𝐵, 𝐵, 0)) = if(0 ≤ 𝐵, (-𝐵 + 𝐵), (-𝐵 + 0))
9370negne0bd 10984 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (𝐵 ≠ 0 ↔ -𝐵 ≠ 0))
9493biimpa 477 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → -𝐵 ≠ 0)
951le0neg2d 11206 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥𝐴) → (0 ≤ 𝐵 ↔ -𝐵 ≤ 0))
96 leloe 10721 . . . . . . . . . . . . . . . . . . . . 21 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝐵 ≤ 0 ↔ (-𝐵 < 0 ∨ -𝐵 = 0)))
9715, 8, 96sylancl 586 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥𝐴) → (-𝐵 ≤ 0 ↔ (-𝐵 < 0 ∨ -𝐵 = 0)))
9895, 97bitrd 280 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (0 ≤ 𝐵 ↔ (-𝐵 < 0 ∨ -𝐵 = 0)))
99 df-ne 3022 . . . . . . . . . . . . . . . . . . . . 21 (-𝐵 ≠ 0 ↔ ¬ -𝐵 = 0)
100 biorf 932 . . . . . . . . . . . . . . . . . . . . 21 (¬ -𝐵 = 0 → (-𝐵 < 0 ↔ (-𝐵 = 0 ∨ -𝐵 < 0)))
10199, 100sylbi 218 . . . . . . . . . . . . . . . . . . . 20 (-𝐵 ≠ 0 → (-𝐵 < 0 ↔ (-𝐵 = 0 ∨ -𝐵 < 0)))
102 orcom 866 . . . . . . . . . . . . . . . . . . . 20 ((-𝐵 = 0 ∨ -𝐵 < 0) ↔ (-𝐵 < 0 ∨ -𝐵 = 0))
103101, 102syl6rbb 289 . . . . . . . . . . . . . . . . . . 19 (-𝐵 ≠ 0 → ((-𝐵 < 0 ∨ -𝐵 = 0) ↔ -𝐵 < 0))
10498, 103sylan9bb 510 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ -𝐵 ≠ 0) → (0 ≤ 𝐵 ↔ -𝐵 < 0))
10594, 104syldan 591 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (0 ≤ 𝐵 ↔ -𝐵 < 0))
106 ltnle 10714 . . . . . . . . . . . . . . . . . . 19 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝐵 < 0 ↔ ¬ 0 ≤ -𝐵))
10715, 8, 106sylancl 586 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (-𝐵 < 0 ↔ ¬ 0 ≤ -𝐵))
108107adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (-𝐵 < 0 ↔ ¬ 0 ≤ -𝐵))
109105, 108bitrd 280 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (0 ≤ 𝐵 ↔ ¬ 0 ≤ -𝐵))
11074, 70addcomd 10836 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (-𝐵 + 𝐵) = (𝐵 + -𝐵))
11170negidd 10981 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (𝐵 + -𝐵) = 0)
112110, 111eqtrd 2861 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (-𝐵 + 𝐵) = 0)
113112adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (-𝐵 + 𝐵) = 0)
11474addid1d 10834 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (-𝐵 + 0) = -𝐵)
115114adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (-𝐵 + 0) = -𝐵)
116109, 113, 115ifbieq12d 4497 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → if(0 ≤ 𝐵, (-𝐵 + 𝐵), (-𝐵 + 0)) = if(¬ 0 ≤ -𝐵, 0, -𝐵))
117 ifnot 4520 . . . . . . . . . . . . . . 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 3109 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (-𝐵 + if(0 ≤ 𝐵, 𝐵, 0)) = if(0 ≤ -𝐵, -𝐵, 0))
121 negeq 10872 . . . . . . . . . . . . . . . 16 (𝐶 = 0 → -𝐶 = -0)
122121, 78syl6eq 2877 . . . . . . . . . . . . . . 15 (𝐶 = 0 → -𝐶 = 0)
12380, 122breqtrrid 5101 . . . . . . . . . . . . . . . 16 (𝐶 = 0 → 0 ≤ -𝐶)
124123iftrued 4478 . . . . . . . . . . . . . . 15 (𝐶 = 0 → if(0 ≤ -𝐶, -𝐶, 0) = -𝐶)
125 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝐶 = 0 → 𝐶 = 0)
12680, 125breqtrrid 5101 . . . . . . . . . . . . . . . . . . 19 (𝐶 = 0 → 0 ≤ 𝐶)
127126iftrued 4478 . . . . . . . . . . . . . . . . . 18 (𝐶 = 0 → if(0 ≤ 𝐶, 𝐶, 0) = 𝐶)
128127, 125eqtrd 2861 . . . . . . . . . . . . . . . . 17 (𝐶 = 0 → if(0 ≤ 𝐶, 𝐶, 0) = 0)
129122, 128oveq12d 7168 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝐶 = 0) → (-𝐶 + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ -𝐶, -𝐶, 0))
133 ovif2 7246 . . . . . . . . . . . . . 14 (-𝐶 + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐶, (-𝐶 + 𝐶), (-𝐶 + 0))
13471negne0bd 10984 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (𝐶 ≠ 0 ↔ -𝐶 ≠ 0))
135134biimpa 477 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → -𝐶 ≠ 0)
1364le0neg2d 11206 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥𝐴) → (0 ≤ 𝐶 ↔ -𝐶 ≤ 0))
137 leloe 10721 . . . . . . . . . . . . . . . . . . . . 21 ((-𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝐶 ≤ 0 ↔ (-𝐶 < 0 ∨ -𝐶 = 0)))
13819, 8, 137sylancl 586 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥𝐴) → (-𝐶 ≤ 0 ↔ (-𝐶 < 0 ∨ -𝐶 = 0)))
139136, 138bitrd 280 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (0 ≤ 𝐶 ↔ (-𝐶 < 0 ∨ -𝐶 = 0)))
140 df-ne 3022 . . . . . . . . . . . . . . . . . . . . 21 (-𝐶 ≠ 0 ↔ ¬ -𝐶 = 0)
141 biorf 932 . . . . . . . . . . . . . . . . . . . . 21 (¬ -𝐶 = 0 → (-𝐶 < 0 ↔ (-𝐶 = 0 ∨ -𝐶 < 0)))
142140, 141sylbi 218 . . . . . . . . . . . . . . . . . . . 20 (-𝐶 ≠ 0 → (-𝐶 < 0 ↔ (-𝐶 = 0 ∨ -𝐶 < 0)))
143 orcom 866 . . . . . . . . . . . . . . . . . . . 20 ((-𝐶 = 0 ∨ -𝐶 < 0) ↔ (-𝐶 < 0 ∨ -𝐶 = 0))
144142, 143syl6rbb 289 . . . . . . . . . . . . . . . . . . 19 (-𝐶 ≠ 0 → ((-𝐶 < 0 ∨ -𝐶 = 0) ↔ -𝐶 < 0))
145139, 144sylan9bb 510 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ -𝐶 ≠ 0) → (0 ≤ 𝐶 ↔ -𝐶 < 0))
146135, 145syldan 591 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (0 ≤ 𝐶 ↔ -𝐶 < 0))
147 ltnle 10714 . . . . . . . . . . . . . . . . . . 19 ((-𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝐶 < 0 ↔ ¬ 0 ≤ -𝐶))
14819, 8, 147sylancl 586 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (-𝐶 < 0 ↔ ¬ 0 ≤ -𝐶))
149148adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (-𝐶 < 0 ↔ ¬ 0 ≤ -𝐶))
150146, 149bitrd 280 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (0 ≤ 𝐶 ↔ ¬ 0 ≤ -𝐶))
15175, 71addcomd 10836 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (-𝐶 + 𝐶) = (𝐶 + -𝐶))
15271negidd 10981 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (𝐶 + -𝐶) = 0)
153151, 152eqtrd 2861 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (-𝐶 + 𝐶) = 0)
154153adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (-𝐶 + 𝐶) = 0)
15575addid1d 10834 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (-𝐶 + 0) = -𝐶)
156155adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (-𝐶 + 0) = -𝐶)
157150, 154, 156ifbieq12d 4497 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → if(0 ≤ 𝐶, (-𝐶 + 𝐶), (-𝐶 + 0)) = if(¬ 0 ≤ -𝐶, 0, -𝐶))
158 ifnot 4520 . . . . . . . . . . . . . . 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 3109 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (-𝐶 + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ -𝐶, -𝐶, 0))
162120, 161oveq12d 7168 . . . . . . . . . . 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 5158 . . . . . . . . 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 24185 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 ↦ -𝐵) ∈ MblFn)
1664, 58mbfneg 24185 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 ↦ -𝐶) ∈ MblFn)
16772mpteq2dva 5158 . . . . . . . . . . 11 (𝜑 → (𝑥𝐴 ↦ -(𝐵 + 𝐶)) = (𝑥𝐴 ↦ (-𝐵 + -𝐶)))
168167, 69eqeltrrd 2919 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 ↦ (-𝐵 + -𝐶)) ∈ MblFn)
169165, 15, 166, 19, 168mbfposadd 34825 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) ∈ MblFn)
170164, 169eqeltrd 2918 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ (-(𝐵 + 𝐶) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))) ∈ MblFn)
17169, 28, 59, 32, 170mbfposadd 34825 . . . . . . 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 2919 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))) ∈ MblFn)
173 max1 12573 . . . . . . 7 ((0 ∈ ℝ ∧ -(𝐵 + 𝐶) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0))
1748, 28, 173sylancr 587 . . . . . 6 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0))
17530, 48, 32, 60, 172, 30, 32, 174, 65itgaddnclem1 34836 . . . . 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 495 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0)) ∈ 𝐿1)
17750simprd 496 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)
17853simprd 496 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)) ∈ 𝐿1)
17917, 177, 21, 178, 169ibladdnc 34835 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) ∈ 𝐿1)
180 max1 12573 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
1818, 15, 180sylancr 587 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
182 max1 12573 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ -𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
1838, 19, 182sylancr 587 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
18417, 21, 181, 183addge0d 11210 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 0 ≤ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)))
185184iftrued 4478 . . . . . . . . 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 7166 . . . . . . . 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 5158 . . . . . . 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 10862 . . . . . . . . . . 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 7168 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝐵 = 0) → (𝐵 + if(0 ≤ -𝐵, -𝐵, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
194 ovif2 7246 . . . . . . . . . . . . . 14 (𝐵 + if(0 ≤ -𝐵, -𝐵, 0)) = if(0 ≤ -𝐵, (𝐵 + -𝐵), (𝐵 + 0))
1951le0neg1d 11205 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
196 leloe 10721 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐵 ≤ 0 ↔ (𝐵 < 0 ∨ 𝐵 = 0)))
1971, 8, 196sylancl 586 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (𝐵 ≤ 0 ↔ (𝐵 < 0 ∨ 𝐵 = 0)))
198195, 197bitr3d 282 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (0 ≤ -𝐵 ↔ (𝐵 < 0 ∨ 𝐵 = 0)))
199 df-ne 3022 . . . . . . . . . . . . . . . . . . . 20 (𝐵 ≠ 0 ↔ ¬ 𝐵 = 0)
200 biorf 932 . . . . . . . . . . . . . . . . . . . 20 𝐵 = 0 → (𝐵 < 0 ↔ (𝐵 = 0 ∨ 𝐵 < 0)))
201199, 200sylbi 218 . . . . . . . . . . . . . . . . . . 19 (𝐵 ≠ 0 → (𝐵 < 0 ↔ (𝐵 = 0 ∨ 𝐵 < 0)))
202 orcom 866 . . . . . . . . . . . . . . . . . . 19 ((𝐵 = 0 ∨ 𝐵 < 0) ↔ (𝐵 < 0 ∨ 𝐵 = 0))
203201, 202syl6rbb 289 . . . . . . . . . . . . . . . . . 18 (𝐵 ≠ 0 → ((𝐵 < 0 ∨ 𝐵 = 0) ↔ 𝐵 < 0))
204198, 203sylan9bb 510 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (0 ≤ -𝐵𝐵 < 0))
205 ltnle 10714 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐵 < 0 ↔ ¬ 0 ≤ 𝐵))
2061, 8, 205sylancl 586 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (𝐵 < 0 ↔ ¬ 0 ≤ 𝐵))
207206adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (𝐵 < 0 ↔ ¬ 0 ≤ 𝐵))
208204, 207bitrd 280 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (0 ≤ -𝐵 ↔ ¬ 0 ≤ 𝐵))
209111adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (𝐵 + -𝐵) = 0)
21070addid1d 10834 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (𝐵 + 0) = 𝐵)
211210adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → (𝐵 + 0) = 𝐵)
212208, 209, 211ifbieq12d 4497 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝐵 ≠ 0) → if(0 ≤ -𝐵, (𝐵 + -𝐵), (𝐵 + 0)) = if(¬ 0 ≤ 𝐵, 0, 𝐵))
213 ifnot 4520 . . . . . . . . . . . . . . 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 3109 . . . . . . . . . . . 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 7168 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝐶 = 0) → (𝐶 + if(0 ≤ -𝐶, -𝐶, 0)) = if(0 ≤ 𝐶, 𝐶, 0))
222 ovif2 7246 . . . . . . . . . . . . . 14 (𝐶 + if(0 ≤ -𝐶, -𝐶, 0)) = if(0 ≤ -𝐶, (𝐶 + -𝐶), (𝐶 + 0))
2234le0neg1d 11205 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (𝐶 ≤ 0 ↔ 0 ≤ -𝐶))
224 leloe 10721 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐶 ≤ 0 ↔ (𝐶 < 0 ∨ 𝐶 = 0)))
2254, 8, 224sylancl 586 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → (𝐶 ≤ 0 ↔ (𝐶 < 0 ∨ 𝐶 = 0)))
226223, 225bitr3d 282 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (0 ≤ -𝐶 ↔ (𝐶 < 0 ∨ 𝐶 = 0)))
227 df-ne 3022 . . . . . . . . . . . . . . . . . . . 20 (𝐶 ≠ 0 ↔ ¬ 𝐶 = 0)
228 biorf 932 . . . . . . . . . . . . . . . . . . . 20 𝐶 = 0 → (𝐶 < 0 ↔ (𝐶 = 0 ∨ 𝐶 < 0)))
229227, 228sylbi 218 . . . . . . . . . . . . . . . . . . 19 (𝐶 ≠ 0 → (𝐶 < 0 ↔ (𝐶 = 0 ∨ 𝐶 < 0)))
230 orcom 866 . . . . . . . . . . . . . . . . . . 19 ((𝐶 = 0 ∨ 𝐶 < 0) ↔ (𝐶 < 0 ∨ 𝐶 = 0))
231229, 230syl6rbb 289 . . . . . . . . . . . . . . . . . 18 (𝐶 ≠ 0 → ((𝐶 < 0 ∨ 𝐶 = 0) ↔ 𝐶 < 0))
232226, 231sylan9bb 510 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (0 ≤ -𝐶𝐶 < 0))
233 ltnle 10714 . . . . . . . . . . . . . . . . . . 19 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐶 < 0 ↔ ¬ 0 ≤ 𝐶))
2344, 8, 233sylancl 586 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (𝐶 < 0 ↔ ¬ 0 ≤ 𝐶))
235234adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (𝐶 < 0 ↔ ¬ 0 ≤ 𝐶))
236232, 235bitrd 280 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (0 ≤ -𝐶 ↔ ¬ 0 ≤ 𝐶))
237152adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (𝐶 + -𝐶) = 0)
23871addid1d 10834 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (𝐶 + 0) = 𝐶)
239238adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → (𝐶 + 0) = 𝐶)
240236, 237, 239ifbieq12d 4497 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝐶 ≠ 0) → if(0 ≤ -𝐶, (𝐶 + -𝐶), (𝐶 + 0)) = if(¬ 0 ≤ 𝐶, 0, 𝐶))
241 ifnot 4520 . . . . . . . . . . . . . . 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 3109 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐶 + if(0 ≤ -𝐶, -𝐶, 0)) = if(0 ≤ 𝐶, 𝐶, 0))
245216, 244oveq12d 7168 . . . . . . . . . . 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 5158 . . . . . . . . 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 2918 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ ((𝐵 + 𝐶) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)))) ∈ MblFn)
24944, 24, 169, 37, 248mbfposadd 34825 . . . . . . 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 2919 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)))) ∈ MblFn)
251 max1 12573 . . . . . . 7 ((0 ∈ ℝ ∧ (𝐵 + 𝐶) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0))
2528, 24, 251sylancr 587 . . . . . 6 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0))
25335, 176, 37, 179, 250, 35, 37, 252, 184itgaddnclem1 34836 . . . . 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 24318 . . . . 5 (𝜑 → ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥 ∈ ℂ)
25610, 51, 13, 54, 59, 10, 13, 62, 64itgaddnclem1 34836 . . . . . 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 24318 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 ∈ ℂ)
25813, 54itgcl 24318 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 ∈ ℂ)
259257, 258addcld 10654 . . . . . 6 (𝜑 → (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥) ∈ ℂ)
260256, 259eqeltrd 2918 . . . . 5 (𝜑 → ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 ∈ ℂ)
26135, 176itgcl 24318 . . . . 5 (𝜑 → ∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 ∈ ℂ)
26217, 177, 21, 178, 169, 17, 21, 181, 183itgaddnclem1 34836 . . . . . 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 24318 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 ∈ ℂ)
26421, 178itgcl 24318 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥 ∈ ℂ)
265263, 264addcld 10654 . . . . . 6 (𝜑 → (∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥) ∈ ℂ)
266262, 265eqeltrd 2918 . . . . 5 (𝜑 → ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥 ∈ ℂ)
267255, 260, 261, 266addsubeq4d 11042 . . . 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 233 . . 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 7168 . . 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 11038 . . 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 24331 . 2 (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 − ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥))
2731, 42itgreval 24331 . . 3 (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥))
2744, 43itgreval 24331 . . 3 (𝜑 → ∫𝐴𝐶 d𝑥 = (∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥))
275273, 274oveq12d 7168 . 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 207  wa 396  wo 843   = wceq 1530  wcel 2107  wne 3021  ifcif 4470   class class class wbr 5063  cmpt 5143  (class class class)co 7150  cc 10529  cr 10530  0cc0 10531   + caddc 10534   < clt 10669  cle 10670  cmin 10864  -cneg 10865  MblFncmbf 24149  𝐿1cibl 24152  citg 24153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609  ax-addf 10610
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-disj 5029  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-ofr 7404  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8284  df-map 8403  df-pm 8404  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fi 8869  df-sup 8900  df-inf 8901  df-oi 8968  df-dju 9324  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-n0 11892  df-z 11976  df-uz 12238  df-q 12343  df-rp 12385  df-xneg 12502  df-xadd 12503  df-xmul 12504  df-ioo 12737  df-ico 12739  df-icc 12740  df-fz 12888  df-fzo 13029  df-fl 13157  df-mod 13233  df-seq 13365  df-exp 13425  df-hash 13686  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-sum 15038  df-rest 16691  df-topgen 16712  df-psmet 20472  df-xmet 20473  df-met 20474  df-bl 20475  df-mopn 20476  df-top 21437  df-topon 21454  df-bases 21489  df-cmp 21930  df-ovol 23999  df-vol 24000  df-mbf 24154  df-itg1 24155  df-itg2 24156  df-ibl 24157  df-itg 24158  df-0p 24205
This theorem is referenced by:  itgaddnc  34838
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