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Theorem mbfposb 24181
Description: A function is measurable iff its positive and negative parts are measurable. (Contributed by Mario Carneiro, 11-Aug-2014.)
Hypothesis
Ref Expression
mbfpos.1 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
Assertion
Ref Expression
mbfposb (𝜑 → ((𝑥𝐴𝐵) ∈ MblFn ↔ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem mbfposb
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2974 . . . . . . . . 9 𝑥0
2 nfcv 2974 . . . . . . . . 9 𝑥
3 nffvmpt1 6674 . . . . . . . . 9 𝑥((𝑥𝐴𝐵)‘𝑦)
41, 2, 3nfbr 5104 . . . . . . . 8 𝑥0 ≤ ((𝑥𝐴𝐵)‘𝑦)
54, 3, 1nfif 4492 . . . . . . 7 𝑥if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)
6 nfcv 2974 . . . . . . 7 𝑦if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0)
7 fveq2 6663 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
87breq2d 5069 . . . . . . . 8 (𝑦 = 𝑥 → (0 ≤ ((𝑥𝐴𝐵)‘𝑦) ↔ 0 ≤ ((𝑥𝐴𝐵)‘𝑥)))
98, 7ifbieq1d 4486 . . . . . . 7 (𝑦 = 𝑥 → if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0) = if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0))
105, 6, 9cbvmpt 5158 . . . . . 6 (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0))
11 simpr 485 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥𝐴)
12 mbfpos.1 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
13 eqid 2818 . . . . . . . . . . 11 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
1413fvmpt2 6771 . . . . . . . . . 10 ((𝑥𝐴𝐵 ∈ ℝ) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1511, 12, 14syl2anc 584 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1615breq2d 5069 . . . . . . . 8 ((𝜑𝑥𝐴) → (0 ≤ ((𝑥𝐴𝐵)‘𝑥) ↔ 0 ≤ 𝐵))
1716, 15ifbieq1d 4486 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0) = if(0 ≤ 𝐵, 𝐵, 0))
1817mpteq2dva 5152 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
1910, 18syl5eq 2865 . . . . 5 (𝜑 → (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
2019adantr 481 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
2112fmpttd 6871 . . . . . . 7 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)
2221adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴𝐵):𝐴⟶ℝ)
2322ffvelrnda 6843 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) ∧ 𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
24 nfcv 2974 . . . . . . . . 9 𝑦((𝑥𝐴𝐵)‘𝑥)
253, 24, 7cbvmpt 5158 . . . . . . . 8 (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) = (𝑥𝐴 ↦ ((𝑥𝐴𝐵)‘𝑥))
2615mpteq2dva 5152 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ ((𝑥𝐴𝐵)‘𝑥)) = (𝑥𝐴𝐵))
2725, 26syl5eq 2865 . . . . . . 7 (𝜑 → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) = (𝑥𝐴𝐵))
2827eleq1d 2894 . . . . . 6 (𝜑 → ((𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn ↔ (𝑥𝐴𝐵) ∈ MblFn))
2928biimpar 478 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn)
3023, 29mbfpos 24179 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) ∈ MblFn)
3120, 30eqeltrrd 2911 . . 3 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
323nfneg 10870 . . . . . . . . 9 𝑥-((𝑥𝐴𝐵)‘𝑦)
331, 2, 32nfbr 5104 . . . . . . . 8 𝑥0 ≤ -((𝑥𝐴𝐵)‘𝑦)
3433, 32, 1nfif 4492 . . . . . . 7 𝑥if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)
35 nfcv 2974 . . . . . . 7 𝑦if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0)
367negeqd 10868 . . . . . . . . 9 (𝑦 = 𝑥 → -((𝑥𝐴𝐵)‘𝑦) = -((𝑥𝐴𝐵)‘𝑥))
3736breq2d 5069 . . . . . . . 8 (𝑦 = 𝑥 → (0 ≤ -((𝑥𝐴𝐵)‘𝑦) ↔ 0 ≤ -((𝑥𝐴𝐵)‘𝑥)))
3837, 36ifbieq1d 4486 . . . . . . 7 (𝑦 = 𝑥 → if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0) = if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0))
3934, 35, 38cbvmpt 5158 . . . . . 6 (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0))
4015negeqd 10868 . . . . . . . . 9 ((𝜑𝑥𝐴) → -((𝑥𝐴𝐵)‘𝑥) = -𝐵)
4140breq2d 5069 . . . . . . . 8 ((𝜑𝑥𝐴) → (0 ≤ -((𝑥𝐴𝐵)‘𝑥) ↔ 0 ≤ -𝐵))
4241, 40ifbieq1d 4486 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0) = if(0 ≤ -𝐵, -𝐵, 0))
4342mpteq2dva 5152 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
4439, 43syl5eq 2865 . . . . 5 (𝜑 → (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
4544adantr 481 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
4623renegcld 11055 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) ∧ 𝑦𝐴) → -((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
4723, 29mbfneg 24178 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ -((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn)
4846, 47mbfpos 24179 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) ∈ MblFn)
4945, 48eqeltrrd 2911 . . 3 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)
5031, 49jca 512 . 2 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn))
5127adantr 481 . . 3 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) = (𝑥𝐴𝐵))
5221ffvelrnda 6843 . . . . 5 ((𝜑𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
5352adantlr 711 . . . 4 (((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) ∧ 𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
5419adantr 481 . . . . 5 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
55 simprl 767 . . . . 5 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
5654, 55eqeltrd 2910 . . . 4 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) ∈ MblFn)
5744adantr 481 . . . . 5 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
58 simprr 769 . . . . 5 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)
5957, 58eqeltrd 2910 . . . 4 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) ∈ MblFn)
6053, 56, 59mbfposr 24180 . . 3 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn)
6151, 60eqeltrrd 2911 . 2 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑥𝐴𝐵) ∈ MblFn)
6250, 61impbida 797 1 (𝜑 → ((𝑥𝐴𝐵) ∈ MblFn ↔ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  ifcif 4463   class class class wbr 5057  cmpt 5137  wf 6344  cfv 6348  cr 10524  0cc0 10525  cle 10664  -cneg 10859  MblFncmbf 24142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xadd 12496  df-ioo 12730  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031  df-xmet 20466  df-met 20467  df-ovol 23992  df-vol 23993  df-mbf 24147
This theorem is referenced by:  iblre  24321
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