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Theorem mbfposb 23360
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 2761 . . . . . . . . 9 𝑥0
2 nfcv 2761 . . . . . . . . 9 𝑥
3 nffvmpt1 6166 . . . . . . . . 9 𝑥((𝑥𝐴𝐵)‘𝑦)
41, 2, 3nfbr 4669 . . . . . . . 8 𝑥0 ≤ ((𝑥𝐴𝐵)‘𝑦)
54, 3, 1nfif 4093 . . . . . . 7 𝑥if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)
6 nfcv 2761 . . . . . . 7 𝑦if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0)
7 fveq2 6158 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
87breq2d 4635 . . . . . . . 8 (𝑦 = 𝑥 → (0 ≤ ((𝑥𝐴𝐵)‘𝑦) ↔ 0 ≤ ((𝑥𝐴𝐵)‘𝑥)))
98, 7ifbieq1d 4087 . . . . . . 7 (𝑦 = 𝑥 → if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0) = if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0))
105, 6, 9cbvmpt 4719 . . . . . 6 (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0))
11 simpr 477 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥𝐴)
12 mbfpos.1 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
13 eqid 2621 . . . . . . . . . . 11 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
1413fvmpt2 6258 . . . . . . . . . 10 ((𝑥𝐴𝐵 ∈ ℝ) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1511, 12, 14syl2anc 692 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1615breq2d 4635 . . . . . . . 8 ((𝜑𝑥𝐴) → (0 ≤ ((𝑥𝐴𝐵)‘𝑥) ↔ 0 ≤ 𝐵))
1716, 15ifbieq1d 4087 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0) = if(0 ≤ 𝐵, 𝐵, 0))
1817mpteq2dva 4714 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
1910, 18syl5eq 2667 . . . . 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)))
2112, 13fmptd 6351 . . . . . . 7 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)
2221adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴𝐵):𝐴⟶ℝ)
2322ffvelrnda 6325 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) ∧ 𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
24 nfcv 2761 . . . . . . . . 9 𝑦((𝑥𝐴𝐵)‘𝑥)
253, 24, 7cbvmpt 4719 . . . . . . . 8 (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) = (𝑥𝐴 ↦ ((𝑥𝐴𝐵)‘𝑥))
2615mpteq2dva 4714 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ ((𝑥𝐴𝐵)‘𝑥)) = (𝑥𝐴𝐵))
2725, 26syl5eq 2667 . . . . . . 7 (𝜑 → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) = (𝑥𝐴𝐵))
2827eleq1d 2683 . . . . . 6 (𝜑 → ((𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn ↔ (𝑥𝐴𝐵) ∈ MblFn))
2928biimpar 502 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn)
3023, 29mbfpos 23358 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) ∈ MblFn)
3120, 30eqeltrrd 2699 . . 3 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
323nfneg 10237 . . . . . . . . 9 𝑥-((𝑥𝐴𝐵)‘𝑦)
331, 2, 32nfbr 4669 . . . . . . . 8 𝑥0 ≤ -((𝑥𝐴𝐵)‘𝑦)
3433, 32, 1nfif 4093 . . . . . . 7 𝑥if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)
35 nfcv 2761 . . . . . . 7 𝑦if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0)
367negeqd 10235 . . . . . . . . 9 (𝑦 = 𝑥 → -((𝑥𝐴𝐵)‘𝑦) = -((𝑥𝐴𝐵)‘𝑥))
3736breq2d 4635 . . . . . . . 8 (𝑦 = 𝑥 → (0 ≤ -((𝑥𝐴𝐵)‘𝑦) ↔ 0 ≤ -((𝑥𝐴𝐵)‘𝑥)))
3837, 36ifbieq1d 4087 . . . . . . 7 (𝑦 = 𝑥 → if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0) = if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0))
3934, 35, 38cbvmpt 4719 . . . . . 6 (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0))
4015negeqd 10235 . . . . . . . . 9 ((𝜑𝑥𝐴) → -((𝑥𝐴𝐵)‘𝑥) = -𝐵)
4140breq2d 4635 . . . . . . . 8 ((𝜑𝑥𝐴) → (0 ≤ -((𝑥𝐴𝐵)‘𝑥) ↔ 0 ≤ -𝐵))
4241, 40ifbieq1d 4087 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0) = if(0 ≤ -𝐵, -𝐵, 0))
4342mpteq2dva 4714 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
4439, 43syl5eq 2667 . . . . 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 10417 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) ∧ 𝑦𝐴) → -((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
4723, 29mbfneg 23357 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ -((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn)
4846, 47mbfpos 23358 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) ∈ MblFn)
4945, 48eqeltrrd 2699 . . 3 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)
5031, 49jca 554 . 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 6325 . . . . 5 ((𝜑𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
5352adantlr 750 . . . 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 793 . . . . 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 2698 . . . 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 795 . . . . 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 2698 . . . 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 23359 . . 3 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn)
6151, 60eqeltrrd 2699 . 2 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑥𝐴𝐵) ∈ MblFn)
6250, 61impbida 876 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 196  wa 384   = wceq 1480  wcel 1987  ifcif 4064   class class class wbr 4623  cmpt 4683  wf 5853  cfv 5857  cr 9895  0cc0 9896  cle 10035  -cneg 10227  MblFncmbf 23323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-z 11338  df-uz 11648  df-q 11749  df-rp 11793  df-xadd 11907  df-ioo 12137  df-ico 12139  df-icc 12140  df-fz 12285  df-fzo 12423  df-fl 12549  df-seq 12758  df-exp 12817  df-hash 13074  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-sum 14367  df-xmet 19679  df-met 19680  df-ovol 23173  df-vol 23174  df-mbf 23328
This theorem is referenced by:  iblre  23500
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