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Theorem mbfposb 23590
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 2890 . . . . . . . . 9 𝑥0
2 nfcv 2890 . . . . . . . . 9 𝑥
3 nffvmpt1 6348 . . . . . . . . 9 𝑥((𝑥𝐴𝐵)‘𝑦)
41, 2, 3nfbr 4839 . . . . . . . 8 𝑥0 ≤ ((𝑥𝐴𝐵)‘𝑦)
54, 3, 1nfif 4247 . . . . . . 7 𝑥if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)
6 nfcv 2890 . . . . . . 7 𝑦if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0)
7 fveq2 6340 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
87breq2d 4804 . . . . . . . 8 (𝑦 = 𝑥 → (0 ≤ ((𝑥𝐴𝐵)‘𝑦) ↔ 0 ≤ ((𝑥𝐴𝐵)‘𝑥)))
98, 7ifbieq1d 4241 . . . . . . 7 (𝑦 = 𝑥 → if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0) = if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0))
105, 6, 9cbvmpt 4889 . . . . . 6 (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0))
11 simpr 479 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥𝐴)
12 mbfpos.1 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
13 eqid 2748 . . . . . . . . . . 11 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
1413fvmpt2 6441 . . . . . . . . . 10 ((𝑥𝐴𝐵 ∈ ℝ) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1511, 12, 14syl2anc 696 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1615breq2d 4804 . . . . . . . 8 ((𝜑𝑥𝐴) → (0 ≤ ((𝑥𝐴𝐵)‘𝑥) ↔ 0 ≤ 𝐵))
1716, 15ifbieq1d 4241 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0) = if(0 ≤ 𝐵, 𝐵, 0))
1817mpteq2dva 4884 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
1910, 18syl5eq 2794 . . . . 5 (𝜑 → (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
2019adantr 472 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
2112, 13fmptd 6536 . . . . . . 7 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)
2221adantr 472 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴𝐵):𝐴⟶ℝ)
2322ffvelrnda 6510 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) ∧ 𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
24 nfcv 2890 . . . . . . . . 9 𝑦((𝑥𝐴𝐵)‘𝑥)
253, 24, 7cbvmpt 4889 . . . . . . . 8 (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) = (𝑥𝐴 ↦ ((𝑥𝐴𝐵)‘𝑥))
2615mpteq2dva 4884 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ ((𝑥𝐴𝐵)‘𝑥)) = (𝑥𝐴𝐵))
2725, 26syl5eq 2794 . . . . . . 7 (𝜑 → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) = (𝑥𝐴𝐵))
2827eleq1d 2812 . . . . . 6 (𝜑 → ((𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn ↔ (𝑥𝐴𝐵) ∈ MblFn))
2928biimpar 503 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn)
3023, 29mbfpos 23588 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) ∈ MblFn)
3120, 30eqeltrrd 2828 . . 3 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
323nfneg 10440 . . . . . . . . 9 𝑥-((𝑥𝐴𝐵)‘𝑦)
331, 2, 32nfbr 4839 . . . . . . . 8 𝑥0 ≤ -((𝑥𝐴𝐵)‘𝑦)
3433, 32, 1nfif 4247 . . . . . . 7 𝑥if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)
35 nfcv 2890 . . . . . . 7 𝑦if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0)
367negeqd 10438 . . . . . . . . 9 (𝑦 = 𝑥 → -((𝑥𝐴𝐵)‘𝑦) = -((𝑥𝐴𝐵)‘𝑥))
3736breq2d 4804 . . . . . . . 8 (𝑦 = 𝑥 → (0 ≤ -((𝑥𝐴𝐵)‘𝑦) ↔ 0 ≤ -((𝑥𝐴𝐵)‘𝑥)))
3837, 36ifbieq1d 4241 . . . . . . 7 (𝑦 = 𝑥 → if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0) = if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0))
3934, 35, 38cbvmpt 4889 . . . . . 6 (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0))
4015negeqd 10438 . . . . . . . . 9 ((𝜑𝑥𝐴) → -((𝑥𝐴𝐵)‘𝑥) = -𝐵)
4140breq2d 4804 . . . . . . . 8 ((𝜑𝑥𝐴) → (0 ≤ -((𝑥𝐴𝐵)‘𝑥) ↔ 0 ≤ -𝐵))
4241, 40ifbieq1d 4241 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0) = if(0 ≤ -𝐵, -𝐵, 0))
4342mpteq2dva 4884 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
4439, 43syl5eq 2794 . . . . 5 (𝜑 → (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
4544adantr 472 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
4623renegcld 10620 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) ∧ 𝑦𝐴) → -((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
4723, 29mbfneg 23587 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ -((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn)
4846, 47mbfpos 23588 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) ∈ MblFn)
4945, 48eqeltrrd 2828 . . 3 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)
5031, 49jca 555 . 2 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn))
5127adantr 472 . . 3 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) = (𝑥𝐴𝐵))
5221ffvelrnda 6510 . . . . 5 ((𝜑𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
5352adantlr 753 . . . 4 (((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) ∧ 𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
5419adantr 472 . . . . 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 811 . . . . 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 2827 . . . 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 472 . . . . 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 813 . . . . 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 2827 . . . 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 23589 . . 3 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn)
6151, 60eqeltrrd 2828 . 2 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑥𝐴𝐵) ∈ MblFn)
6250, 61impbida 913 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 383   = wceq 1620  wcel 2127  ifcif 4218   class class class wbr 4792  cmpt 4869  wf 6033  cfv 6037  cr 10098  0cc0 10099  cle 10238  -cneg 10430  MblFncmbf 23553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-inf2 8699  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176  ax-pre-sup 10177
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-fal 1626  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-se 5214  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-isom 6046  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-of 7050  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7899  df-map 8013  df-pm 8014  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8501  df-inf 8502  df-oi 8568  df-card 8926  df-cda 9153  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-div 10848  df-nn 11184  df-2 11242  df-3 11243  df-n0 11456  df-z 11541  df-uz 11851  df-q 11953  df-rp 11997  df-xadd 12111  df-ioo 12343  df-ico 12345  df-icc 12346  df-fz 12491  df-fzo 12631  df-fl 12758  df-seq 12967  df-exp 13026  df-hash 13283  df-cj 14009  df-re 14010  df-im 14011  df-sqrt 14145  df-abs 14146  df-clim 14389  df-sum 14587  df-xmet 19912  df-met 19913  df-ovol 23404  df-vol 23405  df-mbf 23558
This theorem is referenced by:  iblre  23730
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