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Theorem mbfposb 25017
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 2907 . . . . . . . . 9 𝑥0
2 nfcv 2907 . . . . . . . . 9 𝑥
3 nffvmpt1 6853 . . . . . . . . 9 𝑥((𝑥𝐴𝐵)‘𝑦)
41, 2, 3nfbr 5152 . . . . . . . 8 𝑥0 ≤ ((𝑥𝐴𝐵)‘𝑦)
54, 3, 1nfif 4516 . . . . . . 7 𝑥if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)
6 nfcv 2907 . . . . . . 7 𝑦if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0)
7 fveq2 6842 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
87breq2d 5117 . . . . . . . 8 (𝑦 = 𝑥 → (0 ≤ ((𝑥𝐴𝐵)‘𝑦) ↔ 0 ≤ ((𝑥𝐴𝐵)‘𝑥)))
98, 7ifbieq1d 4510 . . . . . . 7 (𝑦 = 𝑥 → if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0) = if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0))
105, 6, 9cbvmpt 5216 . . . . . 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 2736 . . . . . . . . . . 11 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
1413fvmpt2 6959 . . . . . . . . . 10 ((𝑥𝐴𝐵 ∈ ℝ) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1511, 12, 14syl2anc 584 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1615breq2d 5117 . . . . . . . 8 ((𝜑𝑥𝐴) → (0 ≤ ((𝑥𝐴𝐵)‘𝑥) ↔ 0 ≤ 𝐵))
1716, 15ifbieq1d 4510 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0) = if(0 ≤ 𝐵, 𝐵, 0))
1817mpteq2dva 5205 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
1910, 18eqtrid 2788 . . . . 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 7063 . . . . . . 7 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)
2221adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴𝐵):𝐴⟶ℝ)
2322ffvelcdmda 7035 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) ∧ 𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
24 nfcv 2907 . . . . . . . . 9 𝑦((𝑥𝐴𝐵)‘𝑥)
253, 24, 7cbvmpt 5216 . . . . . . . 8 (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) = (𝑥𝐴 ↦ ((𝑥𝐴𝐵)‘𝑥))
2615mpteq2dva 5205 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ ((𝑥𝐴𝐵)‘𝑥)) = (𝑥𝐴𝐵))
2725, 26eqtrid 2788 . . . . . . 7 (𝜑 → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) = (𝑥𝐴𝐵))
2827eleq1d 2822 . . . . . 6 (𝜑 → ((𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn ↔ (𝑥𝐴𝐵) ∈ MblFn))
2928biimpar 478 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn)
3023, 29mbfpos 25015 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) ∈ MblFn)
3120, 30eqeltrrd 2839 . . 3 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
323nfneg 11397 . . . . . . . . 9 𝑥-((𝑥𝐴𝐵)‘𝑦)
331, 2, 32nfbr 5152 . . . . . . . 8 𝑥0 ≤ -((𝑥𝐴𝐵)‘𝑦)
3433, 32, 1nfif 4516 . . . . . . 7 𝑥if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)
35 nfcv 2907 . . . . . . 7 𝑦if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0)
367negeqd 11395 . . . . . . . . 9 (𝑦 = 𝑥 → -((𝑥𝐴𝐵)‘𝑦) = -((𝑥𝐴𝐵)‘𝑥))
3736breq2d 5117 . . . . . . . 8 (𝑦 = 𝑥 → (0 ≤ -((𝑥𝐴𝐵)‘𝑦) ↔ 0 ≤ -((𝑥𝐴𝐵)‘𝑥)))
3837, 36ifbieq1d 4510 . . . . . . 7 (𝑦 = 𝑥 → if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0) = if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0))
3934, 35, 38cbvmpt 5216 . . . . . 6 (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0))
4015negeqd 11395 . . . . . . . . 9 ((𝜑𝑥𝐴) → -((𝑥𝐴𝐵)‘𝑥) = -𝐵)
4140breq2d 5117 . . . . . . . 8 ((𝜑𝑥𝐴) → (0 ≤ -((𝑥𝐴𝐵)‘𝑥) ↔ 0 ≤ -𝐵))
4241, 40ifbieq1d 4510 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0) = if(0 ≤ -𝐵, -𝐵, 0))
4342mpteq2dva 5205 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
4439, 43eqtrid 2788 . . . . 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 11582 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) ∧ 𝑦𝐴) → -((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
4723, 29mbfneg 25014 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ -((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn)
4846, 47mbfpos 25015 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) ∈ MblFn)
4945, 48eqeltrrd 2839 . . 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))
5221ffvelcdmda 7035 . . . . 5 ((𝜑𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
5352adantlr 713 . . . 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 769 . . . . 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 2838 . . . 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 771 . . . . 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 2838 . . . 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 25016 . . 3 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn)
6151, 60eqeltrrd 2839 . 2 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑥𝐴𝐵) ∈ MblFn)
6250, 61impbida 799 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 205  wa 396   = wceq 1541  wcel 2106  ifcif 4486   class class class wbr 5105  cmpt 5188  wf 6492  cfv 6496  cr 11050  0cc0 11051  cle 11190  -cneg 11386  MblFncmbf 24978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xadd 13034  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-xmet 20789  df-met 20790  df-ovol 24828  df-vol 24829  df-mbf 24983
This theorem is referenced by:  iblre  25158
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