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Theorem mbfposb 25688
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 2905 . . . . . . . . 9 𝑥0
2 nfcv 2905 . . . . . . . . 9 𝑥
3 nffvmpt1 6917 . . . . . . . . 9 𝑥((𝑥𝐴𝐵)‘𝑦)
41, 2, 3nfbr 5190 . . . . . . . 8 𝑥0 ≤ ((𝑥𝐴𝐵)‘𝑦)
54, 3, 1nfif 4556 . . . . . . 7 𝑥if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)
6 nfcv 2905 . . . . . . 7 𝑦if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0)
7 fveq2 6906 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
87breq2d 5155 . . . . . . . 8 (𝑦 = 𝑥 → (0 ≤ ((𝑥𝐴𝐵)‘𝑦) ↔ 0 ≤ ((𝑥𝐴𝐵)‘𝑥)))
98, 7ifbieq1d 4550 . . . . . . 7 (𝑦 = 𝑥 → if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0) = if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0))
105, 6, 9cbvmpt 5253 . . . . . 6 (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0))
11 simpr 484 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥𝐴)
12 mbfpos.1 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
13 eqid 2737 . . . . . . . . . . 11 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
1413fvmpt2 7027 . . . . . . . . . 10 ((𝑥𝐴𝐵 ∈ ℝ) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1511, 12, 14syl2anc 584 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1615breq2d 5155 . . . . . . . 8 ((𝜑𝑥𝐴) → (0 ≤ ((𝑥𝐴𝐵)‘𝑥) ↔ 0 ≤ 𝐵))
1716, 15ifbieq1d 4550 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0) = if(0 ≤ 𝐵, 𝐵, 0))
1817mpteq2dva 5242 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑥), ((𝑥𝐴𝐵)‘𝑥), 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
1910, 18eqtrid 2789 . . . . 5 (𝜑 → (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
2019adantr 480 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
2112fmpttd 7135 . . . . . . 7 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)
2221adantr 480 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴𝐵):𝐴⟶ℝ)
2322ffvelcdmda 7104 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) ∧ 𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
24 nfcv 2905 . . . . . . . . 9 𝑦((𝑥𝐴𝐵)‘𝑥)
253, 24, 7cbvmpt 5253 . . . . . . . 8 (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) = (𝑥𝐴 ↦ ((𝑥𝐴𝐵)‘𝑥))
2615mpteq2dva 5242 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ ((𝑥𝐴𝐵)‘𝑥)) = (𝑥𝐴𝐵))
2725, 26eqtrid 2789 . . . . . . 7 (𝜑 → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) = (𝑥𝐴𝐵))
2827eleq1d 2826 . . . . . 6 (𝜑 → ((𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn ↔ (𝑥𝐴𝐵) ∈ MblFn))
2928biimpar 477 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn)
3023, 29mbfpos 25686 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ ((𝑥𝐴𝐵)‘𝑦), ((𝑥𝐴𝐵)‘𝑦), 0)) ∈ MblFn)
3120, 30eqeltrrd 2842 . . 3 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
323nfneg 11504 . . . . . . . . 9 𝑥-((𝑥𝐴𝐵)‘𝑦)
331, 2, 32nfbr 5190 . . . . . . . 8 𝑥0 ≤ -((𝑥𝐴𝐵)‘𝑦)
3433, 32, 1nfif 4556 . . . . . . 7 𝑥if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)
35 nfcv 2905 . . . . . . 7 𝑦if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0)
367negeqd 11502 . . . . . . . . 9 (𝑦 = 𝑥 → -((𝑥𝐴𝐵)‘𝑦) = -((𝑥𝐴𝐵)‘𝑥))
3736breq2d 5155 . . . . . . . 8 (𝑦 = 𝑥 → (0 ≤ -((𝑥𝐴𝐵)‘𝑦) ↔ 0 ≤ -((𝑥𝐴𝐵)‘𝑥)))
3837, 36ifbieq1d 4550 . . . . . . 7 (𝑦 = 𝑥 → if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0) = if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0))
3934, 35, 38cbvmpt 5253 . . . . . 6 (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0))
4015negeqd 11502 . . . . . . . . 9 ((𝜑𝑥𝐴) → -((𝑥𝐴𝐵)‘𝑥) = -𝐵)
4140breq2d 5155 . . . . . . . 8 ((𝜑𝑥𝐴) → (0 ≤ -((𝑥𝐴𝐵)‘𝑥) ↔ 0 ≤ -𝐵))
4241, 40ifbieq1d 4550 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0) = if(0 ≤ -𝐵, -𝐵, 0))
4342mpteq2dva 5242 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑥), -((𝑥𝐴𝐵)‘𝑥), 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
4439, 43eqtrid 2789 . . . . 5 (𝜑 → (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
4544adantr 480 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
4623renegcld 11690 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) ∧ 𝑦𝐴) → -((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
4723, 29mbfneg 25685 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ -((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn)
4846, 47mbfpos 25686 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑦𝐴 ↦ if(0 ≤ -((𝑥𝐴𝐵)‘𝑦), -((𝑥𝐴𝐵)‘𝑦), 0)) ∈ MblFn)
4945, 48eqeltrrd 2842 . . 3 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)
5031, 49jca 511 . 2 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn))
5127adantr 480 . . 3 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) = (𝑥𝐴𝐵))
5221ffvelcdmda 7104 . . . . 5 ((𝜑𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
5352adantlr 715 . . . 4 (((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) ∧ 𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) ∈ ℝ)
5419adantr 480 . . . . 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 771 . . . . 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 2841 . . . 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 480 . . . . 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 773 . . . . 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 2841 . . . 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 25687 . . 3 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦𝐴 ↦ ((𝑥𝐴𝐵)‘𝑦)) ∈ MblFn)
6151, 60eqeltrrd 2842 . 2 ((𝜑 ∧ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑥𝐴𝐵) ∈ MblFn)
6250, 61impbida 801 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 206  wa 395   = wceq 1540  wcel 2108  ifcif 4525   class class class wbr 5143  cmpt 5225  wf 6557  cfv 6561  cr 11154  0cc0 11155  cle 11296  -cneg 11493  MblFncmbf 25649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xadd 13155  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-xmet 21357  df-met 21358  df-ovol 25499  df-vol 25500  df-mbf 25654
This theorem is referenced by:  iblre  25829
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