MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ismbf2d Structured version   Unicode version

Theorem ismbf2d 19525
Description: Deduction to prove measurability of a real function. (Contributed by Mario Carneiro, 18-Jun-2014.)
Hypotheses
Ref Expression
ismbf2d.1  |-  ( ph  ->  F : A --> RR )
ismbf2d.2  |-  ( ph  ->  A  e.  dom  vol )
ismbf2d.3  |-  ( (
ph  /\  x  e.  RR )  ->  ( `' F " ( x (,)  +oo ) )  e. 
dom  vol )
ismbf2d.4  |-  ( (
ph  /\  x  e.  RR )  ->  ( `' F " (  -oo (,) x ) )  e. 
dom  vol )
Assertion
Ref Expression
ismbf2d  |-  ( ph  ->  F  e. MblFn )
Distinct variable groups:    x, F    ph, x
Allowed substitution hint:    A( x)

Proof of Theorem ismbf2d
StepHypRef Expression
1 ismbf2d.1 . 2  |-  ( ph  ->  F : A --> RR )
2 elxr 10708 . . 3  |-  ( x  e.  RR*  <->  ( x  e.  RR  \/  x  = 
+oo  \/  x  =  -oo ) )
3 ismbf2d.3 . . . 4  |-  ( (
ph  /\  x  e.  RR )  ->  ( `' F " ( x (,)  +oo ) )  e. 
dom  vol )
4 oveq1 6080 . . . . . . . 8  |-  ( x  =  +oo  ->  (
x (,)  +oo )  =  (  +oo (,)  +oo ) )
5 iooid 10936 . . . . . . . 8  |-  (  +oo (,) 
+oo )  =  (/)
64, 5syl6eq 2483 . . . . . . 7  |-  ( x  =  +oo  ->  (
x (,)  +oo )  =  (/) )
76imaeq2d 5195 . . . . . 6  |-  ( x  =  +oo  ->  ( `' F " ( x (,)  +oo ) )  =  ( `' F " (/) ) )
8 ima0 5213 . . . . . . 7  |-  ( `' F " (/) )  =  (/)
9 0mbl 19426 . . . . . . 7  |-  (/)  e.  dom  vol
108, 9eqeltri 2505 . . . . . 6  |-  ( `' F " (/) )  e. 
dom  vol
117, 10syl6eqel 2523 . . . . 5  |-  ( x  =  +oo  ->  ( `' F " ( x (,)  +oo ) )  e. 
dom  vol )
1211adantl 453 . . . 4  |-  ( (
ph  /\  x  =  +oo )  ->  ( `' F " ( x (,)  +oo ) )  e. 
dom  vol )
13 fimacnv 5854 . . . . . . . 8  |-  ( F : A --> RR  ->  ( `' F " RR )  =  A )
141, 13syl 16 . . . . . . 7  |-  ( ph  ->  ( `' F " RR )  =  A
)
15 ismbf2d.2 . . . . . . 7  |-  ( ph  ->  A  e.  dom  vol )
1614, 15eqeltrd 2509 . . . . . 6  |-  ( ph  ->  ( `' F " RR )  e.  dom  vol )
17 oveq1 6080 . . . . . . . . 9  |-  ( x  =  -oo  ->  (
x (,)  +oo )  =  (  -oo (,)  +oo ) )
18 ioomax 10977 . . . . . . . . 9  |-  (  -oo (,) 
+oo )  =  RR
1917, 18syl6eq 2483 . . . . . . . 8  |-  ( x  =  -oo  ->  (
x (,)  +oo )  =  RR )
2019imaeq2d 5195 . . . . . . 7  |-  ( x  =  -oo  ->  ( `' F " ( x (,)  +oo ) )  =  ( `' F " RR ) )
2120eleq1d 2501 . . . . . 6  |-  ( x  =  -oo  ->  (
( `' F "
( x (,)  +oo ) )  e.  dom  vol  <->  ( `' F " RR )  e.  dom  vol )
)
2216, 21syl5ibrcom 214 . . . . 5  |-  ( ph  ->  ( x  =  -oo  ->  ( `' F "
( x (,)  +oo ) )  e.  dom  vol ) )
2322imp 419 . . . 4  |-  ( (
ph  /\  x  =  -oo )  ->  ( `' F " ( x (,)  +oo ) )  e. 
dom  vol )
243, 12, 233jaodan 1250 . . 3  |-  ( (
ph  /\  ( x  e.  RR  \/  x  = 
+oo  \/  x  =  -oo ) )  ->  ( `' F " ( x (,)  +oo ) )  e. 
dom  vol )
252, 24sylan2b 462 . 2  |-  ( (
ph  /\  x  e.  RR* )  ->  ( `' F " ( x (,) 
+oo ) )  e. 
dom  vol )
26 ismbf2d.4 . . . 4  |-  ( (
ph  /\  x  e.  RR )  ->  ( `' F " (  -oo (,) x ) )  e. 
dom  vol )
27 oveq2 6081 . . . . . . . . 9  |-  ( x  =  +oo  ->  (  -oo (,) x )  =  (  -oo (,)  +oo ) )
2827, 18syl6eq 2483 . . . . . . . 8  |-  ( x  =  +oo  ->  (  -oo (,) x )  =  RR )
2928imaeq2d 5195 . . . . . . 7  |-  ( x  =  +oo  ->  ( `' F " (  -oo (,) x ) )  =  ( `' F " RR ) )
3029eleq1d 2501 . . . . . 6  |-  ( x  =  +oo  ->  (
( `' F "
(  -oo (,) x ) )  e.  dom  vol  <->  ( `' F " RR )  e.  dom  vol )
)
3116, 30syl5ibrcom 214 . . . . 5  |-  ( ph  ->  ( x  =  +oo  ->  ( `' F "
(  -oo (,) x ) )  e.  dom  vol ) )
3231imp 419 . . . 4  |-  ( (
ph  /\  x  =  +oo )  ->  ( `' F " (  -oo (,) x ) )  e. 
dom  vol )
33 oveq2 6081 . . . . . . . 8  |-  ( x  =  -oo  ->  (  -oo (,) x )  =  (  -oo (,)  -oo ) )
34 iooid 10936 . . . . . . . 8  |-  (  -oo (,) 
-oo )  =  (/)
3533, 34syl6eq 2483 . . . . . . 7  |-  ( x  =  -oo  ->  (  -oo (,) x )  =  (/) )
3635imaeq2d 5195 . . . . . 6  |-  ( x  =  -oo  ->  ( `' F " (  -oo (,) x ) )  =  ( `' F " (/) ) )
3736, 10syl6eqel 2523 . . . . 5  |-  ( x  =  -oo  ->  ( `' F " (  -oo (,) x ) )  e. 
dom  vol )
3837adantl 453 . . . 4  |-  ( (
ph  /\  x  =  -oo )  ->  ( `' F " (  -oo (,) x ) )  e. 
dom  vol )
3926, 32, 383jaodan 1250 . . 3  |-  ( (
ph  /\  ( x  e.  RR  \/  x  = 
+oo  \/  x  =  -oo ) )  ->  ( `' F " (  -oo (,) x ) )  e. 
dom  vol )
402, 39sylan2b 462 . 2  |-  ( (
ph  /\  x  e.  RR* )  ->  ( `' F " (  -oo (,) x ) )  e. 
dom  vol )
411, 25, 40ismbfd 19524 1  |-  ( ph  ->  F  e. MblFn )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725   (/)c0 3620   `'ccnv 4869   dom cdm 4870   "cima 4873   -->wf 5442  (class class class)co 6073   RRcr 8981    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111   (,)cioo 10908   volcvol 19352  MblFncmbf 19498
This theorem is referenced by:  mbfres  19528  mbfmulc2lem  19531  mbfposr  19536  ismbf3d  19538  iblabsnclem  26258  ftc1anclem1  26270  ftc1anclem6  26275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-xadd 10703  df-ioo 10912  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-xmet 16687  df-met 16688  df-ovol 19353  df-vol 19354  df-mbf 19504
  Copyright terms: Public domain W3C validator