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Theorem ismbf2d 19393
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 10641 . . 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 6020 . . . . . . . 8  |-  ( x  =  +oo  ->  (
x (,)  +oo )  =  (  +oo (,)  +oo ) )
5 iooid 10869 . . . . . . . 8  |-  (  +oo (,) 
+oo )  =  (/)
64, 5syl6eq 2428 . . . . . . 7  |-  ( x  =  +oo  ->  (
x (,)  +oo )  =  (/) )
76imaeq2d 5136 . . . . . 6  |-  ( x  =  +oo  ->  ( `' F " ( x (,)  +oo ) )  =  ( `' F " (/) ) )
8 ima0 5154 . . . . . . 7  |-  ( `' F " (/) )  =  (/)
9 0mbl 19294 . . . . . . 7  |-  (/)  e.  dom  vol
108, 9eqeltri 2450 . . . . . 6  |-  ( `' F " (/) )  e. 
dom  vol
117, 10syl6eqel 2468 . . . . 5  |-  ( x  =  +oo  ->  ( `' F " ( x (,)  +oo ) )  e. 
dom  vol )
1211adantl 453 . . . 4  |-  ( (
ph  /\  x  =  +oo )  ->  ( `' F " ( x (,)  +oo ) )  e. 
dom  vol )
13 fimacnv 5794 . . . . . . . 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 2454 . . . . . 6  |-  ( ph  ->  ( `' F " RR )  e.  dom  vol )
17 oveq1 6020 . . . . . . . . 9  |-  ( x  =  -oo  ->  (
x (,)  +oo )  =  (  -oo (,)  +oo ) )
18 ioomax 10910 . . . . . . . . 9  |-  (  -oo (,) 
+oo )  =  RR
1917, 18syl6eq 2428 . . . . . . . 8  |-  ( x  =  -oo  ->  (
x (,)  +oo )  =  RR )
2019imaeq2d 5136 . . . . . . 7  |-  ( x  =  -oo  ->  ( `' F " ( x (,)  +oo ) )  =  ( `' F " RR ) )
2120eleq1d 2446 . . . . . 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 6021 . . . . . . . . 9  |-  ( x  =  +oo  ->  (  -oo (,) x )  =  (  -oo (,)  +oo ) )
2827, 18syl6eq 2428 . . . . . . . 8  |-  ( x  =  +oo  ->  (  -oo (,) x )  =  RR )
2928imaeq2d 5136 . . . . . . 7  |-  ( x  =  +oo  ->  ( `' F " (  -oo (,) x ) )  =  ( `' F " RR ) )
3029eleq1d 2446 . . . . . 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 6021 . . . . . . . 8  |-  ( x  =  -oo  ->  (  -oo (,) x )  =  (  -oo (,)  -oo ) )
34 iooid 10869 . . . . . . . 8  |-  (  -oo (,) 
-oo )  =  (/)
3533, 34syl6eq 2428 . . . . . . 7  |-  ( x  =  -oo  ->  (  -oo (,) x )  =  (/) )
3635imaeq2d 5136 . . . . . 6  |-  ( x  =  -oo  ->  ( `' F " (  -oo (,) x ) )  =  ( `' F " (/) ) )
3736, 10syl6eqel 2468 . . . . 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 19392 1  |-  ( ph  ->  F  e. MblFn )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    \/ w3o 935    = wceq 1649    e. wcel 1717   (/)c0 3564   `'ccnv 4810   dom cdm 4811   "cima 4814   -->wf 5383  (class class class)co 6013   RRcr 8915    +oocpnf 9043    -oocmnf 9044   RR*cxr 9045   (,)cioo 10841   volcvol 19220  MblFncmbf 19366
This theorem is referenced by:  mbfres  19396  mbfmulc2lem  19399  mbfposr  19404  ismbf3d  19406  iblabsnclem  25961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-map 6949  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-cda 7974  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-q 10500  df-rp 10538  df-xadd 10636  df-ioo 10845  df-ico 10847  df-icc 10848  df-fz 10969  df-fzo 11059  df-fl 11122  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-sum 12400  df-xmet 16612  df-met 16613  df-ovol 19221  df-vol 19222  df-mbf 19372
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