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Theorem ismbf2d 22025
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 11335 . . 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 6288 . . . . . . . 8  |-  ( x  = +oo  ->  (
x (,) +oo )  =  ( +oo (,) +oo ) )
5 iooid 11567 . . . . . . . 8  |-  ( +oo (,) +oo )  =  (/)
64, 5syl6eq 2500 . . . . . . 7  |-  ( x  = +oo  ->  (
x (,) +oo )  =  (/) )
76imaeq2d 5327 . . . . . 6  |-  ( x  = +oo  ->  ( `' F " ( x (,) +oo ) )  =  ( `' F "
(/) ) )
8 ima0 5342 . . . . . . 7  |-  ( `' F " (/) )  =  (/)
9 0mbl 21927 . . . . . . 7  |-  (/)  e.  dom  vol
108, 9eqeltri 2527 . . . . . 6  |-  ( `' F " (/) )  e. 
dom  vol
117, 10syl6eqel 2539 . . . . 5  |-  ( x  = +oo  ->  ( `' F " ( x (,) +oo ) )  e.  dom  vol )
1211adantl 466 . . . 4  |-  ( (
ph  /\  x  = +oo )  ->  ( `' F " ( x (,) +oo ) )  e.  dom  vol )
13 fimacnv 6004 . . . . . . . 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 2531 . . . . . 6  |-  ( ph  ->  ( `' F " RR )  e.  dom  vol )
17 oveq1 6288 . . . . . . . . 9  |-  ( x  = -oo  ->  (
x (,) +oo )  =  ( -oo (,) +oo ) )
18 ioomax 11609 . . . . . . . . 9  |-  ( -oo (,) +oo )  =  RR
1917, 18syl6eq 2500 . . . . . . . 8  |-  ( x  = -oo  ->  (
x (,) +oo )  =  RR )
2019imaeq2d 5327 . . . . . . 7  |-  ( x  = -oo  ->  ( `' F " ( x (,) +oo ) )  =  ( `' F " RR ) )
2120eleq1d 2512 . . . . . 6  |-  ( x  = -oo  ->  (
( `' F "
( x (,) +oo ) )  e.  dom  vol  <->  ( `' F " RR )  e.  dom  vol )
)
2216, 21syl5ibrcom 222 . . . . 5  |-  ( ph  ->  ( x  = -oo  ->  ( `' F "
( x (,) +oo ) )  e.  dom  vol ) )
2322imp 429 . . . 4  |-  ( (
ph  /\  x  = -oo )  ->  ( `' F " ( x (,) +oo ) )  e.  dom  vol )
243, 12, 233jaodan 1295 . . 3  |-  ( (
ph  /\  ( x  e.  RR  \/  x  = +oo  \/  x  = -oo ) )  -> 
( `' F "
( x (,) +oo ) )  e.  dom  vol )
252, 24sylan2b 475 . 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 6289 . . . . . . . . 9  |-  ( x  = +oo  ->  ( -oo (,) x )  =  ( -oo (,) +oo ) )
2827, 18syl6eq 2500 . . . . . . . 8  |-  ( x  = +oo  ->  ( -oo (,) x )  =  RR )
2928imaeq2d 5327 . . . . . . 7  |-  ( x  = +oo  ->  ( `' F " ( -oo (,) x ) )  =  ( `' F " RR ) )
3029eleq1d 2512 . . . . . 6  |-  ( x  = +oo  ->  (
( `' F "
( -oo (,) x ) )  e.  dom  vol  <->  ( `' F " RR )  e.  dom  vol )
)
3116, 30syl5ibrcom 222 . . . . 5  |-  ( ph  ->  ( x  = +oo  ->  ( `' F "
( -oo (,) x ) )  e.  dom  vol ) )
3231imp 429 . . . 4  |-  ( (
ph  /\  x  = +oo )  ->  ( `' F " ( -oo (,) x ) )  e. 
dom  vol )
33 oveq2 6289 . . . . . . . 8  |-  ( x  = -oo  ->  ( -oo (,) x )  =  ( -oo (,) -oo ) )
34 iooid 11567 . . . . . . . 8  |-  ( -oo (,) -oo )  =  (/)
3533, 34syl6eq 2500 . . . . . . 7  |-  ( x  = -oo  ->  ( -oo (,) x )  =  (/) )
3635imaeq2d 5327 . . . . . 6  |-  ( x  = -oo  ->  ( `' F " ( -oo (,) x ) )  =  ( `' F " (/) ) )
3736, 10syl6eqel 2539 . . . . 5  |-  ( x  = -oo  ->  ( `' F " ( -oo (,) x ) )  e. 
dom  vol )
3837adantl 466 . . . 4  |-  ( (
ph  /\  x  = -oo )  ->  ( `' F " ( -oo (,) x ) )  e. 
dom  vol )
3926, 32, 383jaodan 1295 . . 3  |-  ( (
ph  /\  ( x  e.  RR  \/  x  = +oo  \/  x  = -oo ) )  -> 
( `' F "
( -oo (,) x ) )  e.  dom  vol )
402, 39sylan2b 475 . 2  |-  ( (
ph  /\  x  e.  RR* )  ->  ( `' F " ( -oo (,) x ) )  e. 
dom  vol )
411, 25, 40ismbfd 22024 1  |-  ( ph  ->  F  e. MblFn )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    \/ w3o 973    = wceq 1383    e. wcel 1804   (/)c0 3770   `'ccnv 4988   dom cdm 4989   "cima 4992   -->wf 5574  (class class class)co 6281   RRcr 9494   +oocpnf 9628   -oocmnf 9629   RR*cxr 9630   (,)cioo 11539   volcvol 21852  MblFncmbf 22000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11092  df-q 11193  df-rp 11231  df-xadd 11329  df-ioo 11543  df-ico 11545  df-icc 11546  df-fz 11683  df-fzo 11806  df-fl 11910  df-seq 12089  df-exp 12148  df-hash 12387  df-cj 12913  df-re 12914  df-im 12915  df-sqrt 13049  df-abs 13050  df-clim 13292  df-sum 13490  df-xmet 18390  df-met 18391  df-ovol 21853  df-vol 21854  df-mbf 22005
This theorem is referenced by:  mbfres  22028  mbfmulc2lem  22031  mbfposr  22036  ismbf3d  22038  iblabsnclem  30053  ftc1anclem1  30065  ftc1anclem6  30070
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