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Theorem mbfsub 21937
Description: The difference of two measurable functions is measurable. (Contributed by Mario Carneiro, 5-Sep-2014.)
Hypotheses
Ref Expression
mbfadd.1  |-  ( ph  ->  F  e. MblFn )
mbfadd.2  |-  ( ph  ->  G  e. MblFn )
Assertion
Ref Expression
mbfsub  |-  ( ph  ->  ( F  oF  -  G )  e. MblFn
)

Proof of Theorem mbfsub
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mbfadd.1 . . . . . . . 8  |-  ( ph  ->  F  e. MblFn )
2 mbff 21902 . . . . . . . 8  |-  ( F  e. MblFn  ->  F : dom  F --> CC )
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  F : dom  F --> CC )
4 elin 3692 . . . . . . . 8  |-  ( x  e.  ( dom  F  i^i  dom  G )  <->  ( x  e.  dom  F  /\  x  e.  dom  G ) )
54simplbi 460 . . . . . . 7  |-  ( x  e.  ( dom  F  i^i  dom  G )  ->  x  e.  dom  F )
6 ffvelrn 6030 . . . . . . 7  |-  ( ( F : dom  F --> CC  /\  x  e.  dom  F )  ->  ( F `  x )  e.  CC )
73, 5, 6syl2an 477 . . . . . 6  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  ( F `  x )  e.  CC )
8 mbfadd.2 . . . . . . . 8  |-  ( ph  ->  G  e. MblFn )
9 mbff 21902 . . . . . . . 8  |-  ( G  e. MblFn  ->  G : dom  G --> CC )
108, 9syl 16 . . . . . . 7  |-  ( ph  ->  G : dom  G --> CC )
114simprbi 464 . . . . . . 7  |-  ( x  e.  ( dom  F  i^i  dom  G )  ->  x  e.  dom  G )
12 ffvelrn 6030 . . . . . . 7  |-  ( ( G : dom  G --> CC  /\  x  e.  dom  G )  ->  ( G `  x )  e.  CC )
1310, 11, 12syl2an 477 . . . . . 6  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  ( G `  x )  e.  CC )
147, 13negsubd 9948 . . . . 5  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  (
( F `  x
)  +  -u ( G `  x )
)  =  ( ( F `  x )  -  ( G `  x ) ) )
1514eqcomd 2475 . . . 4  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  (
( F `  x
)  -  ( G `
 x ) )  =  ( ( F `
 x )  + 
-u ( G `  x ) ) )
1615mpteq2dva 4539 . . 3  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x )  -  ( G `  x ) ) )  =  ( x  e.  ( dom 
F  i^i  dom  G ) 
|->  ( ( F `  x )  +  -u ( G `  x ) ) ) )
17 ffn 5737 . . . . 5  |-  ( F : dom  F --> CC  ->  F  Fn  dom  F )
183, 17syl 16 . . . 4  |-  ( ph  ->  F  Fn  dom  F
)
19 ffn 5737 . . . . 5  |-  ( G : dom  G --> CC  ->  G  Fn  dom  G )
2010, 19syl 16 . . . 4  |-  ( ph  ->  G  Fn  dom  G
)
21 mbfdm 21903 . . . . 5  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
221, 21syl 16 . . . 4  |-  ( ph  ->  dom  F  e.  dom  vol )
23 mbfdm 21903 . . . . 5  |-  ( G  e. MblFn  ->  dom  G  e.  dom  vol )
248, 23syl 16 . . . 4  |-  ( ph  ->  dom  G  e.  dom  vol )
25 eqid 2467 . . . 4  |-  ( dom 
F  i^i  dom  G )  =  ( dom  F  i^i  dom  G )
26 eqidd 2468 . . . 4  |-  ( (
ph  /\  x  e.  dom  F )  ->  ( F `  x )  =  ( F `  x ) )
27 eqidd 2468 . . . 4  |-  ( (
ph  /\  x  e.  dom  G )  ->  ( G `  x )  =  ( G `  x ) )
2818, 20, 22, 24, 25, 26, 27offval 6542 . . 3  |-  ( ph  ->  ( F  oF  -  G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x )  -  ( G `  x ) ) ) )
29 inmbl 21820 . . . . 5  |-  ( ( dom  F  e.  dom  vol 
/\  dom  G  e.  dom  vol )  ->  ( dom  F  i^i  dom  G
)  e.  dom  vol )
3022, 24, 29syl2anc 661 . . . 4  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  e.  dom  vol )
3113negcld 9929 . . . 4  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  -u ( G `  x )  e.  CC )
32 eqidd 2468 . . . 4  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( F `  x ) )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( F `  x ) ) )
33 eqidd 2468 . . . 4  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  -u ( G `  x ) )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  -u ( G `  x ) ) )
3430, 7, 31, 32, 33offval2 6551 . . 3  |-  ( ph  ->  ( ( x  e.  ( dom  F  i^i  dom 
G )  |->  ( F `
 x ) )  oF  +  ( x  e.  ( dom 
F  i^i  dom  G ) 
|->  -u ( G `  x ) ) )  =  ( x  e.  ( dom  F  i^i  dom 
G )  |->  ( ( F `  x )  +  -u ( G `  x ) ) ) )
3516, 28, 343eqtr4d 2518 . 2  |-  ( ph  ->  ( F  oF  -  G )  =  ( ( x  e.  ( dom  F  i^i  dom 
G )  |->  ( F `
 x ) )  oF  +  ( x  e.  ( dom 
F  i^i  dom  G ) 
|->  -u ( G `  x ) ) ) )
36 inss1 3723 . . . . 5  |-  ( dom 
F  i^i  dom  G ) 
C_  dom  F
37 resmpt 5329 . . . . 5  |-  ( ( dom  F  i^i  dom  G )  C_  dom  F  -> 
( ( x  e. 
dom  F  |->  ( F `
 x ) )  |`  ( dom  F  i^i  dom 
G ) )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( F `  x ) ) )
3836, 37mp1i 12 . . . 4  |-  ( ph  ->  ( ( x  e. 
dom  F  |->  ( F `
 x ) )  |`  ( dom  F  i^i  dom 
G ) )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( F `  x ) ) )
393feqmptd 5927 . . . . . 6  |-  ( ph  ->  F  =  ( x  e.  dom  F  |->  ( F `  x ) ) )
4039, 1eqeltrrd 2556 . . . . 5  |-  ( ph  ->  ( x  e.  dom  F 
|->  ( F `  x
) )  e. MblFn )
41 mbfres 21919 . . . . 5  |-  ( ( ( x  e.  dom  F 
|->  ( F `  x
) )  e. MblFn  /\  ( dom  F  i^i  dom  G
)  e.  dom  vol )  ->  ( ( x  e.  dom  F  |->  ( F `  x ) )  |`  ( dom  F  i^i  dom  G )
)  e. MblFn )
4240, 30, 41syl2anc 661 . . . 4  |-  ( ph  ->  ( ( x  e. 
dom  F  |->  ( F `
 x ) )  |`  ( dom  F  i^i  dom 
G ) )  e. MblFn
)
4338, 42eqeltrrd 2556 . . 3  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( F `  x ) )  e. MblFn
)
44 inss2 3724 . . . . . 6  |-  ( dom 
F  i^i  dom  G ) 
C_  dom  G
45 resmpt 5329 . . . . . 6  |-  ( ( dom  F  i^i  dom  G )  C_  dom  G  -> 
( ( x  e. 
dom  G  |->  ( G `
 x ) )  |`  ( dom  F  i^i  dom 
G ) )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( G `  x ) ) )
4644, 45mp1i 12 . . . . 5  |-  ( ph  ->  ( ( x  e. 
dom  G  |->  ( G `
 x ) )  |`  ( dom  F  i^i  dom 
G ) )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( G `  x ) ) )
4710feqmptd 5927 . . . . . . 7  |-  ( ph  ->  G  =  ( x  e.  dom  G  |->  ( G `  x ) ) )
4847, 8eqeltrrd 2556 . . . . . 6  |-  ( ph  ->  ( x  e.  dom  G 
|->  ( G `  x
) )  e. MblFn )
49 mbfres 21919 . . . . . 6  |-  ( ( ( x  e.  dom  G 
|->  ( G `  x
) )  e. MblFn  /\  ( dom  F  i^i  dom  G
)  e.  dom  vol )  ->  ( ( x  e.  dom  G  |->  ( G `  x ) )  |`  ( dom  F  i^i  dom  G )
)  e. MblFn )
5048, 30, 49syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( x  e. 
dom  G  |->  ( G `
 x ) )  |`  ( dom  F  i^i  dom 
G ) )  e. MblFn
)
5146, 50eqeltrrd 2556 . . . 4  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( G `  x ) )  e. MblFn
)
5213, 51mbfneg 21925 . . 3  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  -u ( G `  x ) )  e. MblFn
)
5343, 52mbfadd 21936 . 2  |-  ( ph  ->  ( ( x  e.  ( dom  F  i^i  dom 
G )  |->  ( F `
 x ) )  oF  +  ( x  e.  ( dom 
F  i^i  dom  G ) 
|->  -u ( G `  x ) ) )  e. MblFn )
5435, 53eqeltrd 2555 1  |-  ( ph  ->  ( F  oF  -  G )  e. MblFn
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    i^i cin 3480    C_ wss 3481    |-> cmpt 4511   dom cdm 5005    |` cres 5007    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    oFcof 6533   CCcc 9502    + caddc 9507    - cmin 9817   -ucneg 9818   volcvol 21743  MblFncmbf 21891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cc 8827  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-acn 8335  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-xadd 11331  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-rlim 13292  df-sum 13489  df-xmet 18282  df-met 18283  df-ovol 21744  df-vol 21745  df-mbf 21896
This theorem is referenced by:  mbfmul  22001  iblulm  22669
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