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Theorem mbfsub 21155
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 21120 . . . . . . . 8  |-  ( F  e. MblFn  ->  F : dom  F --> CC )
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  F : dom  F --> CC )
4 elin 3554 . . . . . . . 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 5856 . . . . . . 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 21120 . . . . . . . 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 5856 . . . . . . 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 9740 . . . . 5  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  (
( F `  x
)  +  -u ( G `  x )
)  =  ( ( F `  x )  -  ( G `  x ) ) )
1514eqcomd 2448 . . . 4  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  (
( F `  x
)  -  ( G `
 x ) )  =  ( ( F `
 x )  + 
-u ( G `  x ) ) )
1615mpteq2dva 4393 . . 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 5574 . . . . 5  |-  ( F : dom  F --> CC  ->  F  Fn  dom  F )
183, 17syl 16 . . . 4  |-  ( ph  ->  F  Fn  dom  F
)
19 ffn 5574 . . . . 5  |-  ( G : dom  G --> CC  ->  G  Fn  dom  G )
2010, 19syl 16 . . . 4  |-  ( ph  ->  G  Fn  dom  G
)
21 mbfdm 21121 . . . . 5  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
221, 21syl 16 . . . 4  |-  ( ph  ->  dom  F  e.  dom  vol )
23 mbfdm 21121 . . . . 5  |-  ( G  e. MblFn  ->  dom  G  e.  dom  vol )
248, 23syl 16 . . . 4  |-  ( ph  ->  dom  G  e.  dom  vol )
25 eqid 2443 . . . 4  |-  ( dom 
F  i^i  dom  G )  =  ( dom  F  i^i  dom  G )
26 eqidd 2444 . . . 4  |-  ( (
ph  /\  x  e.  dom  F )  ->  ( F `  x )  =  ( F `  x ) )
27 eqidd 2444 . . . 4  |-  ( (
ph  /\  x  e.  dom  G )  ->  ( G `  x )  =  ( G `  x ) )
2818, 20, 22, 24, 25, 26, 27offval 6342 . . 3  |-  ( ph  ->  ( F  oF  -  G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x )  -  ( G `  x ) ) ) )
29 inmbl 21038 . . . . 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 9721 . . . 4  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  -u ( G `  x )  e.  CC )
32 eqidd 2444 . . . 4  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( F `  x ) )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( F `  x ) ) )
33 eqidd 2444 . . . 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 6351 . . 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 2485 . 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 3585 . . . . 5  |-  ( dom 
F  i^i  dom  G ) 
C_  dom  F
37 resmpt 5171 . . . . 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 5759 . . . . . 6  |-  ( ph  ->  F  =  ( x  e.  dom  F  |->  ( F `  x ) ) )
4039, 1eqeltrrd 2518 . . . . 5  |-  ( ph  ->  ( x  e.  dom  F 
|->  ( F `  x
) )  e. MblFn )
41 mbfres 21137 . . . . 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 2518 . . 3  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( F `  x ) )  e. MblFn
)
44 inss2 3586 . . . . . 6  |-  ( dom 
F  i^i  dom  G ) 
C_  dom  G
45 resmpt 5171 . . . . . 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 5759 . . . . . . 7  |-  ( ph  ->  G  =  ( x  e.  dom  G  |->  ( G `  x ) ) )
4847, 8eqeltrrd 2518 . . . . . 6  |-  ( ph  ->  ( x  e.  dom  G 
|->  ( G `  x
) )  e. MblFn )
49 mbfres 21137 . . . . . 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 2518 . . . 4  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( G `  x ) )  e. MblFn
)
5213, 51mbfneg 21143 . . 3  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  -u ( G `  x ) )  e. MblFn
)
5343, 52mbfadd 21154 . 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 2517 1  |-  ( ph  ->  ( F  oF  -  G )  e. MblFn
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3342    C_ wss 3343    e. cmpt 4365   dom cdm 4855    |` cres 4857    Fn wfn 5428   -->wf 5429   ` cfv 5433  (class class class)co 6106    oFcof 6333   CCcc 9295    + caddc 9300    - cmin 9610   -ucneg 9611   volcvol 20962  MblFncmbf 21109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-inf2 7862  ax-cc 8619  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-disj 4278  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-se 4695  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-of 6335  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-2o 6936  df-oadd 6939  df-omul 6940  df-er 7116  df-map 7231  df-pm 7232  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-sup 7706  df-oi 7739  df-card 8124  df-acn 8127  df-cda 8352  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-n0 10595  df-z 10662  df-uz 10877  df-q 10969  df-rp 11007  df-xadd 11105  df-ioo 11319  df-ioc 11320  df-ico 11321  df-icc 11322  df-fz 11453  df-fzo 11564  df-fl 11657  df-seq 11822  df-exp 11881  df-hash 12119  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740  df-clim 12981  df-rlim 12982  df-sum 13179  df-xmet 17825  df-met 17826  df-ovol 20963  df-vol 20964  df-mbf 21114
This theorem is referenced by:  mbfmul  21219  iblulm  21887
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