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Theorem mbfsub 19033
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  o F  -  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 18998 . . . . . . . 8  |-  ( F  e. MblFn  ->  F : dom  F --> CC )
31, 2syl 15 . . . . . . 7  |-  ( ph  ->  F : dom  F --> CC )
4 elin 3371 . . . . . . . 8  |-  ( x  e.  ( dom  F  i^i  dom  G )  <->  ( x  e.  dom  F  /\  x  e.  dom  G ) )
54simplbi 446 . . . . . . 7  |-  ( x  e.  ( dom  F  i^i  dom  G )  ->  x  e.  dom  F )
6 ffvelrn 5679 . . . . . . 7  |-  ( ( F : dom  F --> CC  /\  x  e.  dom  F )  ->  ( F `  x )  e.  CC )
73, 5, 6syl2an 463 . . . . . 6  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  ( F `  x )  e.  CC )
8 mbfadd.2 . . . . . . . 8  |-  ( ph  ->  G  e. MblFn )
9 mbff 18998 . . . . . . . 8  |-  ( G  e. MblFn  ->  G : dom  G --> CC )
108, 9syl 15 . . . . . . 7  |-  ( ph  ->  G : dom  G --> CC )
114simprbi 450 . . . . . . 7  |-  ( x  e.  ( dom  F  i^i  dom  G )  ->  x  e.  dom  G )
12 ffvelrn 5679 . . . . . . 7  |-  ( ( G : dom  G --> CC  /\  x  e.  dom  G )  ->  ( G `  x )  e.  CC )
1310, 11, 12syl2an 463 . . . . . 6  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  ( G `  x )  e.  CC )
147, 13negsubd 9179 . . . . 5  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  (
( F `  x
)  +  -u ( G `  x )
)  =  ( ( F `  x )  -  ( G `  x ) ) )
1514eqcomd 2301 . . . 4  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  (
( F `  x
)  -  ( G `
 x ) )  =  ( ( F `
 x )  + 
-u ( G `  x ) ) )
1615mpteq2dva 4122 . . 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 5405 . . . . 5  |-  ( F : dom  F --> CC  ->  F  Fn  dom  F )
183, 17syl 15 . . . 4  |-  ( ph  ->  F  Fn  dom  F
)
19 ffn 5405 . . . . 5  |-  ( G : dom  G --> CC  ->  G  Fn  dom  G )
2010, 19syl 15 . . . 4  |-  ( ph  ->  G  Fn  dom  G
)
21 mbfdm 18999 . . . . 5  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
221, 21syl 15 . . . 4  |-  ( ph  ->  dom  F  e.  dom  vol )
23 mbfdm 18999 . . . . 5  |-  ( G  e. MblFn  ->  dom  G  e.  dom  vol )
248, 23syl 15 . . . 4  |-  ( ph  ->  dom  G  e.  dom  vol )
25 eqid 2296 . . . 4  |-  ( dom 
F  i^i  dom  G )  =  ( dom  F  i^i  dom  G )
26 eqidd 2297 . . . 4  |-  ( (
ph  /\  x  e.  dom  F )  ->  ( F `  x )  =  ( F `  x ) )
27 eqidd 2297 . . . 4  |-  ( (
ph  /\  x  e.  dom  G )  ->  ( G `  x )  =  ( G `  x ) )
2818, 20, 22, 24, 25, 26, 27offval 6101 . . 3  |-  ( ph  ->  ( F  o F  -  G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x )  -  ( G `  x ) ) ) )
29 inmbl 18915 . . . . 5  |-  ( ( dom  F  e.  dom  vol 
/\  dom  G  e.  dom  vol )  ->  ( dom  F  i^i  dom  G
)  e.  dom  vol )
3022, 24, 29syl2anc 642 . . . 4  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  e.  dom  vol )
3113negcld 9160 . . . 4  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  -u ( G `  x )  e.  CC )
32 eqidd 2297 . . . 4  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( F `  x ) )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( F `  x ) ) )
33 eqidd 2297 . . . 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 6111 . . 3  |-  ( ph  ->  ( ( x  e.  ( dom  F  i^i  dom 
G )  |->  ( F `
 x ) )  o F  +  ( 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 2338 . 2  |-  ( ph  ->  ( F  o F  -  G )  =  ( ( x  e.  ( dom  F  i^i  dom 
G )  |->  ( F `
 x ) )  o F  +  ( x  e.  ( dom 
F  i^i  dom  G ) 
|->  -u ( G `  x ) ) ) )
36 inss1 3402 . . . . 5  |-  ( dom 
F  i^i  dom  G ) 
C_  dom  F
37 resmpt 5016 . . . . 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 11 . . . 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 5591 . . . . . 6  |-  ( ph  ->  F  =  ( x  e.  dom  F  |->  ( F `  x ) ) )
4039, 1eqeltrrd 2371 . . . . 5  |-  ( ph  ->  ( x  e.  dom  F 
|->  ( F `  x
) )  e. MblFn )
41 mbfres 19015 . . . . 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 642 . . . 4  |-  ( ph  ->  ( ( x  e. 
dom  F  |->  ( F `
 x ) )  |`  ( dom  F  i^i  dom 
G ) )  e. MblFn
)
4338, 42eqeltrrd 2371 . . 3  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( F `  x ) )  e. MblFn
)
44 inss2 3403 . . . . . 6  |-  ( dom 
F  i^i  dom  G ) 
C_  dom  G
45 resmpt 5016 . . . . . 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 11 . . . . 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 5591 . . . . . . 7  |-  ( ph  ->  G  =  ( x  e.  dom  G  |->  ( G `  x ) ) )
4847, 8eqeltrrd 2371 . . . . . 6  |-  ( ph  ->  ( x  e.  dom  G 
|->  ( G `  x
) )  e. MblFn )
49 mbfres 19015 . . . . . 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 642 . . . . 5  |-  ( ph  ->  ( ( x  e. 
dom  G  |->  ( G `
 x ) )  |`  ( dom  F  i^i  dom 
G ) )  e. MblFn
)
5146, 50eqeltrrd 2371 . . . 4  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( G `  x ) )  e. MblFn
)
5213, 51mbfneg 19021 . . 3  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  -u ( G `  x ) )  e. MblFn
)
5343, 52mbfadd 19032 . 2  |-  ( ph  ->  ( ( x  e.  ( dom  F  i^i  dom 
G )  |->  ( F `
 x ) )  o F  +  ( x  e.  ( dom 
F  i^i  dom  G ) 
|->  -u ( G `  x ) ) )  e. MblFn )
5435, 53eqeltrd 2370 1  |-  ( ph  ->  ( F  o F  -  G )  e. MblFn
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165    e. cmpt 4093   dom cdm 4705    |` cres 4707    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751    + caddc 8756    - cmin 9053   -ucneg 9054   volcvol 18839  MblFncmbf 18985
This theorem is referenced by:  mbfmul  19097  iblulm  19799
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cc 8077  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xadd 10469  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-xmet 16389  df-met 16390  df-ovol 18840  df-vol 18841  df-mbf 18991
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