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Theorem mbfsub 19013
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
)
Dummy variable  x is distinct from all other variables.

Proof of Theorem mbfsub
StepHypRef Expression
1 mbfadd.1 . . . . . . . 8  |-  ( ph  ->  F  e. MblFn )
2 mbff 18978 . . . . . . . 8  |-  ( F  e. MblFn  ->  F : dom  F --> CC )
31, 2syl 17 . . . . . . 7  |-  ( ph  ->  F : dom  F --> CC )
4 elin 3361 . . . . . . . 8  |-  ( x  e.  ( dom  F  i^i  dom  G )  <->  ( x  e.  dom  F  /\  x  e.  dom  G ) )
54simplbi 448 . . . . . . 7  |-  ( x  e.  ( dom  F  i^i  dom  G )  ->  x  e.  dom  F )
6 ffvelrn 5626 . . . . . . 7  |-  ( ( F : dom  F --> CC  /\  x  e.  dom  F )  ->  ( F `  x )  e.  CC )
73, 5, 6syl2an 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  ( F `  x )  e.  CC )
8 mbfadd.2 . . . . . . . 8  |-  ( ph  ->  G  e. MblFn )
9 mbff 18978 . . . . . . . 8  |-  ( G  e. MblFn  ->  G : dom  G --> CC )
108, 9syl 17 . . . . . . 7  |-  ( ph  ->  G : dom  G --> CC )
114simprbi 452 . . . . . . 7  |-  ( x  e.  ( dom  F  i^i  dom  G )  ->  x  e.  dom  G )
12 ffvelrn 5626 . . . . . . 7  |-  ( ( G : dom  G --> CC  /\  x  e.  dom  G )  ->  ( G `  x )  e.  CC )
1310, 11, 12syl2an 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  ( G `  x )  e.  CC )
147, 13negsubd 9160 . . . . 5  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  (
( F `  x
)  +  -u ( G `  x )
)  =  ( ( F `  x )  -  ( G `  x ) ) )
1514eqcomd 2291 . . . 4  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  (
( F `  x
)  -  ( G `
 x ) )  =  ( ( F `
 x )  + 
-u ( G `  x ) ) )
1615mpteq2dva 4109 . . 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 5356 . . . . 5  |-  ( F : dom  F --> CC  ->  F  Fn  dom  F )
183, 17syl 17 . . . 4  |-  ( ph  ->  F  Fn  dom  F
)
19 ffn 5356 . . . . 5  |-  ( G : dom  G --> CC  ->  G  Fn  dom  G )
2010, 19syl 17 . . . 4  |-  ( ph  ->  G  Fn  dom  G
)
21 mbfdm 18979 . . . . 5  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
221, 21syl 17 . . . 4  |-  ( ph  ->  dom  F  e.  dom  vol )
23 mbfdm 18979 . . . . 5  |-  ( G  e. MblFn  ->  dom  G  e.  dom  vol )
248, 23syl 17 . . . 4  |-  ( ph  ->  dom  G  e.  dom  vol )
25 eqid 2286 . . . 4  |-  ( dom 
F  i^i  dom  G )  =  ( dom  F  i^i  dom  G )
26 eqidd 2287 . . . 4  |-  ( (
ph  /\  x  e.  dom  F )  ->  ( F `  x )  =  ( F `  x ) )
27 eqidd 2287 . . . 4  |-  ( (
ph  /\  x  e.  dom  G )  ->  ( G `  x )  =  ( G `  x ) )
2818, 20, 22, 24, 25, 26, 27offval 6048 . . 3  |-  ( ph  ->  ( F  o F  -  G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x )  -  ( G `  x ) ) ) )
29 inmbl 18895 . . . . 5  |-  ( ( dom  F  e.  dom  vol 
/\  dom  G  e.  dom  vol )  ->  ( dom  F  i^i  dom  G
)  e.  dom  vol )
3022, 24, 29syl2anc 644 . . . 4  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  e.  dom  vol )
3113negcld 9141 . . . 4  |-  ( (
ph  /\  x  e.  ( dom  F  i^i  dom  G ) )  ->  -u ( G `  x )  e.  CC )
32 eqidd 2287 . . . 4  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( F `  x ) )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( F `  x ) ) )
33 eqidd 2287 . . . 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 6058 . . 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 2328 . 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 3392 . . . . 5  |-  ( dom 
F  i^i  dom  G ) 
C_  dom  F
37 resmpt 5001 . . . . 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 13 . . . 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 5538 . . . . . 6  |-  ( ph  ->  F  =  ( x  e.  dom  F  |->  ( F `  x ) ) )
4039, 1eqeltrrd 2361 . . . . 5  |-  ( ph  ->  ( x  e.  dom  F 
|->  ( F `  x
) )  e. MblFn )
41 mbfres 18995 . . . . 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 644 . . . 4  |-  ( ph  ->  ( ( x  e. 
dom  F  |->  ( F `
 x ) )  |`  ( dom  F  i^i  dom 
G ) )  e. MblFn
)
4338, 42eqeltrrd 2361 . . 3  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( F `  x ) )  e. MblFn
)
44 inss2 3393 . . . . . 6  |-  ( dom 
F  i^i  dom  G ) 
C_  dom  G
45 resmpt 5001 . . . . . 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 13 . . . . 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 5538 . . . . . . 7  |-  ( ph  ->  G  =  ( x  e.  dom  G  |->  ( G `  x ) ) )
4847, 8eqeltrrd 2361 . . . . . 6  |-  ( ph  ->  ( x  e.  dom  G 
|->  ( G `  x
) )  e. MblFn )
49 mbfres 18995 . . . . . 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 644 . . . . 5  |-  ( ph  ->  ( ( x  e. 
dom  G  |->  ( G `
 x ) )  |`  ( dom  F  i^i  dom 
G ) )  e. MblFn
)
5146, 50eqeltrrd 2361 . . . 4  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( G `  x ) )  e. MblFn
)
5213, 51mbfneg 19001 . . 3  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  -u ( G `  x ) )  e. MblFn
)
5343, 52mbfadd 19012 . 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 2360 1  |-  ( ph  ->  ( F  o F  -  G )  e. MblFn
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1625    e. wcel 1687    i^i cin 3154    C_ wss 3155    e. cmpt 4080   dom cdm 4690    |` cres 4692    Fn wfn 5218   -->wf 5219   ` cfv 5223  (class class class)co 5821    o Fcof 6039   CCcc 8732    + caddc 8737    - cmin 9034   -ucneg 9035   volcvol 18819  MblFncmbf 18965
This theorem is referenced by:  mbfmul  19077  iblulm  19779
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-inf2 7339  ax-cc 8058  ax-cnex 8790  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811  ax-pre-sup 8812
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-int 3866  df-iun 3910  df-disj 3997  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-se 4354  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-isom 5232  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-of 6041  df-1st 6085  df-2nd 6086  df-iota 6254  df-riota 6301  df-recs 6385  df-rdg 6420  df-1o 6476  df-2o 6477  df-oadd 6480  df-omul 6481  df-er 6657  df-map 6771  df-pm 6772  df-en 6861  df-dom 6862  df-sdom 6863  df-fin 6864  df-sup 7191  df-oi 7222  df-card 7569  df-acn 7572  df-cda 7791  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037  df-div 9421  df-nn 9744  df-2 9801  df-3 9802  df-n0 9963  df-z 10022  df-uz 10228  df-q 10314  df-rp 10352  df-xadd 10450  df-ioo 10656  df-ioc 10657  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-fl 10921  df-seq 11043  df-exp 11101  df-hash 11334  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-clim 11958  df-rlim 11959  df-sum 12155  df-xmet 16369  df-met 16370  df-ovol 18820  df-vol 18821  df-mbf 18971
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