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Theorem i1fadd 22228
Description: The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1  |-  ( ph  ->  F  e.  dom  S.1 )
i1fadd.2  |-  ( ph  ->  G  e.  dom  S.1 )
Assertion
Ref Expression
i1fadd  |-  ( ph  ->  ( F  oF  +  G )  e. 
dom  S.1 )

Proof of Theorem i1fadd
Dummy variables  y 
z  w  v  x  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 readdcl 9592 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
21adantl 466 . . 3  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
3 i1fadd.1 . . . 4  |-  ( ph  ->  F  e.  dom  S.1 )
4 i1ff 22209 . . . 4  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
53, 4syl 16 . . 3  |-  ( ph  ->  F : RR --> RR )
6 i1fadd.2 . . . 4  |-  ( ph  ->  G  e.  dom  S.1 )
7 i1ff 22209 . . . 4  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 16 . . 3  |-  ( ph  ->  G : RR --> RR )
9 reex 9600 . . . 4  |-  RR  e.  _V
109a1i 11 . . 3  |-  ( ph  ->  RR  e.  _V )
11 inidm 3703 . . 3  |-  ( RR 
i^i  RR )  =  RR
122, 5, 8, 10, 10, 11off 6553 . 2  |-  ( ph  ->  ( F  oF  +  G ) : RR --> RR )
13 i1frn 22210 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
143, 13syl 16 . . . . 5  |-  ( ph  ->  ran  F  e.  Fin )
15 i1frn 22210 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
166, 15syl 16 . . . . 5  |-  ( ph  ->  ran  G  e.  Fin )
17 xpfi 7809 . . . . 5  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
1814, 16, 17syl2anc 661 . . . 4  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
19 eqid 2457 . . . . . 6  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  +  v ) )  =  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  +  v ) )
20 ovex 6324 . . . . . 6  |-  ( u  +  v )  e. 
_V
2119, 20fnmpt2i 6868 . . . . 5  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  +  v ) )  Fn  ( ran  F  X.  ran  G
)
22 dffn4 5807 . . . . 5  |-  ( ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  +  v ) )  Fn  ( ran  F  X.  ran  G
)  <->  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) ) : ( ran 
F  X.  ran  G
) -onto-> ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) ) )
2321, 22mpbi 208 . . . 4  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  +  v ) ) : ( ran  F  X.  ran  G ) -onto-> ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) )
24 fofi 7824 . . . 4  |-  ( ( ( ran  F  X.  ran  G )  e.  Fin  /\  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  +  v ) ) : ( ran  F  X.  ran  G ) -onto-> ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) ) )  ->  ran  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  +  v ) )  e.  Fin )
2518, 23, 24sylancl 662 . . 3  |-  ( ph  ->  ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) )  e.  Fin )
26 eqid 2457 . . . . . . . . 9  |-  ( x  +  y )  =  ( x  +  y )
27 rspceov 6335 . . . . . . . . 9  |-  ( ( x  e.  ran  F  /\  y  e.  ran  G  /\  ( x  +  y )  =  ( x  +  y ) )  ->  E. u  e.  ran  F E. v  e.  ran  G ( x  +  y )  =  ( u  +  v ) )
2826, 27mp3an3 1313 . . . . . . . 8  |-  ( ( x  e.  ran  F  /\  y  e.  ran  G )  ->  E. u  e.  ran  F E. v  e.  ran  G ( x  +  y )  =  ( u  +  v ) )
29 ovex 6324 . . . . . . . . 9  |-  ( x  +  y )  e. 
_V
30 eqeq1 2461 . . . . . . . . . 10  |-  ( w  =  ( x  +  y )  ->  (
w  =  ( u  +  v )  <->  ( x  +  y )  =  ( u  +  v ) ) )
31302rexbidv 2975 . . . . . . . . 9  |-  ( w  =  ( x  +  y )  ->  ( E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v )  <->  E. u  e.  ran  F E. v  e.  ran  G ( x  +  y )  =  ( u  +  v ) ) )
3229, 31elab 3246 . . . . . . . 8  |-  ( ( x  +  y )  e.  { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) }  <->  E. u  e.  ran  F E. v  e.  ran  G ( x  +  y )  =  ( u  +  v ) )
3328, 32sylibr 212 . . . . . . 7  |-  ( ( x  e.  ran  F  /\  y  e.  ran  G )  ->  ( x  +  y )  e. 
{ w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
3433adantl 466 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ran  F  /\  y  e.  ran  G ) )  ->  ( x  +  y )  e.  {
w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
35 ffn 5737 . . . . . . . 8  |-  ( F : RR --> RR  ->  F  Fn  RR )
365, 35syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  RR )
37 dffn3 5744 . . . . . . 7  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
3836, 37sylib 196 . . . . . 6  |-  ( ph  ->  F : RR --> ran  F
)
39 ffn 5737 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
408, 39syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
41 dffn3 5744 . . . . . . 7  |-  ( G  Fn  RR  <->  G : RR
--> ran  G )
4240, 41sylib 196 . . . . . 6  |-  ( ph  ->  G : RR --> ran  G
)
4334, 38, 42, 10, 10, 11off 6553 . . . . 5  |-  ( ph  ->  ( F  oF  +  G ) : RR --> { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
44 frn 5743 . . . . 5  |-  ( ( F  oF  +  G ) : RR --> { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) }  ->  ran  ( F  oF  +  G )  C_  { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
4543, 44syl 16 . . . 4  |-  ( ph  ->  ran  ( F  oF  +  G )  C_ 
{ w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
4619rnmpt2 6411 . . . 4  |-  ran  (
u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  +  v ) )  =  {
w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) }
4745, 46syl6sseqr 3546 . . 3  |-  ( ph  ->  ran  ( F  oF  +  G )  C_ 
ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) ) )
48 ssfi 7759 . . 3  |-  ( ( ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) )  e.  Fin  /\  ran  ( F  oF  +  G )  C_  ran  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  +  v ) ) )  ->  ran  ( F  oF  +  G )  e. 
Fin )
4925, 47, 48syl2anc 661 . 2  |-  ( ph  ->  ran  ( F  oF  +  G )  e.  Fin )
50 frn 5743 . . . . . . . 8  |-  ( ( F  oF  +  G ) : RR --> RR  ->  ran  ( F  oF  +  G
)  C_  RR )
5112, 50syl 16 . . . . . . 7  |-  ( ph  ->  ran  ( F  oF  +  G )  C_  RR )
5251ssdifssd 3638 . . . . . 6  |-  ( ph  ->  ( ran  ( F  oF  +  G
)  \  { 0 } )  C_  RR )
5352sselda 3499 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  y  e.  RR )
5453recnd 9639 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  y  e.  CC )
553, 6i1faddlem 22226 . . . 4  |-  ( (
ph  /\  y  e.  CC )  ->  ( `' ( F  oF  +  G ) " { y } )  =  U_ z  e. 
ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )
5654, 55syldan 470 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( `' ( F  oF  +  G ) " {
y } )  = 
U_ z  e.  ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )
5716adantr 465 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ran  G  e. 
Fin )
583ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F  e.  dom  S.1 )
59 i1fmbf 22208 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F  e. MblFn )
6058, 59syl 16 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F  e. MblFn )
615ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F : RR --> RR )
6212ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( F  oF  +  G ) : RR --> RR )
6362, 50syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ran  ( F  oF  +  G
)  C_  RR )
64 eldifi 3622 . . . . . . . . . 10  |-  ( y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } )  ->  y  e.  ran  ( F  oF  +  G )
)
6564ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  y  e.  ran  ( F  oF  +  G ) )
6663, 65sseldd 3500 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  y  e.  RR )
678adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  G : RR
--> RR )
68 frn 5743 . . . . . . . . . 10  |-  ( G : RR --> RR  ->  ran 
G  C_  RR )
6967, 68syl 16 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ran  G  C_  RR )
7069sselda 3499 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  z  e.  RR )
7166, 70resubcld 10008 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( y  -  z )  e.  RR )
72 mbfimasn 22167 . . . . . . 7  |-  ( ( F  e. MblFn  /\  F : RR
--> RR  /\  ( y  -  z )  e.  RR )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
7360, 61, 71, 72syl3anc 1228 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
746ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G  e.  dom  S.1 )
75 i1fmbf 22208 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  G  e. MblFn )
7674, 75syl 16 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G  e. MblFn )
778ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G : RR --> RR )
78 mbfimasn 22167 . . . . . . 7  |-  ( ( G  e. MblFn  /\  G : RR
--> RR  /\  z  e.  RR )  ->  ( `' G " { z } )  e.  dom  vol )
7976, 77, 70, 78syl3anc 1228 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' G " { z } )  e.  dom  vol )
80 inmbl 22078 . . . . . 6  |-  ( ( ( `' F " { ( y  -  z ) } )  e.  dom  vol  /\  ( `' G " { z } )  e.  dom  vol )  ->  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
8173, 79, 80syl2anc 661 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
8281ralrimiva 2871 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  A. z  e.  ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
83 finiunmbl 22080 . . . 4  |-  ( ( ran  G  e.  Fin  /\ 
A. z  e.  ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )  ->  U_ z  e.  ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
8457, 82, 83syl2anc 661 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  U_ z  e. 
ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
8556, 84eqeltrd 2545 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( `' ( F  oF  +  G ) " {
y } )  e. 
dom  vol )
86 mblvol 22067 . . . 4  |-  ( ( `' ( F  oF  +  G ) " { y } )  e.  dom  vol  ->  ( vol `  ( `' ( F  oF  +  G ) " { y } ) )  =  ( vol* `  ( `' ( F  oF  +  G ) " {
y } ) ) )
8785, 86syl 16 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  oF  +  G
) " { y } ) )  =  ( vol* `  ( `' ( F  oF  +  G ) " { y } ) ) )
88 mblss 22068 . . . . 5  |-  ( ( `' ( F  oF  +  G ) " { y } )  e.  dom  vol  ->  ( `' ( F  oF  +  G ) " { y } ) 
C_  RR )
8985, 88syl 16 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( `' ( F  oF  +  G ) " {
y } )  C_  RR )
90 inss1 3714 . . . . . . . . 9  |-  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { ( y  -  z ) } )
9190a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { ( y  -  z ) } ) )
9273adantrr 716 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
93 mblss 22068 . . . . . . . . 9  |-  ( ( `' F " { ( y  -  z ) } )  e.  dom  vol 
->  ( `' F " { ( y  -  z ) } ) 
C_  RR )
9492, 93syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  C_  RR )
95 mblvol 22067 . . . . . . . . . 10  |-  ( ( `' F " { ( y  -  z ) } )  e.  dom  vol 
->  ( vol `  ( `' F " { ( y  -  z ) } ) )  =  ( vol* `  ( `' F " { ( y  -  z ) } ) ) )
9692, 95syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  =  ( vol* `  ( `' F " { ( y  -  z ) } ) ) )
97 simprr 757 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  z  =  0 )
9897oveq2d 6312 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  z )  =  ( y  - 
0 ) )
9954adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  y  e.  CC )
10099subid1d 9939 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  0 )  =  y )
10198, 100eqtrd 2498 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  z )  =  y )
102101sneqd 4044 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  { ( y  -  z ) }  =  { y } )
103102imaeq2d 5347 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  =  ( `' F " { y } ) )
104103fveq2d 5876 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  =  ( vol `  ( `' F " { y } ) ) )
105 i1fima2sn 22213 . . . . . . . . . . . 12  |-  ( ( F  e.  dom  S.1  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
1063, 105sylan 471 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
107106adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
108104, 107eqeltrd 2545 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  e.  RR )
10996, 108eqeltrrd 2546 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol* `  ( `' F " { ( y  -  z ) } ) )  e.  RR )
110 ovolsscl 22023 . . . . . . . 8  |-  ( ( ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { ( y  -  z ) } )  /\  ( `' F " { ( y  -  z ) } )  C_  RR  /\  ( vol* `  ( `' F " { ( y  -  z ) } ) )  e.  RR )  ->  ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
11191, 94, 109, 110syl3anc 1228 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
112111expr 615 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( z  =  0  ->  ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR ) )
113 eldifsn 4157 . . . . . . . 8  |-  ( z  e.  ( ran  G  \  { 0 } )  <-> 
( z  e.  ran  G  /\  z  =/=  0
) )
114 inss2 3715 . . . . . . . . . 10  |-  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } )
115114a1i 11 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  (
( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } ) )
116 eldifi 3622 . . . . . . . . . 10  |-  ( z  e.  ( ran  G  \  { 0 } )  ->  z  e.  ran  G )
117 mblss 22068 . . . . . . . . . . 11  |-  ( ( `' G " { z } )  e.  dom  vol 
->  ( `' G " { z } ) 
C_  RR )
11879, 117syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' G " { z } ) 
C_  RR )
119116, 118sylan2 474 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  C_  RR )
120 i1fima 22211 . . . . . . . . . . . . 13  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
1216, 120syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
122121ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  e.  dom  vol )
123 mblvol 22067 . . . . . . . . . . 11  |-  ( ( `' G " { z } )  e.  dom  vol 
->  ( vol `  ( `' G " { z } ) )  =  ( vol* `  ( `' G " { z } ) ) )
124122, 123syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol `  ( `' G " { z } ) )  =  ( vol* `  ( `' G " { z } ) ) )
1256adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  G  e.  dom  S.1 )
126 i1fima2sn 22213 . . . . . . . . . . 11  |-  ( ( G  e.  dom  S.1  /\  z  e.  ( ran 
G  \  { 0 } ) )  -> 
( vol `  ( `' G " { z } ) )  e.  RR )
127125, 126sylan 471 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol `  ( `' G " { z } ) )  e.  RR )
128124, 127eqeltrrd 2546 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol* `  ( `' G " { z } ) )  e.  RR )
129 ovolsscl 22023 . . . . . . . . 9  |-  ( ( ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } )  /\  ( `' G " { z } )  C_  RR  /\  ( vol* `  ( `' G " { z } ) )  e.  RR )  ->  ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
130115, 119, 128, 129syl3anc 1228 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
131113, 130sylan2br 476 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =/=  0
) )  ->  ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
132131expr 615 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( z  =/=  0  ->  ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR ) )
133112, 132pm2.61dne 2774 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
13457, 133fsumrecl 13568 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  sum_ z  e. 
ran  G ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
13556fveq2d 5876 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol* `  ( `' ( F  oF  +  G ) " {
y } ) )  =  ( vol* `  U_ z  e.  ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
136114, 118syl5ss 3510 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  RR )
137136, 133jca 532 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  RR  /\  ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR ) )
138137ralrimiva 2871 . . . . . 6  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  A. z  e.  ran  G ( ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  RR  /\  ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR ) )
139 ovolfiniun 22038 . . . . . 6  |-  ( ( ran  G  e.  Fin  /\ 
A. z  e.  ran  G ( ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  RR  /\  ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR ) )  ->  ( vol* `  U_ z  e.  ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  <_  sum_ z  e.  ran  G ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
14057, 138, 139syl2anc 661 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol* `  U_ z  e. 
ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  <_  sum_ z  e.  ran  G ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
141135, 140eqbrtrd 4476 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol* `  ( `' ( F  oF  +  G ) " {
y } ) )  <_  sum_ z  e.  ran  G ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
142 ovollecl 22020 . . . 4  |-  ( ( ( `' ( F  oF  +  G
) " { y } )  C_  RR  /\ 
sum_ z  e.  ran  G ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR  /\  ( vol* `  ( `' ( F  oF  +  G ) " { y } ) )  <_  sum_ z  e. 
ran  G ( vol* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) ) )  ->  ( vol* `  ( `' ( F  oF  +  G ) " {
y } ) )  e.  RR )
14389, 134, 141, 142syl3anc 1228 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol* `  ( `' ( F  oF  +  G ) " {
y } ) )  e.  RR )
14487, 143eqeltrd 2545 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  oF  +  G
) " { y } ) )  e.  RR )
14512, 49, 85, 144i1fd 22214 1  |-  ( ph  ->  ( F  oF  +  G )  e. 
dom  S.1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442    =/= wne 2652   A.wral 2807   E.wrex 2808   _Vcvv 3109    \ cdif 3468    i^i cin 3470    C_ wss 3471   {csn 4032   U_ciun 4332   class class class wbr 4456    X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011    Fn wfn 5589   -->wf 5590   -onto->wfo 5592   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298    oFcof 6537   Fincfn 7535   CCcc 9507   RRcr 9508   0cc0 9509    + caddc 9512    <_ cle 9646    - cmin 9824   sum_csu 13520   vol*covol 22000   volcvol 22001  MblFncmbf 22149   S.1citg1 22150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xadd 11344  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-sum 13521  df-xmet 18539  df-met 18540  df-ovol 22002  df-vol 22003  df-mbf 22154  df-itg1 22155
This theorem is referenced by:  itg1addlem4  22232  i1fsub  22241  itg2splitlem  22281  itg2split  22282  itg2addlem  22291  itg2addnc  30274  ftc1anclem3  30297  ftc1anclem5  30299  ftc1anclem8  30302
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