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Theorem i1fadd 19050
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  o F  +  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 8820 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
21adantl 452 . . 3  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
3 i1fadd.1 . . . 4  |-  ( ph  ->  F  e.  dom  S.1 )
4 i1ff 19031 . . . 4  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
53, 4syl 15 . . 3  |-  ( ph  ->  F : RR --> RR )
6 i1fadd.2 . . . 4  |-  ( ph  ->  G  e.  dom  S.1 )
7 i1ff 19031 . . . 4  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 15 . . 3  |-  ( ph  ->  G : RR --> RR )
9 reex 8828 . . . 4  |-  RR  e.  _V
109a1i 10 . . 3  |-  ( ph  ->  RR  e.  _V )
11 inidm 3378 . . 3  |-  ( RR 
i^i  RR )  =  RR
122, 5, 8, 10, 10, 11off 6093 . 2  |-  ( ph  ->  ( F  o F  +  G ) : RR --> RR )
13 i1frn 19032 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
143, 13syl 15 . . . . 5  |-  ( ph  ->  ran  F  e.  Fin )
15 i1frn 19032 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
166, 15syl 15 . . . . 5  |-  ( ph  ->  ran  G  e.  Fin )
17 xpfi 7128 . . . . 5  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
1814, 16, 17syl2anc 642 . . . 4  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
19 eqid 2283 . . . . . 6  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  +  v ) )  =  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  +  v ) )
20 ovex 5883 . . . . . 6  |-  ( u  +  v )  e. 
_V
2119, 20fnmpt2i 6193 . . . . 5  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  +  v ) )  Fn  ( ran  F  X.  ran  G
)
22 dffn4 5457 . . . . 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 199 . . . 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 7142 . . . 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 643 . . 3  |-  ( ph  ->  ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) )  e.  Fin )
26 eqid 2283 . . . . . . . . 9  |-  ( x  +  y )  =  ( x  +  y )
27 rspceov 5893 . . . . . . . . 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 1266 . . . . . . . 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 5883 . . . . . . . . 9  |-  ( x  +  y )  e. 
_V
30 eqeq1 2289 . . . . . . . . . 10  |-  ( w  =  ( x  +  y )  ->  (
w  =  ( u  +  v )  <->  ( x  +  y )  =  ( u  +  v ) ) )
31302rexbidv 2586 . . . . . . . . 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 2914 . . . . . . . 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 203 . . . . . . 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 452 . . . . . 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 5389 . . . . . . . 8  |-  ( F : RR --> RR  ->  F  Fn  RR )
365, 35syl 15 . . . . . . 7  |-  ( ph  ->  F  Fn  RR )
37 dffn3 5396 . . . . . . 7  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
3836, 37sylib 188 . . . . . 6  |-  ( ph  ->  F : RR --> ran  F
)
39 ffn 5389 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
408, 39syl 15 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
41 dffn3 5396 . . . . . . 7  |-  ( G  Fn  RR  <->  G : RR
--> ran  G )
4240, 41sylib 188 . . . . . 6  |-  ( ph  ->  G : RR --> ran  G
)
4334, 38, 42, 10, 10, 11off 6093 . . . . 5  |-  ( ph  ->  ( F  o F  +  G ) : RR --> { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
44 frn 5395 . . . . 5  |-  ( ( F  o F  +  G ) : RR --> { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) }  ->  ran  ( F  o F  +  G )  C_  { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
4543, 44syl 15 . . . 4  |-  ( ph  ->  ran  ( F  o F  +  G )  C_ 
{ w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
4619rnmpt2 5954 . . . 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 3225 . . 3  |-  ( ph  ->  ran  ( F  o F  +  G )  C_ 
ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) ) )
48 ssfi 7083 . . 3  |-  ( ( ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) )  e.  Fin  /\  ran  ( F  o F  +  G )  C_  ran  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  +  v ) ) )  ->  ran  ( F  o F  +  G )  e. 
Fin )
4925, 47, 48syl2anc 642 . 2  |-  ( ph  ->  ran  ( F  o F  +  G )  e.  Fin )
50 difss 3303 . . . . . . 7  |-  ( ran  ( F  o F  +  G )  \  { 0 } ) 
C_  ran  ( F  o F  +  G
)
51 frn 5395 . . . . . . . 8  |-  ( ( F  o F  +  G ) : RR --> RR  ->  ran  ( F  o F  +  G
)  C_  RR )
5212, 51syl 15 . . . . . . 7  |-  ( ph  ->  ran  ( F  o F  +  G )  C_  RR )
5350, 52syl5ss 3190 . . . . . 6  |-  ( ph  ->  ( ran  ( F  o F  +  G
)  \  { 0 } )  C_  RR )
5453sselda 3180 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  y  e.  RR )
5554recnd 8861 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  y  e.  CC )
563, 6i1faddlem 19048 . . . 4  |-  ( (
ph  /\  y  e.  CC )  ->  ( `' ( F  o F  +  G ) " { y } )  =  U_ z  e. 
ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )
5755, 56syldan 456 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( `' ( F  o F  +  G ) " {
y } )  = 
U_ z  e.  ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )
5816adantr 451 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ran  G  e. 
Fin )
593ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F  e.  dom  S.1 )
60 i1fmbf 19030 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F  e. MblFn )
6159, 60syl 15 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F  e. MblFn )
625ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F : RR --> RR )
6312ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( F  o F  +  G ) : RR --> RR )
6463, 51syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ran  ( F  o F  +  G
)  C_  RR )
65 eldifi 3298 . . . . . . . . . 10  |-  ( y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } )  ->  y  e.  ran  ( F  o F  +  G )
)
6665ad2antlr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  y  e.  ran  ( F  o F  +  G ) )
6764, 66sseldd 3181 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  y  e.  RR )
688adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  G : RR
--> RR )
69 frn 5395 . . . . . . . . . 10  |-  ( G : RR --> RR  ->  ran 
G  C_  RR )
7068, 69syl 15 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ran  G  C_  RR )
7170sselda 3180 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  z  e.  RR )
7267, 71resubcld 9211 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( y  -  z )  e.  RR )
73 mbfimasn 18989 . . . . . . 7  |-  ( ( F  e. MblFn  /\  F : RR
--> RR  /\  ( y  -  z )  e.  RR )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
7461, 62, 72, 73syl3anc 1182 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
756ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G  e.  dom  S.1 )
76 i1fmbf 19030 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  G  e. MblFn )
7775, 76syl 15 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G  e. MblFn )
788ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G : RR --> RR )
79 mbfimasn 18989 . . . . . . 7  |-  ( ( G  e. MblFn  /\  G : RR
--> RR  /\  z  e.  RR )  ->  ( `' G " { z } )  e.  dom  vol )
8077, 78, 71, 79syl3anc 1182 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' G " { z } )  e.  dom  vol )
81 inmbl 18899 . . . . . 6  |-  ( ( ( `' F " { ( y  -  z ) } )  e.  dom  vol  /\  ( `' G " { z } )  e.  dom  vol )  ->  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
8274, 80, 81syl2anc 642 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
8382ralrimiva 2626 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  A. z  e.  ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
84 finiunmbl 18901 . . . 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 )
8558, 83, 84syl2anc 642 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  U_ z  e. 
ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
8657, 85eqeltrd 2357 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( `' ( F  o F  +  G ) " {
y } )  e. 
dom  vol )
87 mblvol 18889 . . . 4  |-  ( ( `' ( F  o F  +  G ) " { y } )  e.  dom  vol  ->  ( vol `  ( `' ( F  o F  +  G ) " { y } ) )  =  ( vol
* `  ( `' ( F  o F  +  G ) " {
y } ) ) )
8886, 87syl 15 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  o F  +  G
) " { y } ) )  =  ( vol * `  ( `' ( F  o F  +  G ) " { y } ) ) )
89 mblss 18890 . . . . 5  |-  ( ( `' ( F  o F  +  G ) " { y } )  e.  dom  vol  ->  ( `' ( F  o F  +  G ) " { y } ) 
C_  RR )
9086, 89syl 15 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( `' ( F  o F  +  G ) " {
y } )  C_  RR )
91 inss1 3389 . . . . . . . . 9  |-  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { ( y  -  z ) } )
9291a1i 10 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { ( y  -  z ) } ) )
9374adantrr 697 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
94 mblss 18890 . . . . . . . . 9  |-  ( ( `' F " { ( y  -  z ) } )  e.  dom  vol 
->  ( `' F " { ( y  -  z ) } ) 
C_  RR )
9593, 94syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  C_  RR )
96 mblvol 18889 . . . . . . . . . 10  |-  ( ( `' F " { ( y  -  z ) } )  e.  dom  vol 
->  ( vol `  ( `' F " { ( y  -  z ) } ) )  =  ( vol * `  ( `' F " { ( y  -  z ) } ) ) )
9793, 96syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  =  ( vol
* `  ( `' F " { ( y  -  z ) } ) ) )
98 simprr 733 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  z  =  0 )
9998oveq2d 5874 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  z )  =  ( y  - 
0 ) )
10055adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  y  e.  CC )
101100subid1d 9146 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  0 )  =  y )
10299, 101eqtrd 2315 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  z )  =  y )
103102sneqd 3653 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  { ( y  -  z ) }  =  { y } )
104103imaeq2d 5012 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  =  ( `' F " { y } ) )
105104fveq2d 5529 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  =  ( vol `  ( `' F " { y } ) ) )
106 i1fima2sn 19035 . . . . . . . . . . . 12  |-  ( ( F  e.  dom  S.1  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
1073, 106sylan 457 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
108107adantr 451 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
109105, 108eqeltrd 2357 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  e.  RR )
11097, 109eqeltrrd 2358 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol * `  ( `' F " { ( y  -  z ) } ) )  e.  RR )
111 ovolsscl 18845 . . . . . . . 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 )
11292, 95, 110, 111syl3anc 1182 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol * `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
113112expr 598 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( z  =  0  ->  ( vol * `
 ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR ) )
114 eldifsn 3749 . . . . . . . 8  |-  ( z  e.  ( ran  G  \  { 0 } )  <-> 
( z  e.  ran  G  /\  z  =/=  0
) )
115 inss2 3390 . . . . . . . . . 10  |-  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } )
116115a1i 10 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  (
( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } ) )
117 eldifi 3298 . . . . . . . . . 10  |-  ( z  e.  ( ran  G  \  { 0 } )  ->  z  e.  ran  G )
118 mblss 18890 . . . . . . . . . . 11  |-  ( ( `' G " { z } )  e.  dom  vol 
->  ( `' G " { z } ) 
C_  RR )
11980, 118syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' G " { z } ) 
C_  RR )
120117, 119sylan2 460 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  C_  RR )
121 i1fima 19033 . . . . . . . . . . . . 13  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
1226, 121syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
123122ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  e.  dom  vol )
124 mblvol 18889 . . . . . . . . . . 11  |-  ( ( `' G " { z } )  e.  dom  vol 
->  ( vol `  ( `' G " { z } ) )  =  ( vol * `  ( `' G " { z } ) ) )
125123, 124syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol `  ( `' G " { z } ) )  =  ( vol
* `  ( `' G " { z } ) ) )
1266adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  G  e.  dom  S.1 )
127 i1fima2sn 19035 . . . . . . . . . . 11  |-  ( ( G  e.  dom  S.1  /\  z  e.  ( ran 
G  \  { 0 } ) )  -> 
( vol `  ( `' G " { z } ) )  e.  RR )
128126, 127sylan 457 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol `  ( `' G " { z } ) )  e.  RR )
129125, 128eqeltrrd 2358 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol * `  ( `' G " { z } ) )  e.  RR )
130 ovolsscl 18845 . . . . . . . . 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 )
131116, 120, 129, 130syl3anc 1182 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol * `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
132114, 131sylan2br 462 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =/=  0
) )  ->  ( vol * `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
133132expr 598 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( z  =/=  0  ->  ( vol * `
 ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR ) )
134113, 133pm2.61dne 2523 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( vol * `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
13558, 134fsumrecl 12207 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  sum_ z  e. 
ran  G ( vol
* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
13657fveq2d 5529 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( vol * `
 ( `' ( F  o F  +  G ) " {
y } ) )  =  ( vol * `  U_ z  e.  ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
137115, 119syl5ss 3190 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  RR )
138137, 134jca 518 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  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 ) )
139138ralrimiva 2626 . . . . . 6  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  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 ) )
140 ovolfiniun 18860 . . . . . 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 } ) ) ) )
14158, 139, 140syl2anc 642 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  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 } ) ) ) )
142136, 141eqbrtrd 4043 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( vol * `
 ( `' ( F  o F  +  G ) " {
y } ) )  <_  sum_ z  e.  ran  G ( vol * `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
143 ovollecl 18842 . . . 4  |-  ( ( ( `' ( F  o F  +  G
) " { y } )  C_  RR  /\ 
sum_ z  e.  ran  G ( vol * `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR  /\  ( vol * `  ( `' ( F  o F  +  G ) " { y } ) )  <_  sum_ z  e. 
ran  G ( vol
* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) ) )  ->  ( vol * `
 ( `' ( F  o F  +  G ) " {
y } ) )  e.  RR )
14490, 135, 142, 143syl3anc 1182 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( vol * `
 ( `' ( F  o F  +  G ) " {
y } ) )  e.  RR )
14588, 144eqeltrd 2357 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  o F  +  G
) " { y } ) )  e.  RR )
14612, 49, 86, 145i1fd 19036 1  |-  ( ph  ->  ( F  o F  +  G )  e. 
dom  S.1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   {csn 3640   U_ciun 3905   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    o Fcof 6076   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    <_ cle 8868    - cmin 9037   sum_csu 12158   vol
*covol 18822   volcvol 18823  MblFncmbf 18969   S.1citg1 18970
This theorem is referenced by:  itg1addlem4  19054  i1fsub  19063  itg2splitlem  19103  itg2split  19104  itg2addlem  19113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xadd 10453  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-xmet 16373  df-met 16374  df-ovol 18824  df-vol 18825  df-mbf 18975  df-itg1 18976
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