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Theorem i1fadd 22639
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 9622 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
21adantl 467 . . 3  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
3 i1fadd.1 . . . 4  |-  ( ph  ->  F  e.  dom  S.1 )
4 i1ff 22620 . . . 4  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
53, 4syl 17 . . 3  |-  ( ph  ->  F : RR --> RR )
6 i1fadd.2 . . . 4  |-  ( ph  ->  G  e.  dom  S.1 )
7 i1ff 22620 . . . 4  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 17 . . 3  |-  ( ph  ->  G : RR --> RR )
9 reex 9630 . . . 4  |-  RR  e.  _V
109a1i 11 . . 3  |-  ( ph  ->  RR  e.  _V )
11 inidm 3671 . . 3  |-  ( RR 
i^i  RR )  =  RR
122, 5, 8, 10, 10, 11off 6556 . 2  |-  ( ph  ->  ( F  oF  +  G ) : RR --> RR )
13 i1frn 22621 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
143, 13syl 17 . . . . 5  |-  ( ph  ->  ran  F  e.  Fin )
15 i1frn 22621 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
166, 15syl 17 . . . . 5  |-  ( ph  ->  ran  G  e.  Fin )
17 xpfi 7844 . . . . 5  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
1814, 16, 17syl2anc 665 . . . 4  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
19 eqid 2422 . . . . . 6  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  +  v ) )  =  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  +  v ) )
20 ovex 6329 . . . . . 6  |-  ( u  +  v )  e. 
_V
2119, 20fnmpt2i 6872 . . . . 5  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  +  v ) )  Fn  ( ran  F  X.  ran  G
)
22 dffn4 5812 . . . . 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 211 . . . 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 7862 . . . 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 666 . . 3  |-  ( ph  ->  ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) )  e.  Fin )
26 eqid 2422 . . . . . . . . 9  |-  ( x  +  y )  =  ( x  +  y )
27 rspceov 6340 . . . . . . . . 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 1349 . . . . . . . 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 6329 . . . . . . . . 9  |-  ( x  +  y )  e. 
_V
30 eqeq1 2426 . . . . . . . . . 10  |-  ( w  =  ( x  +  y )  ->  (
w  =  ( u  +  v )  <->  ( x  +  y )  =  ( u  +  v ) ) )
31302rexbidv 2946 . . . . . . . . 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 3218 . . . . . . . 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 215 . . . . . . 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 467 . . . . . 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 5742 . . . . . . . 8  |-  ( F : RR --> RR  ->  F  Fn  RR )
365, 35syl 17 . . . . . . 7  |-  ( ph  ->  F  Fn  RR )
37 dffn3 5749 . . . . . . 7  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
3836, 37sylib 199 . . . . . 6  |-  ( ph  ->  F : RR --> ran  F
)
39 ffn 5742 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
408, 39syl 17 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
41 dffn3 5749 . . . . . . 7  |-  ( G  Fn  RR  <->  G : RR
--> ran  G )
4240, 41sylib 199 . . . . . 6  |-  ( ph  ->  G : RR --> ran  G
)
4334, 38, 42, 10, 10, 11off 6556 . . . . 5  |-  ( ph  ->  ( F  oF  +  G ) : RR --> { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
44 frn 5748 . . . . 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 17 . . . 4  |-  ( ph  ->  ran  ( F  oF  +  G )  C_ 
{ w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
4619rnmpt2 6416 . . . 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 3511 . . 3  |-  ( ph  ->  ran  ( F  oF  +  G )  C_ 
ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) ) )
48 ssfi 7794 . . 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 665 . 2  |-  ( ph  ->  ran  ( F  oF  +  G )  e.  Fin )
50 frn 5748 . . . . . . . 8  |-  ( ( F  oF  +  G ) : RR --> RR  ->  ran  ( F  oF  +  G
)  C_  RR )
5112, 50syl 17 . . . . . . 7  |-  ( ph  ->  ran  ( F  oF  +  G )  C_  RR )
5251ssdifssd 3603 . . . . . 6  |-  ( ph  ->  ( ran  ( F  oF  +  G
)  \  { 0 } )  C_  RR )
5352sselda 3464 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  y  e.  RR )
5453recnd 9669 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  y  e.  CC )
553, 6i1faddlem 22637 . . . 4  |-  ( (
ph  /\  y  e.  CC )  ->  ( `' ( F  oF  +  G ) " { y } )  =  U_ z  e. 
ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )
5654, 55syldan 472 . . 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 466 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ran  G  e. 
Fin )
583ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F  e.  dom  S.1 )
59 i1fmbf 22619 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F  e. MblFn )
6058, 59syl 17 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F  e. MblFn )
615ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F : RR --> RR )
6212ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( F  oF  +  G ) : RR --> RR )
6362, 50syl 17 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ran  ( F  oF  +  G
)  C_  RR )
64 eldifi 3587 . . . . . . . . . 10  |-  ( y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } )  ->  y  e.  ran  ( F  oF  +  G )
)
6564ad2antlr 731 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  y  e.  ran  ( F  oF  +  G ) )
6663, 65sseldd 3465 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  y  e.  RR )
678adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  G : RR
--> RR )
68 frn 5748 . . . . . . . . . 10  |-  ( G : RR --> RR  ->  ran 
G  C_  RR )
6967, 68syl 17 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ran  G  C_  RR )
7069sselda 3464 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  z  e.  RR )
7166, 70resubcld 10047 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( y  -  z )  e.  RR )
72 mbfimasn 22576 . . . . . . 7  |-  ( ( F  e. MblFn  /\  F : RR
--> RR  /\  ( y  -  z )  e.  RR )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
7360, 61, 71, 72syl3anc 1264 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
746ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G  e.  dom  S.1 )
75 i1fmbf 22619 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  G  e. MblFn )
7674, 75syl 17 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G  e. MblFn )
778ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G : RR --> RR )
78 mbfimasn 22576 . . . . . . 7  |-  ( ( G  e. MblFn  /\  G : RR
--> RR  /\  z  e.  RR )  ->  ( `' G " { z } )  e.  dom  vol )
7976, 77, 70, 78syl3anc 1264 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' G " { z } )  e.  dom  vol )
80 inmbl 22481 . . . . . 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 665 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
8281ralrimiva 2839 . . . 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 22483 . . . 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 665 . . 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 2510 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( `' ( F  oF  +  G ) " {
y } )  e. 
dom  vol )
86 mblvol 22470 . . . 4  |-  ( ( `' ( F  oF  +  G ) " { y } )  e.  dom  vol  ->  ( vol `  ( `' ( F  oF  +  G ) " { y } ) )  =  ( vol* `  ( `' ( F  oF  +  G ) " {
y } ) ) )
8785, 86syl 17 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  oF  +  G
) " { y } ) )  =  ( vol* `  ( `' ( F  oF  +  G ) " { y } ) ) )
88 mblss 22471 . . . . 5  |-  ( ( `' ( F  oF  +  G ) " { y } )  e.  dom  vol  ->  ( `' ( F  oF  +  G ) " { y } ) 
C_  RR )
8985, 88syl 17 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( `' ( F  oF  +  G ) " {
y } )  C_  RR )
90 inss1 3682 . . . . . . . . 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 721 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
93 mblss 22471 . . . . . . . . 9  |-  ( ( `' F " { ( y  -  z ) } )  e.  dom  vol 
->  ( `' F " { ( y  -  z ) } ) 
C_  RR )
9492, 93syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  C_  RR )
95 mblvol 22470 . . . . . . . . . 10  |-  ( ( `' F " { ( y  -  z ) } )  e.  dom  vol 
->  ( vol `  ( `' F " { ( y  -  z ) } ) )  =  ( vol* `  ( `' F " { ( y  -  z ) } ) ) )
9692, 95syl 17 . . . . . . . . 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 764 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  z  =  0 )
9897oveq2d 6317 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  z )  =  ( y  - 
0 ) )
9954adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  y  e.  CC )
10099subid1d 9975 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  0 )  =  y )
10198, 100eqtrd 2463 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  z )  =  y )
102101sneqd 4008 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  { ( y  -  z ) }  =  { y } )
103102imaeq2d 5183 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  =  ( `' F " { y } ) )
104103fveq2d 5881 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  =  ( vol `  ( `' F " { y } ) ) )
105 i1fima2sn 22624 . . . . . . . . . . . 12  |-  ( ( F  e.  dom  S.1  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
1063, 105sylan 473 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
107106adantr 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
108104, 107eqeltrd 2510 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  e.  RR )
10996, 108eqeltrrd 2511 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol* `  ( `' F " { ( y  -  z ) } ) )  e.  RR )
110 ovolsscl 22425 . . . . . . . 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 1264 . . . . . . 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 618 . . . . . 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 4122 . . . . . . . 8  |-  ( z  e.  ( ran  G  \  { 0 } )  <-> 
( z  e.  ran  G  /\  z  =/=  0
) )
114 inss2 3683 . . . . . . . . . 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 3587 . . . . . . . . . 10  |-  ( z  e.  ( ran  G  \  { 0 } )  ->  z  e.  ran  G )
117 mblss 22471 . . . . . . . . . . 11  |-  ( ( `' G " { z } )  e.  dom  vol 
->  ( `' G " { z } ) 
C_  RR )
11879, 117syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' G " { z } ) 
C_  RR )
119116, 118sylan2 476 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  C_  RR )
120 i1fima 22622 . . . . . . . . . . . . 13  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
1216, 120syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
122121ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  e.  dom  vol )
123 mblvol 22470 . . . . . . . . . . 11  |-  ( ( `' G " { z } )  e.  dom  vol 
->  ( vol `  ( `' G " { z } ) )  =  ( vol* `  ( `' G " { z } ) ) )
124122, 123syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol `  ( `' G " { z } ) )  =  ( vol* `  ( `' G " { z } ) ) )
1256adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  G  e.  dom  S.1 )
126 i1fima2sn 22624 . . . . . . . . . . 11  |-  ( ( G  e.  dom  S.1  /\  z  e.  ( ran 
G  \  { 0 } ) )  -> 
( vol `  ( `' G " { z } ) )  e.  RR )
127125, 126sylan 473 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol `  ( `' G " { z } ) )  e.  RR )
128124, 127eqeltrrd 2511 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol* `  ( `' G " { z } ) )  e.  RR )
129 ovolsscl 22425 . . . . . . . . 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 1264 . . . . . . . 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 478 . . . . . . 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 618 . . . . . 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 2741 . . . . 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 13787 . . . 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 5881 . . . . 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 3475 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  RR )
137136, 133jca 534 . . . . . . 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 2839 . . . . . 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 22440 . . . . . 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 665 . . . . 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 4441 . . . 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 22422 . . . 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 1264 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol* `  ( `' ( F  oF  +  G ) " {
y } ) )  e.  RR )
14487, 143eqeltrd 2510 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  oF  +  G
) " { y } ) )  e.  RR )
14512, 49, 85, 144i1fd 22625 1  |-  ( ph  ->  ( F  oF  +  G )  e. 
dom  S.1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868   {cab 2407    =/= wne 2618   A.wral 2775   E.wrex 2776   _Vcvv 3081    \ cdif 3433    i^i cin 3435    C_ wss 3436   {csn 3996   U_ciun 4296   class class class wbr 4420    X. cxp 4847   `'ccnv 4848   dom cdm 4849   ran crn 4850   "cima 4852    Fn wfn 5592   -->wf 5593   -onto->wfo 5595   ` cfv 5597  (class class class)co 6301    |-> cmpt2 6303    oFcof 6539   Fincfn 7573   CCcc 9537   RRcr 9538   0cc0 9539    + caddc 9542    <_ cle 9676    - cmin 9860   sum_csu 13739   vol*covol 22399   volcvol 22401  MblFncmbf 22558   S.1citg1 22559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-pm 7479  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xadd 11410  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12027  df-seq 12213  df-exp 12272  df-hash 12515  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-clim 13539  df-sum 13740  df-xmet 18950  df-met 18951  df-ovol 22402  df-vol 22404  df-mbf 22563  df-itg1 22564
This theorem is referenced by:  itg1addlem4  22643  i1fsub  22652  itg2splitlem  22692  itg2split  22693  itg2addlem  22702  itg2addnc  31909  ftc1anclem3  31932  ftc1anclem5  31934  ftc1anclem8  31937
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