MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  i1fadd Structured version   Visualization version   Unicode version

Theorem i1fadd 22732
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 9640 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
21adantl 473 . . 3  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
3 i1fadd.1 . . . 4  |-  ( ph  ->  F  e.  dom  S.1 )
4 i1ff 22713 . . . 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 22713 . . . 4  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 17 . . 3  |-  ( ph  ->  G : RR --> RR )
9 reex 9648 . . . 4  |-  RR  e.  _V
109a1i 11 . . 3  |-  ( ph  ->  RR  e.  _V )
11 inidm 3632 . . 3  |-  ( RR 
i^i  RR )  =  RR
122, 5, 8, 10, 10, 11off 6565 . 2  |-  ( ph  ->  ( F  oF  +  G ) : RR --> RR )
13 i1frn 22714 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
143, 13syl 17 . . . . 5  |-  ( ph  ->  ran  F  e.  Fin )
15 i1frn 22714 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
166, 15syl 17 . . . . 5  |-  ( ph  ->  ran  G  e.  Fin )
17 xpfi 7860 . . . . 5  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
1814, 16, 17syl2anc 673 . . . 4  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
19 eqid 2471 . . . . . 6  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  +  v ) )  =  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  +  v ) )
20 ovex 6336 . . . . . 6  |-  ( u  +  v )  e. 
_V
2119, 20fnmpt2i 6881 . . . . 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 213 . . . 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 7878 . . . 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 675 . . 3  |-  ( ph  ->  ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) )  e.  Fin )
26 eqid 2471 . . . . . . . . 9  |-  ( x  +  y )  =  ( x  +  y )
27 rspceov 6347 . . . . . . . . 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 1379 . . . . . . . 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 6336 . . . . . . . . 9  |-  ( x  +  y )  e. 
_V
30 eqeq1 2475 . . . . . . . . . 10  |-  ( w  =  ( x  +  y )  ->  (
w  =  ( u  +  v )  <->  ( x  +  y )  =  ( u  +  v ) ) )
31302rexbidv 2897 . . . . . . . . 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 3173 . . . . . . . 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 217 . . . . . . 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 473 . . . . . 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 5739 . . . . . . . 8  |-  ( F : RR --> RR  ->  F  Fn  RR )
365, 35syl 17 . . . . . . 7  |-  ( ph  ->  F  Fn  RR )
37 dffn3 5748 . . . . . . 7  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
3836, 37sylib 201 . . . . . 6  |-  ( ph  ->  F : RR --> ran  F
)
39 ffn 5739 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
408, 39syl 17 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
41 dffn3 5748 . . . . . . 7  |-  ( G  Fn  RR  <->  G : RR
--> ran  G )
4240, 41sylib 201 . . . . . 6  |-  ( ph  ->  G : RR --> ran  G
)
4334, 38, 42, 10, 10, 11off 6565 . . . . 5  |-  ( ph  ->  ( F  oF  +  G ) : RR --> { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
44 frn 5747 . . . . 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 6425 . . . 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 3465 . . 3  |-  ( ph  ->  ran  ( F  oF  +  G )  C_ 
ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) ) )
48 ssfi 7810 . . 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 673 . 2  |-  ( ph  ->  ran  ( F  oF  +  G )  e.  Fin )
50 frn 5747 . . . . . . . 8  |-  ( ( F  oF  +  G ) : RR --> RR  ->  ran  ( F  oF  +  G
)  C_  RR )
5112, 50syl 17 . . . . . . 7  |-  ( ph  ->  ran  ( F  oF  +  G )  C_  RR )
5251ssdifssd 3560 . . . . . 6  |-  ( ph  ->  ( ran  ( F  oF  +  G
)  \  { 0 } )  C_  RR )
5352sselda 3418 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  y  e.  RR )
5453recnd 9687 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  y  e.  CC )
553, 6i1faddlem 22730 . . . 4  |-  ( (
ph  /\  y  e.  CC )  ->  ( `' ( F  oF  +  G ) " { y } )  =  U_ z  e. 
ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )
5654, 55syldan 478 . . 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 472 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ran  G  e. 
Fin )
583ad2antrr 740 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F  e.  dom  S.1 )
59 i1fmbf 22712 . . . . . . . 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 740 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F : RR --> RR )
6212ad2antrr 740 . . . . . . . . . 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 3544 . . . . . . . . . 10  |-  ( y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } )  ->  y  e.  ran  ( F  oF  +  G )
)
6564ad2antlr 741 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  y  e.  ran  ( F  oF  +  G ) )
6663, 65sseldd 3419 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  y  e.  RR )
678adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  G : RR
--> RR )
68 frn 5747 . . . . . . . . . 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 3418 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  z  e.  RR )
7166, 70resubcld 10068 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( y  -  z )  e.  RR )
72 mbfimasn 22669 . . . . . . 7  |-  ( ( F  e. MblFn  /\  F : RR
--> RR  /\  ( y  -  z )  e.  RR )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
7360, 61, 71, 72syl3anc 1292 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
746ad2antrr 740 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G  e.  dom  S.1 )
75 i1fmbf 22712 . . . . . . . 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 740 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G : RR --> RR )
78 mbfimasn 22669 . . . . . . 7  |-  ( ( G  e. MblFn  /\  G : RR
--> RR  /\  z  e.  RR )  ->  ( `' G " { z } )  e.  dom  vol )
7976, 77, 70, 78syl3anc 1292 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' G " { z } )  e.  dom  vol )
80 inmbl 22574 . . . . . 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 673 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
8281ralrimiva 2809 . . . 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 22576 . . . 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 673 . . 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 2549 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( `' ( F  oF  +  G ) " {
y } )  e. 
dom  vol )
86 mblvol 22562 . . . 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 22563 . . . . 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 3643 . . . . . . . . 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 731 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
93 mblss 22563 . . . . . . . . 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 22562 . . . . . . . . . 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 774 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  z  =  0 )
9897oveq2d 6324 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  z )  =  ( y  - 
0 ) )
9954adantr 472 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  y  e.  CC )
10099subid1d 9994 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  0 )  =  y )
10198, 100eqtrd 2505 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  z )  =  y )
102101sneqd 3971 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  { ( y  -  z ) }  =  { y } )
103102imaeq2d 5174 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  =  ( `' F " { y } ) )
104103fveq2d 5883 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  =  ( vol `  ( `' F " { y } ) ) )
105 i1fima2sn 22717 . . . . . . . . . . . 12  |-  ( ( F  e.  dom  S.1  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
1063, 105sylan 479 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
107106adantr 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
108104, 107eqeltrd 2549 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  e.  RR )
10996, 108eqeltrrd 2550 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol* `  ( `' F " { ( y  -  z ) } ) )  e.  RR )
110 ovolsscl 22517 . . . . . . . 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 1292 . . . . . . 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 626 . . . . . 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 4088 . . . . . . . 8  |-  ( z  e.  ( ran  G  \  { 0 } )  <-> 
( z  e.  ran  G  /\  z  =/=  0
) )
114 inss2 3644 . . . . . . . . . 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 3544 . . . . . . . . . 10  |-  ( z  e.  ( ran  G  \  { 0 } )  ->  z  e.  ran  G )
117 mblss 22563 . . . . . . . . . . 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 482 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  C_  RR )
120 i1fima 22715 . . . . . . . . . . . . 13  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
1216, 120syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
122121ad2antrr 740 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  e.  dom  vol )
123 mblvol 22562 . . . . . . . . . . 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 472 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  G  e.  dom  S.1 )
126 i1fima2sn 22717 . . . . . . . . . . 11  |-  ( ( G  e.  dom  S.1  /\  z  e.  ( ran 
G  \  { 0 } ) )  -> 
( vol `  ( `' G " { z } ) )  e.  RR )
127125, 126sylan 479 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol `  ( `' G " { z } ) )  e.  RR )
128124, 127eqeltrrd 2550 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol* `  ( `' G " { z } ) )  e.  RR )
129 ovolsscl 22517 . . . . . . . . 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 1292 . . . . . . . 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 484 . . . . . . 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 626 . . . . . 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 2729 . . . . 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 13877 . . . 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 5883 . . . . 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 3429 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  RR )
137136, 133jca 541 . . . . . . 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 2809 . . . . . 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 22532 . . . . . 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 673 . . . . 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 4416 . . . 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 22514 . . . 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 1292 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol* `  ( `' ( F  oF  +  G ) " {
y } ) )  e.  RR )
14487, 143eqeltrd 2549 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  oF  +  G
) " { y } ) )  e.  RR )
14512, 49, 85, 144i1fd 22718 1  |-  ( ph  ->  ( F  oF  +  G )  e. 
dom  S.1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   {cab 2457    =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031    \ cdif 3387    i^i cin 3389    C_ wss 3390   {csn 3959   U_ciun 4269   class class class wbr 4395    X. cxp 4837   `'ccnv 4838   dom cdm 4839   ran crn 4840   "cima 4842    Fn wfn 5584   -->wf 5585   -onto->wfo 5587   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310    oFcof 6548   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557    + caddc 9560    <_ cle 9694    - cmin 9880   sum_csu 13829   vol*covol 22491   volcvol 22493  MblFncmbf 22651   S.1citg1 22652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xadd 11433  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-xmet 19040  df-met 19041  df-ovol 22494  df-vol 22496  df-mbf 22656  df-itg1 22657
This theorem is referenced by:  itg1addlem4  22736  i1fsub  22745  itg2splitlem  22785  itg2split  22786  itg2addlem  22795  itg2addnc  32060  ftc1anclem3  32083  ftc1anclem5  32085  ftc1anclem8  32088
  Copyright terms: Public domain W3C validator