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

Theorem i1fadd 21195
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 9386 . . . 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 21176 . . . 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 21176 . . . 4  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 16 . . 3  |-  ( ph  ->  G : RR --> RR )
9 reex 9394 . . . 4  |-  RR  e.  _V
109a1i 11 . . 3  |-  ( ph  ->  RR  e.  _V )
11 inidm 3580 . . 3  |-  ( RR 
i^i  RR )  =  RR
122, 5, 8, 10, 10, 11off 6355 . 2  |-  ( ph  ->  ( F  oF  +  G ) : RR --> RR )
13 i1frn 21177 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
143, 13syl 16 . . . . 5  |-  ( ph  ->  ran  F  e.  Fin )
15 i1frn 21177 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
166, 15syl 16 . . . . 5  |-  ( ph  ->  ran  G  e.  Fin )
17 xpfi 7604 . . . . 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 2443 . . . . . 6  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  +  v ) )  =  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  +  v ) )
20 ovex 6137 . . . . . 6  |-  ( u  +  v )  e. 
_V
2119, 20fnmpt2i 6664 . . . . 5  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  +  v ) )  Fn  ( ran  F  X.  ran  G
)
22 dffn4 5647 . . . . 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 7618 . . . 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 2443 . . . . . . . . 9  |-  ( x  +  y )  =  ( x  +  y )
27 rspceov 6149 . . . . . . . . 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 1303 . . . . . . . 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 6137 . . . . . . . . 9  |-  ( x  +  y )  e. 
_V
30 eqeq1 2449 . . . . . . . . . 10  |-  ( w  =  ( x  +  y )  ->  (
w  =  ( u  +  v )  <->  ( x  +  y )  =  ( u  +  v ) ) )
31302rexbidv 2779 . . . . . . . . 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 3127 . . . . . . . 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 5580 . . . . . . . 8  |-  ( F : RR --> RR  ->  F  Fn  RR )
365, 35syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  RR )
37 dffn3 5587 . . . . . . 7  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
3836, 37sylib 196 . . . . . 6  |-  ( ph  ->  F : RR --> ran  F
)
39 ffn 5580 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
408, 39syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
41 dffn3 5587 . . . . . . 7  |-  ( G  Fn  RR  <->  G : RR
--> ran  G )
4240, 41sylib 196 . . . . . 6  |-  ( ph  ->  G : RR --> ran  G
)
4334, 38, 42, 10, 10, 11off 6355 . . . . 5  |-  ( ph  ->  ( F  oF  +  G ) : RR --> { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
44 frn 5586 . . . . 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 6221 . . . 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 3424 . . 3  |-  ( ph  ->  ran  ( F  oF  +  G )  C_ 
ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) ) )
48 ssfi 7554 . . 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 5586 . . . . . . . 8  |-  ( ( F  oF  +  G ) : RR --> RR  ->  ran  ( F  oF  +  G
)  C_  RR )
5112, 50syl 16 . . . . . . 7  |-  ( ph  ->  ran  ( F  oF  +  G )  C_  RR )
5251ssdifssd 3515 . . . . . 6  |-  ( ph  ->  ( ran  ( F  oF  +  G
)  \  { 0 } )  C_  RR )
5352sselda 3377 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  y  e.  RR )
5453recnd 9433 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  y  e.  CC )
553, 6i1faddlem 21193 . . . 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 21175 . . . . . . . 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 3499 . . . . . . . . . 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 3378 . . . . . . . 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 5586 . . . . . . . . . 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 3377 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  z  e.  RR )
7166, 70resubcld 9797 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( y  -  z )  e.  RR )
72 mbfimasn 21134 . . . . . . 7  |-  ( ( F  e. MblFn  /\  F : RR
--> RR  /\  ( y  -  z )  e.  RR )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
7360, 61, 71, 72syl3anc 1218 . . . . . 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 21175 . . . . . . . 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 21134 . . . . . . 7  |-  ( ( G  e. MblFn  /\  G : RR
--> RR  /\  z  e.  RR )  ->  ( `' G " { z } )  e.  dom  vol )
7976, 77, 70, 78syl3anc 1218 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' G " { z } )  e.  dom  vol )
80 inmbl 21045 . . . . . 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 2820 . . . 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 21047 . . . 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 2517 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( `' ( F  oF  +  G ) " {
y } )  e. 
dom  vol )
86 mblvol 21035 . . . 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 21036 . . . . 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 3591 . . . . . . . . 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 21036 . . . . . . . . 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 21035 . . . . . . . . . 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 756 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  z  =  0 )
9897oveq2d 6128 . . . . . . . . . . . . . 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 9729 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  0 )  =  y )
10198, 100eqtrd 2475 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  z )  =  y )
102101sneqd 3910 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  { ( y  -  z ) }  =  { y } )
103102imaeq2d 5190 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  =  ( `' F " { y } ) )
104103fveq2d 5716 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  =  ( vol `  ( `' F " { y } ) ) )
105 i1fima2sn 21180 . . . . . . . . . . . 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 2517 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  e.  RR )
10996, 108eqeltrrd 2518 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol* `  ( `' F " { ( y  -  z ) } ) )  e.  RR )
110 ovolsscl 20991 . . . . . . . 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 1218 . . . . . . 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 4021 . . . . . . . 8  |-  ( z  e.  ( ran  G  \  { 0 } )  <-> 
( z  e.  ran  G  /\  z  =/=  0
) )
114 inss2 3592 . . . . . . . . . 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 3499 . . . . . . . . . 10  |-  ( z  e.  ( ran  G  \  { 0 } )  ->  z  e.  ran  G )
117 mblss 21036 . . . . . . . . . . 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 21178 . . . . . . . . . . . . 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 21035 . . . . . . . . . . 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 21180 . . . . . . . . . . 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 2518 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol* `  ( `' G " { z } ) )  e.  RR )
129 ovolsscl 20991 . . . . . . . . 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 1218 . . . . . . . 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 2712 . . . . 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 13232 . . . 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 5716 . . . . 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 3388 . . . . . . . 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 2820 . . . . . 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 21006 . . . . . 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 4333 . . . 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 20988 . . . 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 1218 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol* `  ( `' ( F  oF  +  G ) " {
y } ) )  e.  RR )
14487, 143eqeltrd 2517 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  oF  +  G
) " { y } ) )  e.  RR )
14512, 49, 85, 144i1fd 21181 1  |-  ( ph  ->  ( F  oF  +  G )  e. 
dom  S.1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429    =/= wne 2620   A.wral 2736   E.wrex 2737   _Vcvv 2993    \ cdif 3346    i^i cin 3348    C_ wss 3349   {csn 3898   U_ciun 4192   class class class wbr 4313    X. cxp 4859   `'ccnv 4860   dom cdm 4861   ran crn 4862   "cima 4864    Fn wfn 5434   -->wf 5435   -onto->wfo 5437   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114    oFcof 6339   Fincfn 7331   CCcc 9301   RRcr 9302   0cc0 9303    + caddc 9306    <_ cle 9440    - cmin 9616   sum_csu 13184   vol*covol 20968   volcvol 20969  MblFncmbf 21116   S.1citg1 21117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-oi 7745  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-q 10975  df-rp 11013  df-xadd 11111  df-ioo 11325  df-ico 11327  df-icc 11328  df-fz 11459  df-fzo 11570  df-fl 11663  df-seq 11828  df-exp 11887  df-hash 12125  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-clim 12987  df-sum 13185  df-xmet 17832  df-met 17833  df-ovol 20970  df-vol 20971  df-mbf 21121  df-itg1 21122
This theorem is referenced by:  itg1addlem4  21199  i1fsub  21208  itg2splitlem  21248  itg2split  21249  itg2addlem  21258  itg2addnc  28472  ftc1anclem3  28495  ftc1anclem5  28497  ftc1anclem8  28500
  Copyright terms: Public domain W3C validator