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Theorem i1fadd 22646
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 9619 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
21adantl 468 . . 3  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
3 i1fadd.1 . . . 4  |-  ( ph  ->  F  e.  dom  S.1 )
4 i1ff 22627 . . . 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 22627 . . . 4  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 17 . . 3  |-  ( ph  ->  G : RR --> RR )
9 reex 9627 . . . 4  |-  RR  e.  _V
109a1i 11 . . 3  |-  ( ph  ->  RR  e.  _V )
11 inidm 3640 . . 3  |-  ( RR 
i^i  RR )  =  RR
122, 5, 8, 10, 10, 11off 6543 . 2  |-  ( ph  ->  ( F  oF  +  G ) : RR --> RR )
13 i1frn 22628 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
143, 13syl 17 . . . . 5  |-  ( ph  ->  ran  F  e.  Fin )
15 i1frn 22628 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
166, 15syl 17 . . . . 5  |-  ( ph  ->  ran  G  e.  Fin )
17 xpfi 7839 . . . . 5  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
1814, 16, 17syl2anc 666 . . . 4  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
19 eqid 2450 . . . . . 6  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  +  v ) )  =  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  +  v ) )
20 ovex 6316 . . . . . 6  |-  ( u  +  v )  e. 
_V
2119, 20fnmpt2i 6859 . . . . 5  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  +  v ) )  Fn  ( ran  F  X.  ran  G
)
22 dffn4 5797 . . . . 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 212 . . . 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 7857 . . . 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 667 . . 3  |-  ( ph  ->  ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) )  e.  Fin )
26 eqid 2450 . . . . . . . . 9  |-  ( x  +  y )  =  ( x  +  y )
27 rspceov 6327 . . . . . . . . 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 1352 . . . . . . . 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 6316 . . . . . . . . 9  |-  ( x  +  y )  e. 
_V
30 eqeq1 2454 . . . . . . . . . 10  |-  ( w  =  ( x  +  y )  ->  (
w  =  ( u  +  v )  <->  ( x  +  y )  =  ( u  +  v ) ) )
31302rexbidv 2907 . . . . . . . . 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 3184 . . . . . . . 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 216 . . . . . . 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 468 . . . . . 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 5726 . . . . . . . 8  |-  ( F : RR --> RR  ->  F  Fn  RR )
365, 35syl 17 . . . . . . 7  |-  ( ph  ->  F  Fn  RR )
37 dffn3 5734 . . . . . . 7  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
3836, 37sylib 200 . . . . . 6  |-  ( ph  ->  F : RR --> ran  F
)
39 ffn 5726 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
408, 39syl 17 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
41 dffn3 5734 . . . . . . 7  |-  ( G  Fn  RR  <->  G : RR
--> ran  G )
4240, 41sylib 200 . . . . . 6  |-  ( ph  ->  G : RR --> ran  G
)
4334, 38, 42, 10, 10, 11off 6543 . . . . 5  |-  ( ph  ->  ( F  oF  +  G ) : RR --> { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
44 frn 5733 . . . . 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 6403 . . . 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 3478 . . 3  |-  ( ph  ->  ran  ( F  oF  +  G )  C_ 
ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) ) )
48 ssfi 7789 . . 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 666 . 2  |-  ( ph  ->  ran  ( F  oF  +  G )  e.  Fin )
50 frn 5733 . . . . . . . 8  |-  ( ( F  oF  +  G ) : RR --> RR  ->  ran  ( F  oF  +  G
)  C_  RR )
5112, 50syl 17 . . . . . . 7  |-  ( ph  ->  ran  ( F  oF  +  G )  C_  RR )
5251ssdifssd 3570 . . . . . 6  |-  ( ph  ->  ( ran  ( F  oF  +  G
)  \  { 0 } )  C_  RR )
5352sselda 3431 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  y  e.  RR )
5453recnd 9666 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  y  e.  CC )
553, 6i1faddlem 22644 . . . 4  |-  ( (
ph  /\  y  e.  CC )  ->  ( `' ( F  oF  +  G ) " { y } )  =  U_ z  e. 
ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )
5654, 55syldan 473 . . 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 467 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ran  G  e. 
Fin )
583ad2antrr 731 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F  e.  dom  S.1 )
59 i1fmbf 22626 . . . . . . . 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 731 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F : RR --> RR )
6212ad2antrr 731 . . . . . . . . . 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 3554 . . . . . . . . . 10  |-  ( y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } )  ->  y  e.  ran  ( F  oF  +  G )
)
6564ad2antlr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  y  e.  ran  ( F  oF  +  G ) )
6663, 65sseldd 3432 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  y  e.  RR )
678adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  G : RR
--> RR )
68 frn 5733 . . . . . . . . . 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 3431 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  z  e.  RR )
7166, 70resubcld 10044 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( y  -  z )  e.  RR )
72 mbfimasn 22583 . . . . . . 7  |-  ( ( F  e. MblFn  /\  F : RR
--> RR  /\  ( y  -  z )  e.  RR )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
7360, 61, 71, 72syl3anc 1267 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
746ad2antrr 731 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G  e.  dom  S.1 )
75 i1fmbf 22626 . . . . . . . 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 731 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G : RR --> RR )
78 mbfimasn 22583 . . . . . . 7  |-  ( ( G  e. MblFn  /\  G : RR
--> RR  /\  z  e.  RR )  ->  ( `' G " { z } )  e.  dom  vol )
7976, 77, 70, 78syl3anc 1267 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' G " { z } )  e.  dom  vol )
80 inmbl 22488 . . . . . 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 666 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
8281ralrimiva 2801 . . . 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 22490 . . . 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 666 . . 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 2528 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( `' ( F  oF  +  G ) " {
y } )  e. 
dom  vol )
86 mblvol 22477 . . . 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 22478 . . . . 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 3651 . . . . . . . . 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 722 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
93 mblss 22478 . . . . . . . . 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 22477 . . . . . . . . . 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 765 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  z  =  0 )
9897oveq2d 6304 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  z )  =  ( y  - 
0 ) )
9954adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  y  e.  CC )
10099subid1d 9972 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  0 )  =  y )
10198, 100eqtrd 2484 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  z )  =  y )
102101sneqd 3979 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  { ( y  -  z ) }  =  { y } )
103102imaeq2d 5167 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  =  ( `' F " { y } ) )
104103fveq2d 5867 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  =  ( vol `  ( `' F " { y } ) ) )
105 i1fima2sn 22631 . . . . . . . . . . . 12  |-  ( ( F  e.  dom  S.1  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
1063, 105sylan 474 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
107106adantr 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
108104, 107eqeltrd 2528 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  e.  RR )
10996, 108eqeltrrd 2529 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol* `  ( `' F " { ( y  -  z ) } ) )  e.  RR )
110 ovolsscl 22432 . . . . . . . 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 1267 . . . . . . 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 619 . . . . . 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 4096 . . . . . . . 8  |-  ( z  e.  ( ran  G  \  { 0 } )  <-> 
( z  e.  ran  G  /\  z  =/=  0
) )
114 inss2 3652 . . . . . . . . . 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 3554 . . . . . . . . . 10  |-  ( z  e.  ( ran  G  \  { 0 } )  ->  z  e.  ran  G )
117 mblss 22478 . . . . . . . . . . 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 477 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  C_  RR )
120 i1fima 22629 . . . . . . . . . . . . 13  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
1216, 120syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
122121ad2antrr 731 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  e.  dom  vol )
123 mblvol 22477 . . . . . . . . . . 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 467 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  G  e.  dom  S.1 )
126 i1fima2sn 22631 . . . . . . . . . . 11  |-  ( ( G  e.  dom  S.1  /\  z  e.  ( ran 
G  \  { 0 } ) )  -> 
( vol `  ( `' G " { z } ) )  e.  RR )
127125, 126sylan 474 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol `  ( `' G " { z } ) )  e.  RR )
128124, 127eqeltrrd 2529 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol* `  ( `' G " { z } ) )  e.  RR )
129 ovolsscl 22432 . . . . . . . . 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 1267 . . . . . . . 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 479 . . . . . . 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 619 . . . . . 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 2709 . . . . 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 13793 . . . 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 5867 . . . . 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 3442 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  RR )
137136, 133jca 535 . . . . . . 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 2801 . . . . . 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 22447 . . . . . 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 666 . . . . 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 4422 . . . 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 22429 . . . 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 1267 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol* `  ( `' ( F  oF  +  G ) " {
y } ) )  e.  RR )
14487, 143eqeltrd 2528 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  oF  +  G
) " { y } ) )  e.  RR )
14512, 49, 85, 144i1fd 22632 1  |-  ( ph  ->  ( F  oF  +  G )  e. 
dom  S.1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1443    e. wcel 1886   {cab 2436    =/= wne 2621   A.wral 2736   E.wrex 2737   _Vcvv 3044    \ cdif 3400    i^i cin 3402    C_ wss 3403   {csn 3967   U_ciun 4277   class class class wbr 4401    X. cxp 4831   `'ccnv 4832   dom cdm 4833   ran crn 4834   "cima 4836    Fn wfn 5576   -->wf 5577   -onto->wfo 5579   ` cfv 5581  (class class class)co 6288    |-> cmpt2 6290    oFcof 6526   Fincfn 7566   CCcc 9534   RRcr 9535   0cc0 9536    + caddc 9539    <_ cle 9673    - cmin 9857   sum_csu 13745   vol*covol 22406   volcvol 22408  MblFncmbf 22565   S.1citg1 22566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-q 11262  df-rp 11300  df-xadd 11407  df-ioo 11636  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-sum 13746  df-xmet 18956  df-met 18957  df-ovol 22409  df-vol 22411  df-mbf 22570  df-itg1 22571
This theorem is referenced by:  itg1addlem4  22650  i1fsub  22659  itg2splitlem  22699  itg2split  22700  itg2addlem  22709  itg2addnc  31989  ftc1anclem3  32012  ftc1anclem5  32014  ftc1anclem8  32017
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