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Theorem i1fadd 19589
Description: The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1  |-  ( ph  ->  F  e.  dom  S.1 )
i1fadd.2  |-  ( ph  ->  G  e.  dom  S.1 )
Assertion
Ref Expression
i1fadd  |-  ( ph  ->  ( F  o F  +  G )  e. 
dom  S.1 )

Proof of Theorem i1fadd
Dummy variables  y 
z  w  v  x  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 readdcl 9075 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
21adantl 454 . . 3  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
3 i1fadd.1 . . . 4  |-  ( ph  ->  F  e.  dom  S.1 )
4 i1ff 19570 . . . 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 19570 . . . 4  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 16 . . 3  |-  ( ph  ->  G : RR --> RR )
9 reex 9083 . . . 4  |-  RR  e.  _V
109a1i 11 . . 3  |-  ( ph  ->  RR  e.  _V )
11 inidm 3552 . . 3  |-  ( RR 
i^i  RR )  =  RR
122, 5, 8, 10, 10, 11off 6322 . 2  |-  ( ph  ->  ( F  o F  +  G ) : RR --> RR )
13 i1frn 19571 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
143, 13syl 16 . . . . 5  |-  ( ph  ->  ran  F  e.  Fin )
15 i1frn 19571 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
166, 15syl 16 . . . . 5  |-  ( ph  ->  ran  G  e.  Fin )
17 xpfi 7380 . . . . 5  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
1814, 16, 17syl2anc 644 . . . 4  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
19 eqid 2438 . . . . . 6  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  +  v ) )  =  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  +  v ) )
20 ovex 6108 . . . . . 6  |-  ( u  +  v )  e. 
_V
2119, 20fnmpt2i 6422 . . . . 5  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  +  v ) )  Fn  ( ran  F  X.  ran  G
)
22 dffn4 5661 . . . . 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 201 . . . 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 7394 . . . 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 645 . . 3  |-  ( ph  ->  ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) )  e.  Fin )
26 eqid 2438 . . . . . . . . 9  |-  ( x  +  y )  =  ( x  +  y )
27 rspceov 6118 . . . . . . . . 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 1269 . . . . . . . 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 6108 . . . . . . . . 9  |-  ( x  +  y )  e. 
_V
30 eqeq1 2444 . . . . . . . . . 10  |-  ( w  =  ( x  +  y )  ->  (
w  =  ( u  +  v )  <->  ( x  +  y )  =  ( u  +  v ) ) )
31302rexbidv 2750 . . . . . . . . 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 3084 . . . . . . . 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 205 . . . . . . 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 454 . . . . . 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 5593 . . . . . . . 8  |-  ( F : RR --> RR  ->  F  Fn  RR )
365, 35syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  RR )
37 dffn3 5600 . . . . . . 7  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
3836, 37sylib 190 . . . . . 6  |-  ( ph  ->  F : RR --> ran  F
)
39 ffn 5593 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
408, 39syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
41 dffn3 5600 . . . . . . 7  |-  ( G  Fn  RR  <->  G : RR
--> ran  G )
4240, 41sylib 190 . . . . . 6  |-  ( ph  ->  G : RR --> ran  G
)
4334, 38, 42, 10, 10, 11off 6322 . . . . 5  |-  ( ph  ->  ( F  o F  +  G ) : RR --> { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
44 frn 5599 . . . . 5  |-  ( ( F  o F  +  G ) : RR --> { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) }  ->  ran  ( F  o F  +  G )  C_  { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
4543, 44syl 16 . . . 4  |-  ( ph  ->  ran  ( F  o F  +  G )  C_ 
{ w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  +  v ) } )
4619rnmpt2 6182 . . . 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 3397 . . 3  |-  ( ph  ->  ran  ( F  o F  +  G )  C_ 
ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) ) )
48 ssfi 7331 . . 3  |-  ( ( ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  +  v ) )  e.  Fin  /\  ran  ( F  o F  +  G )  C_  ran  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  +  v ) ) )  ->  ran  ( F  o F  +  G )  e. 
Fin )
4925, 47, 48syl2anc 644 . 2  |-  ( ph  ->  ran  ( F  o F  +  G )  e.  Fin )
50 frn 5599 . . . . . . . 8  |-  ( ( F  o F  +  G ) : RR --> RR  ->  ran  ( F  o F  +  G
)  C_  RR )
5112, 50syl 16 . . . . . . 7  |-  ( ph  ->  ran  ( F  o F  +  G )  C_  RR )
5251ssdifssd 3487 . . . . . 6  |-  ( ph  ->  ( ran  ( F  o F  +  G
)  \  { 0 } )  C_  RR )
5352sselda 3350 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  y  e.  RR )
5453recnd 9116 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  y  e.  CC )
553, 6i1faddlem 19587 . . . 4  |-  ( (
ph  /\  y  e.  CC )  ->  ( `' ( F  o F  +  G ) " { y } )  =  U_ z  e. 
ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )
5654, 55syldan 458 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( `' ( F  o F  +  G ) " {
y } )  = 
U_ z  e.  ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )
5716adantr 453 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ran  G  e. 
Fin )
583ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F  e.  dom  S.1 )
59 i1fmbf 19569 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F  e. MblFn )
6058, 59syl 16 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F  e. MblFn )
615ad2antrr 708 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  F : RR --> RR )
6212ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( F  o F  +  G ) : RR --> RR )
6362, 50syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ran  ( F  o F  +  G
)  C_  RR )
64 eldifi 3471 . . . . . . . . . 10  |-  ( y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } )  ->  y  e.  ran  ( F  o F  +  G )
)
6564ad2antlr 709 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  y  e.  ran  ( F  o F  +  G ) )
6663, 65sseldd 3351 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  y  e.  RR )
678adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  G : RR
--> RR )
68 frn 5599 . . . . . . . . . 10  |-  ( G : RR --> RR  ->  ran 
G  C_  RR )
6967, 68syl 16 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ran  G  C_  RR )
7069sselda 3350 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  z  e.  RR )
7166, 70resubcld 9467 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( y  -  z )  e.  RR )
72 mbfimasn 19528 . . . . . . 7  |-  ( ( F  e. MblFn  /\  F : RR
--> RR  /\  ( y  -  z )  e.  RR )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
7360, 61, 71, 72syl3anc 1185 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
746ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G  e.  dom  S.1 )
75 i1fmbf 19569 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  G  e. MblFn )
7674, 75syl 16 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G  e. MblFn )
778ad2antrr 708 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  G : RR --> RR )
78 mbfimasn 19528 . . . . . . 7  |-  ( ( G  e. MblFn  /\  G : RR
--> RR  /\  z  e.  RR )  ->  ( `' G " { z } )  e.  dom  vol )
7976, 77, 70, 78syl3anc 1185 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' G " { z } )  e.  dom  vol )
80 inmbl 19438 . . . . . 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 644 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
8281ralrimiva 2791 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  A. z  e.  ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
83 finiunmbl 19440 . . . 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 644 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  U_ z  e. 
ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
8556, 84eqeltrd 2512 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( `' ( F  o F  +  G ) " {
y } )  e. 
dom  vol )
86 mblvol 19428 . . . 4  |-  ( ( `' ( F  o F  +  G ) " { y } )  e.  dom  vol  ->  ( vol `  ( `' ( F  o F  +  G ) " { y } ) )  =  ( vol
* `  ( `' ( F  o F  +  G ) " {
y } ) ) )
8785, 86syl 16 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  o F  +  G
) " { y } ) )  =  ( vol * `  ( `' ( F  o F  +  G ) " { y } ) ) )
88 mblss 19429 . . . . 5  |-  ( ( `' ( F  o F  +  G ) " { y } )  e.  dom  vol  ->  ( `' ( F  o F  +  G ) " { y } ) 
C_  RR )
8985, 88syl 16 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( `' ( F  o F  +  G ) " {
y } )  C_  RR )
90 inss1 3563 . . . . . . . . 9  |-  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { ( y  -  z ) } )
9190a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { ( y  -  z ) } ) )
9273adantrr 699 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  e.  dom  vol )
93 mblss 19429 . . . . . . . . 9  |-  ( ( `' F " { ( y  -  z ) } )  e.  dom  vol 
->  ( `' F " { ( y  -  z ) } ) 
C_  RR )
9492, 93syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  C_  RR )
95 mblvol 19428 . . . . . . . . . 10  |-  ( ( `' F " { ( y  -  z ) } )  e.  dom  vol 
->  ( vol `  ( `' F " { ( y  -  z ) } ) )  =  ( vol * `  ( `' F " { ( y  -  z ) } ) ) )
9692, 95syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  =  ( vol
* `  ( `' F " { ( y  -  z ) } ) ) )
97 simprr 735 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  z  =  0 )
9897oveq2d 6099 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  z )  =  ( y  - 
0 ) )
9954adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  y  e.  CC )
10099subid1d 9402 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  0 )  =  y )
10198, 100eqtrd 2470 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  (
y  -  z )  =  y )
102101sneqd 3829 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  { ( y  -  z ) }  =  { y } )
103102imaeq2d 5205 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( `' F " { ( y  -  z ) } )  =  ( `' F " { y } ) )
104103fveq2d 5734 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  =  ( vol `  ( `' F " { y } ) ) )
105 i1fima2sn 19574 . . . . . . . . . . . 12  |-  ( ( F  e.  dom  S.1  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
1063, 105sylan 459 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
107106adantr 453 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
108104, 107eqeltrd 2512 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol `  ( `' F " { ( y  -  z ) } ) )  e.  RR )
10996, 108eqeltrrd 2513 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol * `  ( `' F " { ( y  -  z ) } ) )  e.  RR )
110 ovolsscl 19384 . . . . . . . 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 1185 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =  0 ) )  ->  ( vol * `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
112111expr 600 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( z  =  0  ->  ( vol * `
 ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR ) )
113 eldifsn 3929 . . . . . . . 8  |-  ( z  e.  ( ran  G  \  { 0 } )  <-> 
( z  e.  ran  G  /\  z  =/=  0
) )
114 inss2 3564 . . . . . . . . . 10  |-  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } )
115114a1i 11 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  (
( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } ) )
116 eldifi 3471 . . . . . . . . . 10  |-  ( z  e.  ( ran  G  \  { 0 } )  ->  z  e.  ran  G )
117 mblss 19429 . . . . . . . . . . 11  |-  ( ( `' G " { z } )  e.  dom  vol 
->  ( `' G " { z } ) 
C_  RR )
11879, 117syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( `' G " { z } ) 
C_  RR )
119116, 118sylan2 462 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  C_  RR )
120 i1fima 19572 . . . . . . . . . . . . 13  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
1216, 120syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
122121ad2antrr 708 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  e.  dom  vol )
123 mblvol 19428 . . . . . . . . . . 11  |-  ( ( `' G " { z } )  e.  dom  vol 
->  ( vol `  ( `' G " { z } ) )  =  ( vol * `  ( `' G " { z } ) ) )
124122, 123syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol `  ( `' G " { z } ) )  =  ( vol
* `  ( `' G " { z } ) ) )
1256adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  G  e.  dom  S.1 )
126 i1fima2sn 19574 . . . . . . . . . . 11  |-  ( ( G  e.  dom  S.1  /\  z  e.  ( ran 
G  \  { 0 } ) )  -> 
( vol `  ( `' G " { z } ) )  e.  RR )
127125, 126sylan 459 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol `  ( `' G " { z } ) )  e.  RR )
128124, 127eqeltrrd 2513 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol * `  ( `' G " { z } ) )  e.  RR )
129 ovolsscl 19384 . . . . . . . . 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 1185 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol * `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
131113, 130sylan2br 464 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  ( z  e.  ran  G  /\  z  =/=  0
) )  ->  ( vol * `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
132131expr 600 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( z  =/=  0  ->  ( vol * `
 ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR ) )
133112, 132pm2.61dne 2683 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( vol * `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
13457, 133fsumrecl 12530 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  sum_ z  e. 
ran  G ( vol
* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
13556fveq2d 5734 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( vol * `
 ( `' ( F  o F  +  G ) " {
y } ) )  =  ( vol * `  U_ z  e.  ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
136114, 118syl5ss 3361 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  RR )
137136, 133jca 520 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  +  G
)  \  { 0 } ) )  /\  z  e.  ran  G )  ->  ( ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  RR  /\  ( vol * `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR ) )
138137ralrimiva 2791 . . . . . 6  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  A. z  e.  ran  G ( ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) )  C_  RR  /\  ( vol * `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR ) )
139 ovolfiniun 19399 . . . . . 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 644 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( vol * `
 U_ z  e.  ran  G ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  <_  sum_ z  e.  ran  G ( vol
* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
141135, 140eqbrtrd 4234 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( vol * `
 ( `' ( F  o F  +  G ) " {
y } ) )  <_  sum_ z  e.  ran  G ( vol * `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
142 ovollecl 19381 . . . 4  |-  ( ( ( `' ( F  o F  +  G
) " { y } )  C_  RR  /\ 
sum_ z  e.  ran  G ( vol * `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR  /\  ( vol * `  ( `' ( F  o F  +  G ) " { y } ) )  <_  sum_ z  e. 
ran  G ( vol
* `  ( ( `' F " { ( y  -  z ) } )  i^i  ( `' G " { z } ) ) ) )  ->  ( vol * `
 ( `' ( F  o F  +  G ) " {
y } ) )  e.  RR )
14389, 134, 141, 142syl3anc 1185 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( vol * `
 ( `' ( F  o F  +  G ) " {
y } ) )  e.  RR )
14487, 143eqeltrd 2512 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  +  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  o F  +  G
) " { y } ) )  e.  RR )
14512, 49, 85, 144i1fd 19575 1  |-  ( ph  ->  ( F  o F  +  G )  e. 
dom  S.1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424    =/= wne 2601   A.wral 2707   E.wrex 2708   _Vcvv 2958    \ cdif 3319    i^i cin 3321    C_ wss 3322   {csn 3816   U_ciun 4095   class class class wbr 4214    X. cxp 4878   `'ccnv 4879   dom cdm 4880   ran crn 4881   "cima 4883    Fn wfn 5451   -->wf 5452   -onto->wfo 5454   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085    o Fcof 6305   Fincfn 7111   CCcc 8990   RRcr 8991   0cc0 8992    + caddc 8995    <_ cle 9123    - cmin 9293   sum_csu 12481   vol
*covol 19361   volcvol 19362  MblFncmbf 19508   S.1citg1 19509
This theorem is referenced by:  itg1addlem4  19593  i1fsub  19602  itg2splitlem  19642  itg2split  19643  itg2addlem  19652  itg2addnc  26261  ftc1anclem3  26284  ftc1anclem5  26286  ftc1anclem8  26289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-xadd 10713  df-ioo 10922  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-xmet 16697  df-met 16698  df-ovol 19363  df-vol 19364  df-mbf 19514  df-itg1 19515
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