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Theorem sibfof 28155
Description: Applying function operations on simple functions results in simple functions with regard to the destination space, provided the operation fulfills a simple condition. (Contributed by Thierry Arnoux, 12-Mar-2018.)
Hypotheses
Ref Expression
sitgval.b  |-  B  =  ( Base `  W
)
sitgval.j  |-  J  =  ( TopOpen `  W )
sitgval.s  |-  S  =  (sigaGen `  J )
sitgval.0  |-  .0.  =  ( 0g `  W )
sitgval.x  |-  .x.  =  ( .s `  W )
sitgval.h  |-  H  =  (RRHom `  (Scalar `  W
) )
sitgval.1  |-  ( ph  ->  W  e.  V )
sitgval.2  |-  ( ph  ->  M  e.  U. ran measures )
sibfmbl.1  |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )
sibfof.c  |-  C  =  ( Base `  K
)
sibfof.0  |-  ( ph  ->  W  e.  TopSp )
sibfof.1  |-  ( ph  ->  .+  : ( B  X.  B ) --> C )
sibfof.2  |-  ( ph  ->  G  e.  dom  ( Wsitg M ) )
sibfof.3  |-  ( ph  ->  K  e.  TopSp )
sibfof.4  |-  ( ph  ->  J  e.  Fre )
sibfof.5  |-  ( ph  ->  (  .0.  .+  .0.  )  =  ( 0g `  K ) )
Assertion
Ref Expression
sibfof  |-  ( ph  ->  ( F  oF  .+  G )  e. 
dom  ( Ksitg M
) )

Proof of Theorem sibfof
Dummy variables  x  y  z  b  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sibfof.1 . . . . . . . 8  |-  ( ph  ->  .+  : ( B  X.  B ) --> C )
2 sibfof.0 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  TopSp )
3 sitgval.b . . . . . . . . . . . 12  |-  B  =  ( Base `  W
)
4 sitgval.j . . . . . . . . . . . 12  |-  J  =  ( TopOpen `  W )
53, 4tpsuni 19312 . . . . . . . . . . 11  |-  ( W  e.  TopSp  ->  B  =  U. J )
62, 5syl 16 . . . . . . . . . 10  |-  ( ph  ->  B  =  U. J
)
76sqxpeqd 5015 . . . . . . . . 9  |-  ( ph  ->  ( B  X.  B
)  =  ( U. J  X.  U. J ) )
87feq2d 5708 . . . . . . . 8  |-  ( ph  ->  (  .+  : ( B  X.  B ) --> C  <->  .+  : ( U. J  X.  U. J ) --> C ) )
91, 8mpbid 210 . . . . . . 7  |-  ( ph  ->  .+  : ( U. J  X.  U. J ) --> C )
109fovrnda 6431 . . . . . 6  |-  ( (
ph  /\  ( z  e.  U. J  /\  x  e.  U. J ) )  ->  ( z  .+  x )  e.  C
)
11 sitgval.s . . . . . . 7  |-  S  =  (sigaGen `  J )
12 sitgval.0 . . . . . . 7  |-  .0.  =  ( 0g `  W )
13 sitgval.x . . . . . . 7  |-  .x.  =  ( .s `  W )
14 sitgval.h . . . . . . 7  |-  H  =  (RRHom `  (Scalar `  W
) )
15 sitgval.1 . . . . . . 7  |-  ( ph  ->  W  e.  V )
16 sitgval.2 . . . . . . 7  |-  ( ph  ->  M  e.  U. ran measures )
17 sibfmbl.1 . . . . . . 7  |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )
183, 4, 11, 12, 13, 14, 15, 16, 17sibff 28151 . . . . . 6  |-  ( ph  ->  F : U. dom  M --> U. J )
19 sibfof.2 . . . . . . 7  |-  ( ph  ->  G  e.  dom  ( Wsitg M ) )
203, 4, 11, 12, 13, 14, 15, 16, 19sibff 28151 . . . . . 6  |-  ( ph  ->  G : U. dom  M --> U. J )
21 dmexg 6716 . . . . . . 7  |-  ( M  e.  U. ran measures  ->  dom  M  e.  _V )
22 uniexg 6582 . . . . . . 7  |-  ( dom 
M  e.  _V  ->  U.
dom  M  e.  _V )
2316, 21, 223syl 20 . . . . . 6  |-  ( ph  ->  U. dom  M  e. 
_V )
24 inidm 3692 . . . . . 6  |-  ( U. dom  M  i^i  U. dom  M )  =  U. dom  M
2510, 18, 20, 23, 23, 24off 6539 . . . . 5  |-  ( ph  ->  ( F  oF  .+  G ) : U. dom  M --> C )
26 sibfof.3 . . . . . . . 8  |-  ( ph  ->  K  e.  TopSp )
27 sibfof.c . . . . . . . . 9  |-  C  =  ( Base `  K
)
28 eqid 2443 . . . . . . . . 9  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
2927, 28tpsuni 19312 . . . . . . . 8  |-  ( K  e.  TopSp  ->  C  =  U. ( TopOpen `  K )
)
3026, 29syl 16 . . . . . . 7  |-  ( ph  ->  C  =  U. ( TopOpen
`  K ) )
31 fvex 5866 . . . . . . . 8  |-  ( TopOpen `  K )  e.  _V
32 unisg 28016 . . . . . . . 8  |-  ( (
TopOpen `  K )  e. 
_V  ->  U. (sigaGen `  ( TopOpen
`  K ) )  =  U. ( TopOpen `  K ) )
3331, 32ax-mp 5 . . . . . . 7  |-  U. (sigaGen `  ( TopOpen `  K )
)  =  U. ( TopOpen
`  K )
3430, 33syl6eqr 2502 . . . . . 6  |-  ( ph  ->  C  =  U. (sigaGen `  ( TopOpen `  K )
) )
3534feq3d 5709 . . . . 5  |-  ( ph  ->  ( ( F  oF  .+  G ) : U. dom  M --> C  <->  ( F  oF  .+  G ) : U. dom  M --> U. (sigaGen `  ( TopOpen `  K
) ) ) )
3625, 35mpbid 210 . . . 4  |-  ( ph  ->  ( F  oF  .+  G ) : U. dom  M --> U. (sigaGen `  ( TopOpen `  K )
) )
3731a1i 11 . . . . . . 7  |-  ( ph  ->  ( TopOpen `  K )  e.  _V )
3837sgsiga 28015 . . . . . 6  |-  ( ph  ->  (sigaGen `  ( TopOpen `  K
) )  e.  U. ran sigAlgebra )
39 uniexg 6582 . . . . . 6  |-  ( (sigaGen `  ( TopOpen `  K )
)  e.  U. ran sigAlgebra  ->  U. (sigaGen `  ( TopOpen `  K
) )  e.  _V )
4038, 39syl 16 . . . . 5  |-  ( ph  ->  U. (sigaGen `  ( TopOpen
`  K ) )  e.  _V )
41 elmapg 7435 . . . . 5  |-  ( ( U. (sigaGen `  ( TopOpen
`  K ) )  e.  _V  /\  U. dom  M  e.  _V )  ->  ( ( F  oF  .+  G )  e.  ( U. (sigaGen `  ( TopOpen
`  K ) )  ^m  U. dom  M
)  <->  ( F  oF  .+  G ) : U. dom  M --> U. (sigaGen `  ( TopOpen `  K )
) ) )
4240, 23, 41syl2anc 661 . . . 4  |-  ( ph  ->  ( ( F  oF  .+  G )  e.  ( U. (sigaGen `  ( TopOpen
`  K ) )  ^m  U. dom  M
)  <->  ( F  oF  .+  G ) : U. dom  M --> U. (sigaGen `  ( TopOpen `  K )
) ) )
4336, 42mpbird 232 . . 3  |-  ( ph  ->  ( F  oF  .+  G )  e.  ( U. (sigaGen `  ( TopOpen
`  K ) )  ^m  U. dom  M
) )
44 inundif 3892 . . . . . . 7  |-  ( ( b  i^i  ran  ( F  oF  .+  G
) )  u.  (
b  \  ran  ( F  oF  .+  G
) ) )  =  b
4544imaeq2i 5325 . . . . . 6  |-  ( `' ( F  oF  .+  G ) "
( ( b  i^i 
ran  ( F  oF  .+  G ) )  u.  ( b  \  ran  ( F  oF  .+  G ) ) ) )  =  ( `' ( F  oF  .+  G ) "
b )
46 ffun 5723 . . . . . . . 8  |-  ( ( F  oF  .+  G ) : U. dom  M --> C  ->  Fun  ( F  oF  .+  G ) )
47 unpreima 5998 . . . . . . . 8  |-  ( Fun  ( F  oF  .+  G )  -> 
( `' ( F  oF  .+  G
) " ( ( b  i^i  ran  ( F  oF  .+  G
) )  u.  (
b  \  ran  ( F  oF  .+  G
) ) ) )  =  ( ( `' ( F  oF  .+  G ) "
( b  i^i  ran  ( F  oF  .+  G ) ) )  u.  ( `' ( F  oF  .+  G ) " (
b  \  ran  ( F  oF  .+  G
) ) ) ) )
4825, 46, 473syl 20 . . . . . . 7  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " ( ( b  i^i  ran  ( F  oF  .+  G
) )  u.  (
b  \  ran  ( F  oF  .+  G
) ) ) )  =  ( ( `' ( F  oF  .+  G ) "
( b  i^i  ran  ( F  oF  .+  G ) ) )  u.  ( `' ( F  oF  .+  G ) " (
b  \  ran  ( F  oF  .+  G
) ) ) ) )
4948adantr 465 . . . . . 6  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " ( ( b  i^i  ran  ( F  oF  .+  G
) )  u.  (
b  \  ran  ( F  oF  .+  G
) ) ) )  =  ( ( `' ( F  oF  .+  G ) "
( b  i^i  ran  ( F  oF  .+  G ) ) )  u.  ( `' ( F  oF  .+  G ) " (
b  \  ran  ( F  oF  .+  G
) ) ) ) )
5045, 49syl5eqr 2498 . . . . 5  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " b )  =  ( ( `' ( F  oF  .+  G ) "
( b  i^i  ran  ( F  oF  .+  G ) ) )  u.  ( `' ( F  oF  .+  G ) " (
b  \  ran  ( F  oF  .+  G
) ) ) ) )
51 dmmeas 28045 . . . . . . . 8  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
5216, 51syl 16 . . . . . . 7  |-  ( ph  ->  dom  M  e.  U. ran sigAlgebra )
5352adantr 465 . . . . . 6  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  ->  dom  M  e.  U. ran sigAlgebra )
54 imaiun 6142 . . . . . . . 8  |-  ( `' ( F  oF  .+  G ) " U_ z  e.  (
b  i^i  ran  ( F  oF  .+  G
) ) { z } )  =  U_ z  e.  ( b  i^i  ran  ( F  oF  .+  G ) ) ( `' ( F  oF  .+  G
) " { z } )
55 iunid 4370 . . . . . . . . 9  |-  U_ z  e.  ( b  i^i  ran  ( F  oF  .+  G ) ) { z }  =  ( b  i^i  ran  ( F  oF  .+  G
) )
5655imaeq2i 5325 . . . . . . . 8  |-  ( `' ( F  oF  .+  G ) " U_ z  e.  (
b  i^i  ran  ( F  oF  .+  G
) ) { z } )  =  ( `' ( F  oF  .+  G ) "
( b  i^i  ran  ( F  oF  .+  G ) ) )
5754, 56eqtr3i 2474 . . . . . . 7  |-  U_ z  e.  ( b  i^i  ran  ( F  oF  .+  G ) ) ( `' ( F  oF  .+  G ) " { z } )  =  ( `' ( F  oF  .+  G ) " (
b  i^i  ran  ( F  oF  .+  G
) ) )
58 inss2 3704 . . . . . . . . . 10  |-  ( b  i^i  ran  ( F  oF  .+  G ) )  C_  ran  ( F  oF  .+  G
)
596feq3d 5709 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( F : U. dom  M --> B  <->  F : U. dom  M --> U. J
) )
6018, 59mpbird 232 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : U. dom  M --> B )
616feq3d 5709 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G : U. dom  M --> B  <->  G : U. dom  M --> U. J
) )
6220, 61mpbird 232 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : U. dom  M --> B )
63 ffn 5721 . . . . . . . . . . . . . . 15  |-  (  .+  : ( B  X.  B ) --> C  ->  .+  Fn  ( B  X.  B ) )
641, 63syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  .+  Fn  ( B  X.  B ) )
6560, 62, 23, 64ofpreima2 27380 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " { z } )  =  U_ p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
6665adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( `' ( F  oF  .+  G ) " {
z } )  = 
U_ p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )
6752adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  dom  M  e.  U.
ran sigAlgebra )
6852ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  dom  M  e.  U.
ran sigAlgebra )
69 simpll 753 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ph )
70 inss1 3703 . . . . . . . . . . . . . . . . . 18  |-  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  C_  ( `'  .+  " { z } )
71 cnvimass 5347 . . . . . . . . . . . . . . . . . . . 20  |-  ( `' 
.+  " { z } )  C_  dom  .+
72 fdm 5725 . . . . . . . . . . . . . . . . . . . . 21  |-  (  .+  : ( B  X.  B ) --> C  ->  dom  .+  =  ( B  X.  B ) )
731, 72syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  dom  .+  =  ( B  X.  B ) )
7471, 73syl5sseq 3537 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( `'  .+  " {
z } )  C_  ( B  X.  B
) )
7574adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( `'  .+  " { z } ) 
C_  ( B  X.  B ) )
7670, 75syl5ss 3500 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( ( `' 
.+  " { z } )  i^i  ( ran 
F  X.  ran  G
) )  C_  ( B  X.  B ) )
7776sselda 3489 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  p  e.  ( B  X.  B ) )
7852adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  dom  M  e.  U. ran sigAlgebra )
79 sibfof.4 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  J  e.  Fre )
8079sgsiga 28015 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
8111, 80syl5eqel 2535 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
8281adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  S  e.  U. ran sigAlgebra )
833, 4, 11, 12, 13, 14, 15, 16, 17sibfmbl 28150 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
8483adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  F  e.  ( dom  MMblFnM S
) )
854tpstop 19313 . . . . . . . . . . . . . . . . . . . . 21  |-  ( W  e.  TopSp  ->  J  e.  Top )
86 cldssbrsiga 28031 . . . . . . . . . . . . . . . . . . . . 21  |-  ( J  e.  Top  ->  ( Clsd `  J )  C_  (sigaGen `  J ) )
872, 85, 863syl 20 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( Clsd `  J
)  C_  (sigaGen `  J
) )
8887adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( Clsd `  J )  C_  (sigaGen `  J ) )
8979adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  J  e.  Fre )
90 xp1st 6815 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p  e.  ( B  X.  B )  ->  ( 1st `  p )  e.  B )
9190adantl 466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 1st `  p )  e.  B )
926adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  B  =  U. J )
9391, 92eleqtrd 2533 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 1st `  p )  e. 
U. J )
94 eqid 2443 . . . . . . . . . . . . . . . . . . . . 21  |-  U. J  =  U. J
9594t1sncld 19700 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( J  e.  Fre  /\  ( 1st `  p )  e.  U. J )  ->  { ( 1st `  p ) }  e.  ( Clsd `  J )
)
9689, 93, 95syl2anc 661 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 1st `  p ) }  e.  ( Clsd `  J ) )
9788, 96sseldd 3490 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 1st `  p ) }  e.  (sigaGen `  J
) )
9897, 11syl6eleqr 2542 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 1st `  p ) }  e.  S )
9978, 82, 84, 98mbfmcnvima 28101 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( `' F " { ( 1st `  p ) } )  e.  dom  M )
10069, 77, 99syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( `' F " { ( 1st `  p
) } )  e. 
dom  M )
1013, 4, 11, 12, 13, 14, 15, 16, 19sibfmbl 28150 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  G  e.  ( dom 
MMblFnM S ) )
102101adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  G  e.  ( dom  MMblFnM S
) )
103 xp2nd 6816 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p  e.  ( B  X.  B )  ->  ( 2nd `  p )  e.  B )
104103adantl 466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 2nd `  p )  e.  B )
105104, 92eleqtrd 2533 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 2nd `  p )  e. 
U. J )
10694t1sncld 19700 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( J  e.  Fre  /\  ( 2nd `  p )  e.  U. J )  ->  { ( 2nd `  p ) }  e.  ( Clsd `  J )
)
10789, 105, 106syl2anc 661 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 2nd `  p ) }  e.  ( Clsd `  J ) )
10888, 107sseldd 3490 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 2nd `  p ) }  e.  (sigaGen `  J
) )
109108, 11syl6eleqr 2542 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 2nd `  p ) }  e.  S )
11078, 82, 102, 109mbfmcnvima 28101 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( `' G " { ( 2nd `  p ) } )  e.  dom  M )
11169, 77, 110syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( `' G " { ( 2nd `  p
) } )  e. 
dom  M )
112 inelsiga 28008 . . . . . . . . . . . . . . 15  |-  ( ( dom  M  e.  U. ran sigAlgebra  /\  ( `' F " { ( 1st `  p
) } )  e. 
dom  M  /\  ( `' G " { ( 2nd `  p ) } )  e.  dom  M )  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  e. 
dom  M )
11368, 100, 111, 112syl3anc 1229 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  e. 
dom  M )
114113ralrimiva 2857 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  A. p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  e.  dom  M )
1153, 4, 11, 12, 13, 14, 15, 16, 17sibfrn 28152 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ran  F  e.  Fin )
1163, 4, 11, 12, 13, 14, 15, 16, 19sibfrn 28152 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ran  G  e.  Fin )
117 xpfi 7793 . . . . . . . . . . . . . . . . 17  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
118115, 116, 117syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
119 inss2 3704 . . . . . . . . . . . . . . . 16  |-  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  C_  ( ran  F  X.  ran  G
)
120 ssdomg 7563 . . . . . . . . . . . . . . . 16  |-  ( ( ran  F  X.  ran  G )  e.  Fin  ->  ( ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) 
C_  ( ran  F  X.  ran  G )  -> 
( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  ~<_  ( ran  F  X.  ran  G ) ) )
121118, 119, 120mpisyl 18 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  ~<_  ( ran  F  X.  ran  G ) )
122 isfinite 8072 . . . . . . . . . . . . . . . . 17  |-  ( ( ran  F  X.  ran  G )  e.  Fin  <->  ( ran  F  X.  ran  G ) 
~<  om )
123122biimpi 194 . . . . . . . . . . . . . . . 16  |-  ( ( ran  F  X.  ran  G )  e.  Fin  ->  ( ran  F  X.  ran  G )  ~<  om )
124 sdomdom 7545 . . . . . . . . . . . . . . . 16  |-  ( ( ran  F  X.  ran  G )  ~<  om  ->  ( ran  F  X.  ran  G )  ~<_  om )
125118, 123, 1243syl 20 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ran  F  X.  ran  G )  ~<_  om )
126 domtr 7570 . . . . . . . . . . . . . . 15  |-  ( ( ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  ~<_  ( ran  F  X.  ran  G )  /\  ( ran  F  X.  ran  G
)  ~<_  om )  ->  (
( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) )  ~<_  om )
127121, 125, 126syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  ~<_  om )
128127adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( ( `' 
.+  " { z } )  i^i  ( ran 
F  X.  ran  G
) )  ~<_  om )
129 nfcv 2605 . . . . . . . . . . . . . 14  |-  F/_ p
( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )
130129sigaclcuni 27991 . . . . . . . . . . . . 13  |-  ( ( dom  M  e.  U. ran sigAlgebra  /\  A. p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  e.  dom  M  /\  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  ~<_  om )  ->  U_ p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  e.  dom  M )
13167, 114, 128, 130syl3anc 1229 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  U_ p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  e.  dom  M )
13266, 131eqeltrd 2531 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( `' ( F  oF  .+  G ) " {
z } )  e. 
dom  M )
133132ralrimiva 2857 . . . . . . . . . 10  |-  ( ph  ->  A. z  e.  ran  ( F  oF  .+  G ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
134 ssralv 3549 . . . . . . . . . 10  |-  ( ( b  i^i  ran  ( F  oF  .+  G
) )  C_  ran  ( F  oF  .+  G )  ->  ( A. z  e.  ran  ( F  oF  .+  G ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M  ->  A. z  e.  (
b  i^i  ran  ( F  oF  .+  G
) ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M ) )
13558, 133, 134mpsyl 63 . . . . . . . . 9  |-  ( ph  ->  A. z  e.  ( b  i^i  ran  ( F  oF  .+  G
) ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
136135adantr 465 . . . . . . . 8  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  ->  A. z  e.  (
b  i^i  ran  ( F  oF  .+  G
) ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
137 ffun 5723 . . . . . . . . . . . . . 14  |-  (  .+  : ( B  X.  B ) --> C  ->  Fun  .+  )
1381, 137syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  Fun  .+  )
139 imafi 7815 . . . . . . . . . . . . 13  |-  ( ( Fun  .+  /\  ( ran  F  X.  ran  G
)  e.  Fin )  ->  (  .+  " ( ran  F  X.  ran  G
) )  e.  Fin )
140138, 118, 139syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  (  .+  " ( ran  F  X.  ran  G
) )  e.  Fin )
14118, 20, 9, 23ofrn2 27352 . . . . . . . . . . . 12  |-  ( ph  ->  ran  ( F  oF  .+  G )  C_  (  .+  " ( ran 
F  X.  ran  G
) ) )
142 ssfi 7742 . . . . . . . . . . . 12  |-  ( ( (  .+  " ( ran  F  X.  ran  G
) )  e.  Fin  /\ 
ran  ( F  oF  .+  G )  C_  (  .+  " ( ran 
F  X.  ran  G
) ) )  ->  ran  ( F  oF  .+  G )  e. 
Fin )
143140, 141, 142syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( F  oF  .+  G )  e. 
Fin )
144 ssdomg 7563 . . . . . . . . . . 11  |-  ( ran  ( F  oF  .+  G )  e. 
Fin  ->  ( ( b  i^i  ran  ( F  oF  .+  G ) )  C_  ran  ( F  oF  .+  G
)  ->  ( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  ran  ( F  oF  .+  G ) ) )
145143, 58, 144mpisyl 18 . . . . . . . . . 10  |-  ( ph  ->  ( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  ran  ( F  oF  .+  G ) )
146 isfinite 8072 . . . . . . . . . . . 12  |-  ( ran  ( F  oF  .+  G )  e. 
Fin 
<->  ran  ( F  oF  .+  G )  ~<  om )
147143, 146sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( F  oF  .+  G )  ~<  om )
148 sdomdom 7545 . . . . . . . . . . 11  |-  ( ran  ( F  oF  .+  G )  ~<  om  ->  ran  ( F  oF  .+  G )  ~<_  om )
149147, 148syl 16 . . . . . . . . . 10  |-  ( ph  ->  ran  ( F  oF  .+  G )  ~<_  om )
150 domtr 7570 . . . . . . . . . 10  |-  ( ( ( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  ran  ( F  oF  .+  G )  /\  ran  ( F  oF  .+  G )  ~<_  om )  ->  ( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  om )
151145, 149, 150syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  om )
152151adantr 465 . . . . . . . 8  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  om )
153 nfcv 2605 . . . . . . . . 9  |-  F/_ z
( b  i^i  ran  ( F  oF  .+  G ) )
154153sigaclcuni 27991 . . . . . . . 8  |-  ( ( dom  M  e.  U. ran sigAlgebra  /\  A. z  e.  ( b  i^i  ran  ( F  oF  .+  G
) ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M  /\  ( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  om )  ->  U_ z  e.  ( b  i^i  ran  ( F  oF  .+  G ) ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
15553, 136, 152, 154syl3anc 1229 . . . . . . 7  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  ->  U_ z  e.  (
b  i^i  ran  ( F  oF  .+  G
) ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
15657, 155syl5eqelr 2536 . . . . . 6  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " ( b  i^i  ran  ( F  oF  .+  G ) ) )  e.  dom  M )
157 difpreima 6000 . . . . . . . . . 10  |-  ( Fun  ( F  oF  .+  G )  -> 
( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  =  ( ( `' ( F  oF  .+  G
) " b ) 
\  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) ) )
15825, 46, 1573syl 20 . . . . . . . . 9  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  =  ( ( `' ( F  oF  .+  G
) " b ) 
\  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) ) )
159 cnvimarndm 5348 . . . . . . . . . . 11  |-  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) )  =  dom  ( F  oF  .+  G
)
160159difeq2i 3604 . . . . . . . . . 10  |-  ( ( `' ( F  oF  .+  G ) "
b )  \  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) )  =  ( ( `' ( F  oF  .+  G ) "
b )  \  dom  ( F  oF  .+  G ) )
161 cnvimass 5347 . . . . . . . . . . 11  |-  ( `' ( F  oF  .+  G ) "
b )  C_  dom  ( F  oF  .+  G )
162 ssdif0 3871 . . . . . . . . . . 11  |-  ( ( `' ( F  oF  .+  G ) "
b )  C_  dom  ( F  oF  .+  G )  <->  ( ( `' ( F  oF  .+  G ) "
b )  \  dom  ( F  oF  .+  G ) )  =  (/) )
163161, 162mpbi 208 . . . . . . . . . 10  |-  ( ( `' ( F  oF  .+  G ) "
b )  \  dom  ( F  oF  .+  G ) )  =  (/)
164160, 163eqtri 2472 . . . . . . . . 9  |-  ( ( `' ( F  oF  .+  G ) "
b )  \  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) )  =  (/)
165158, 164syl6eq 2500 . . . . . . . 8  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  =  (/) )
166 0elsiga 27987 . . . . . . . . 9  |-  ( dom 
M  e.  U. ran sigAlgebra  ->  (/)  e.  dom  M )
16716, 51, 1663syl 20 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  dom  M )
168165, 167eqeltrd 2531 . . . . . . 7  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  e.  dom  M )
169168adantr 465 . . . . . 6  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  e.  dom  M )
170 unelsiga 28007 . . . . . 6  |-  ( ( dom  M  e.  U. ran sigAlgebra  /\  ( `' ( F  oF  .+  G
) " ( b  i^i  ran  ( F  oF  .+  G ) ) )  e.  dom  M  /\  ( `' ( F  oF  .+  G ) " (
b  \  ran  ( F  oF  .+  G
) ) )  e. 
dom  M )  -> 
( ( `' ( F  oF  .+  G ) " (
b  i^i  ran  ( F  oF  .+  G
) ) )  u.  ( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) ) )  e. 
dom  M )
17153, 156, 169, 170syl3anc 1229 . . . . 5  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( ( `' ( F  oF  .+  G ) " (
b  i^i  ran  ( F  oF  .+  G
) ) )  u.  ( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) ) )  e. 
dom  M )
17250, 171eqeltrd 2531 . . . 4  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " b )  e.  dom  M )
173172ralrimiva 2857 . . 3  |-  ( ph  ->  A. b  e.  (sigaGen `  ( TopOpen `  K )
) ( `' ( F  oF  .+  G ) " b
)  e.  dom  M
)
17452, 38ismbfm 28096 . . 3  |-  ( ph  ->  ( ( F  oF  .+  G )  e.  ( dom  MMblFnM (sigaGen `  ( TopOpen `  K )
) )  <->  ( ( F  oF  .+  G
)  e.  ( U. (sigaGen `  ( TopOpen `  K
) )  ^m  U. dom  M )  /\  A. b  e.  (sigaGen `  ( TopOpen
`  K ) ) ( `' ( F  oF  .+  G
) " b )  e.  dom  M ) ) )
17543, 173, 174mpbir2and 922 . 2  |-  ( ph  ->  ( F  oF  .+  G )  e.  ( dom  MMblFnM (sigaGen `  ( TopOpen `  K )
) ) )
17665adantr 465 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( `' ( F  oF  .+  G ) " {
z } )  = 
U_ p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )
177176fveq2d 5860 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( M `  ( `' ( F  oF  .+  G
) " { z } ) )  =  ( M `  U_ p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) ) )
178 measbasedom 28046 . . . . . . . . 9  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )
17916, 178sylib 196 . . . . . . . 8  |-  ( ph  ->  M  e.  (measures `  dom  M ) )
180179adantr 465 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  M  e.  (measures `  dom  M ) )
181 eldifi 3611 . . . . . . . 8  |-  ( z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } )  ->  z  e.  ran  ( F  oF  .+  G ) )
182181, 114sylan2 474 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  A. p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  e.  dom  M )
183127adantr 465 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  ~<_  om )
184 sneq 4024 . . . . . . . . . . 11  |-  ( x  =  ( 1st `  p
)  ->  { x }  =  { ( 1st `  p ) } )
185184imaeq2d 5327 . . . . . . . . . 10  |-  ( x  =  ( 1st `  p
)  ->  ( `' F " { x }
)  =  ( `' F " { ( 1st `  p ) } ) )
186 sneq 4024 . . . . . . . . . . 11  |-  ( y  =  ( 2nd `  p
)  ->  { y }  =  { ( 2nd `  p ) } )
187186imaeq2d 5327 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  p
)  ->  ( `' G " { y } )  =  ( `' G " { ( 2nd `  p ) } ) )
188 ffun 5723 . . . . . . . . . . . 12  |-  ( F : U. dom  M --> U. J  ->  Fun  F
)
18918, 188syl 16 . . . . . . . . . . 11  |-  ( ph  ->  Fun  F )
190 sndisj 4429 . . . . . . . . . . 11  |- Disj  x  e. 
ran  F { x }
191 disjpreima 27317 . . . . . . . . . . 11  |-  ( ( Fun  F  /\ Disj  x  e. 
ran  F { x } )  -> Disj  x  e. 
ran  F ( `' F " { x } ) )
192189, 190, 191sylancl 662 . . . . . . . . . 10  |-  ( ph  -> Disj  x  e.  ran  F ( `' F " { x } ) )
193 ffun 5723 . . . . . . . . . . . 12  |-  ( G : U. dom  M --> U. J  ->  Fun  G
)
19420, 193syl 16 . . . . . . . . . . 11  |-  ( ph  ->  Fun  G )
195 sndisj 4429 . . . . . . . . . . 11  |- Disj  y  e. 
ran  G { y }
196 disjpreima 27317 . . . . . . . . . . 11  |-  ( ( Fun  G  /\ Disj  y  e. 
ran  G { y } )  -> Disj  y  e. 
ran  G ( `' G " { y } ) )
197194, 195, 196sylancl 662 . . . . . . . . . 10  |-  ( ph  -> Disj  y  e.  ran  G ( `' G " { y } ) )
198185, 187, 192, 197disjxpin 27319 . . . . . . . . 9  |-  ( ph  -> Disj  p  e.  ( ran  F  X.  ran  G ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )
199 disjss1 4413 . . . . . . . . 9  |-  ( ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) )  C_  ( ran  F  X.  ran  G )  ->  (Disj  p  e.  ( ran  F  X.  ran  G ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  -> Disj  p  e.  ( ( `' 
.+  " { z } )  i^i  ( ran 
F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) ) )
200119, 198, 199mpsyl 63 . . . . . . . 8  |-  ( ph  -> Disj  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
201200adantr 465 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  -> Disj  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )
202 measvuni 28058 . . . . . . 7  |-  ( ( M  e.  (measures `  dom  M )  /\  A. p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  e.  dom  M  /\  ( ( ( `' 
.+  " { z } )  i^i  ( ran 
F  X.  ran  G
) )  ~<_  om  /\ Disj  p  e.  ( ( `' 
.+  " { z } )  i^i  ( ran 
F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) ) )  ->  ( M `  U_ p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )  = Σ* p  e.  (
( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) ( M `  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) ) )
203180, 182, 183, 201, 202syl112anc 1233 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( M `  U_ p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )  = Σ* p  e.  (
( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) ( M `  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) ) )
204 ssfi 7742 . . . . . . . . 9  |-  ( ( ( ran  F  X.  ran  G )  e.  Fin  /\  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) 
C_  ( ran  F  X.  ran  G ) )  ->  ( ( `' 
.+  " { z } )  i^i  ( ran 
F  X.  ran  G
) )  e.  Fin )
205118, 119, 204sylancl 662 . . . . . . . 8  |-  ( ph  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  e.  Fin )
206205adantr 465 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  e.  Fin )
207 simpll 753 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ph )
208 simpr 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )
209119, 208sseldi 3487 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  p  e.  ( ran  F  X.  ran  G ) )
210 xp1st 6815 . . . . . . . . 9  |-  ( p  e.  ( ran  F  X.  ran  G )  -> 
( 1st `  p
)  e.  ran  F
)
211209, 210syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( 1st `  p )  e.  ran  F )
212 xp2nd 6816 . . . . . . . . 9  |-  ( p  e.  ( ran  F  X.  ran  G )  -> 
( 2nd `  p
)  e.  ran  G
)
213209, 212syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( 2nd `  p )  e.  ran  G )
214 oveq12 6290 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  .0.  /\  y  =  .0.  )  ->  ( x  .+  y
)  =  (  .0.  .+  .0.  ) )
215 sibfof.5 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  (  .0.  .+  .0.  )  =  ( 0g `  K ) )
216214, 215sylan9eqr 2506 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  =  .0.  /\  y  =  .0.  ) )  -> 
( x  .+  y
)  =  ( 0g
`  K ) )
217216ex 434 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( x  =  .0.  /\  y  =  .0.  )  ->  (
x  .+  y )  =  ( 0g `  K ) ) )
218217necon3ad 2653 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( x  .+  y )  =/=  ( 0g `  K )  ->  -.  ( x  =  .0. 
/\  y  =  .0.  ) ) )
219 neorian 2770 . . . . . . . . . . . . 13  |-  ( ( x  =/=  .0.  \/  y  =/=  .0.  )  <->  -.  (
x  =  .0.  /\  y  =  .0.  )
)
220218, 219syl6ibr 227 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( x  .+  y )  =/=  ( 0g `  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) ) )
221220adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( x  .+  y )  =/=  ( 0g `  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) ) )
222221ralrimivva 2864 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  B  A. y  e.  B  ( ( x  .+  y )  =/=  ( 0g `  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) ) )
223207, 222syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  A. x  e.  B  A. y  e.  B  ( (
x  .+  y )  =/=  ( 0g `  K
)  ->  ( x  =/=  .0.  \/  y  =/= 
.0.  ) ) )
22470a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  C_  ( `'  .+  " { z } ) )
225224sselda 3489 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  p  e.  ( `'  .+  " {
z } ) )
226 fniniseg 5993 . . . . . . . . . . . . 13  |-  (  .+  Fn  ( B  X.  B
)  ->  ( p  e.  ( `'  .+  " {
z } )  <->  ( p  e.  ( B  X.  B
)  /\  (  .+  `  p )  =  z ) ) )
227207, 64, 2263syl 20 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( p  e.  ( `'  .+  " {
z } )  <->  ( p  e.  ( B  X.  B
)  /\  (  .+  `  p )  =  z ) ) )
228225, 227mpbid 210 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( p  e.  ( B  X.  B
)  /\  (  .+  `  p )  =  z ) )
229 simpr 461 . . . . . . . . . . . 12  |-  ( ( p  e.  ( B  X.  B )  /\  (  .+  `  p )  =  z )  -> 
(  .+  `  p )  =  z )
230 1st2nd2 6822 . . . . . . . . . . . . . . 15  |-  ( p  e.  ( B  X.  B )  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
231230fveq2d 5860 . . . . . . . . . . . . . 14  |-  ( p  e.  ( B  X.  B )  ->  (  .+  `  p )  =  (  .+  `  <. ( 1st `  p ) ,  ( 2nd `  p
) >. ) )
232 df-ov 6284 . . . . . . . . . . . . . 14  |-  ( ( 1st `  p ) 
.+  ( 2nd `  p
) )  =  ( 
.+  `  <. ( 1st `  p ) ,  ( 2nd `  p )
>. )
233231, 232syl6eqr 2502 . . . . . . . . . . . . 13  |-  ( p  e.  ( B  X.  B )  ->  (  .+  `  p )  =  ( ( 1st `  p
)  .+  ( 2nd `  p ) ) )
234233adantr 465 . . . . . . . . . . . 12  |-  ( ( p  e.  ( B  X.  B )  /\  (  .+  `  p )  =  z )  -> 
(  .+  `  p )  =  ( ( 1st `  p )  .+  ( 2nd `  p ) ) )
235229, 234eqtr3d 2486 . . . . . . . . . . 11  |-  ( ( p  e.  ( B  X.  B )  /\  (  .+  `  p )  =  z )  -> 
z  =  ( ( 1st `  p ) 
.+  ( 2nd `  p
) ) )
236228, 235syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  z  =  ( ( 1st `  p
)  .+  ( 2nd `  p ) ) )
237 simplr 755 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )
238237eldifbd 3474 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  -.  z  e.  { ( 0g `  K ) } )
239 elsn 4028 . . . . . . . . . . . 12  |-  ( z  e.  { ( 0g
`  K ) }  <-> 
z  =  ( 0g
`  K ) )
240239necon3bbii 2704 . . . . . . . . . . 11  |-  ( -.  z  e.  { ( 0g `  K ) }  <->  z  =/=  ( 0g `  K ) )
241238, 240sylib 196 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  z  =/=  ( 0g `  K ) )
242236, 241eqnetrrd 2737 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( ( 1st `  p )  .+  ( 2nd `  p ) )  =/=  ( 0g
`  K ) )
243181, 77sylanl2 651 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  p  e.  ( B  X.  B
) )
244243, 90syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( 1st `  p )  e.  B
)
245243, 103syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( 2nd `  p )  e.  B
)
246 oveq1 6288 . . . . . . . . . . . . 13  |-  ( x  =  ( 1st `  p
)  ->  ( x  .+  y )  =  ( ( 1st `  p
)  .+  y )
)
247246neeq1d 2720 . . . . . . . . . . . 12  |-  ( x  =  ( 1st `  p
)  ->  ( (
x  .+  y )  =/=  ( 0g `  K
)  <->  ( ( 1st `  p )  .+  y
)  =/=  ( 0g
`  K ) ) )
248 neeq1 2724 . . . . . . . . . . . . 13  |-  ( x  =  ( 1st `  p
)  ->  ( x  =/=  .0.  <->  ( 1st `  p
)  =/=  .0.  )
)
249248orbi1d 702 . . . . . . . . . . . 12  |-  ( x  =  ( 1st `  p
)  ->  ( (
x  =/=  .0.  \/  y  =/=  .0.  )  <->  ( ( 1st `  p )  =/= 
.0.  \/  y  =/=  .0.  ) ) )
250247, 249imbi12d 320 . . . . . . . . . . 11  |-  ( x  =  ( 1st `  p
)  ->  ( (
( x  .+  y
)  =/=  ( 0g
`  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) )  <->  ( (
( 1st `  p
)  .+  y )  =/=  ( 0g `  K
)  ->  ( ( 1st `  p )  =/= 
.0.  \/  y  =/=  .0.  ) ) ) )
251 oveq2 6289 . . . . . . . . . . . . 13  |-  ( y  =  ( 2nd `  p
)  ->  ( ( 1st `  p )  .+  y )  =  ( ( 1st `  p
)  .+  ( 2nd `  p ) ) )
252251neeq1d 2720 . . . . . . . . . . . 12  |-  ( y  =  ( 2nd `  p
)  ->  ( (
( 1st `  p
)  .+  y )  =/=  ( 0g `  K
)  <->  ( ( 1st `  p )  .+  ( 2nd `  p ) )  =/=  ( 0g `  K ) ) )
253 neeq1 2724 . . . . . . . . . . . . 13  |-  ( y  =  ( 2nd `  p
)  ->  ( y  =/=  .0.  <->  ( 2nd `  p
)  =/=  .0.  )
)
254253orbi2d 701 . . . . . . . . . . . 12  |-  ( y  =  ( 2nd `  p
)  ->  ( (
( 1st `  p
)  =/=  .0.  \/  y  =/=  .0.  )  <->  ( ( 1st `  p )  =/= 
.0.  \/  ( 2nd `  p )  =/=  .0.  ) ) )
255252, 254imbi12d 320 . . . . . . . . . . 11  |-  ( y  =  ( 2nd `  p
)  ->  ( (
( ( 1st `  p
)  .+  y )  =/=  ( 0g `  K
)  ->  ( ( 1st `  p )  =/= 
.0.  \/  y  =/=  .0.  ) )  <->  ( (
( 1st `  p
)  .+  ( 2nd `  p ) )  =/=  ( 0g `  K
)  ->  ( ( 1st `  p )  =/= 
.0.  \/  ( 2nd `  p )  =/=  .0.  ) ) ) )
256250, 255rspc2v 3205 . . . . . . . . . 10  |-  ( ( ( 1st `  p
)  e.  B  /\  ( 2nd `  p )  e.  B )  -> 
( A. x  e.  B  A. y  e.  B  ( ( x 
.+  y )  =/=  ( 0g `  K
)  ->  ( x  =/=  .0.  \/  y  =/= 
.0.  ) )  -> 
( ( ( 1st `  p )  .+  ( 2nd `  p ) )  =/=  ( 0g `  K )  ->  (
( 1st `  p
)  =/=  .0.  \/  ( 2nd `  p )  =/=  .0.  ) ) ) )
257244, 245, 256syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( A. x  e.  B  A. y  e.  B  (
( x  .+  y
)  =/=  ( 0g
`  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) )  ->  (
( ( 1st `  p
)  .+  ( 2nd `  p ) )  =/=  ( 0g `  K
)  ->  ( ( 1st `  p )  =/= 
.0.  \/  ( 2nd `  p )  =/=  .0.  ) ) ) )
258223, 242, 257mp2d 45 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( ( 1st `  p )  =/= 
.0.  \/  ( 2nd `  p )  =/=  .0.  ) )
2593, 4, 11, 12, 13, 14, 15, 16, 17, 19, 2, 79sibfinima 28154 . . . . . . . 8  |-  ( ( ( ph  /\  ( 1st `  p )  e. 
ran  F  /\  ( 2nd `  p )  e. 
ran  G )  /\  ( ( 1st `  p
)  =/=  .0.  \/  ( 2nd `  p )  =/=  .0.  ) )  ->  ( M `  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )  e.  ( 0 [,) +oo ) )
260207, 211, 213, 258, 259syl31anc 1232 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( M `  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )  e.  ( 0 [,) +oo ) )
261206, 260esumpfinval 27954 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  -> Σ* p  e.  (
( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) ( M `  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )  =  sum_ p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) ( M `  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) ) )
262177, 203, 2613eqtrd 2488 . . . . 5  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( M `  ( `' ( F  oF  .+  G
) " { z } ) )  = 
sum_ p  e.  (
( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) ( M `  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) ) )
263 rge0ssre 11637 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR
264263, 260sseldi 3487 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( M `  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )  e.  RR )
265206, 264fsumrecl 13535 . . . . 5  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  sum_ p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) ( M `  (
( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )  e.  RR )
266262, 265eqeltrd 2531 . . . 4  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( M `  ( `' ( F  oF  .+  G
) " { z } ) )  e.  RR )
267180adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  M  e.  (measures `  dom  M ) )
268181, 113sylanl2 651 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  e. 
dom  M )
269 measge0 28051 . . . . . . 7  |-  ( ( M  e.  (measures `  dom  M )  /\  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  e. 
dom  M )  -> 
0  <_  ( M `  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) ) )
270267, 268, 269syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  0  <_  ( M `  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) ) )
271206, 264, 270fsumge0 13588 . . . . 5  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  0  <_  sum_
p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) ) ( M `
 ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) ) )
272271, 262breqtrrd 4463 . . . 4  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  0  <_  ( M `  ( `' ( F  oF  .+  G ) " { z } ) ) )
273 elrege0 11636 . . . 4  |-  ( ( M `  ( `' ( F  oF  .+  G ) " { z } ) )  e.  ( 0 [,) +oo )  <->  ( ( M `  ( `' ( F  oF  .+  G ) " {
z } ) )  e.  RR  /\  0  <_  ( M `  ( `' ( F  oF  .+  G ) " { z } ) ) ) )
274266, 272, 273sylanbrc 664 . . 3  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( M `  ( `' ( F  oF  .+  G
) " { z } ) )  e.  ( 0 [,) +oo ) )
275274ralrimiva 2857 . 2  |-  ( ph  ->  A. z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) ( M `  ( `' ( F  oF  .+  G ) " { z } ) )  e.  ( 0 [,) +oo ) )
276 eqid 2443 . . 3  |-  (sigaGen `  ( TopOpen
`  K ) )  =  (sigaGen `  ( TopOpen
`  K ) )
277 eqid 2443 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
278 eqid 2443 . . 3  |-  ( .s
`  K )  =  ( .s `  K
)
279 eqid 2443 . . 3  |-  (RRHom `  (Scalar `  K ) )  =  (RRHom `  (Scalar `  K ) )
28027, 28, 276, 277, 278, 279, 26, 16issibf 28148 . 2  |-  ( ph  ->  ( ( F  oF  .+  G )  e. 
dom  ( Ksitg M
)  <->  ( ( F  oF  .+  G
)  e.  ( dom 
MMblFnM (sigaGen `  ( TopOpen `  K
) ) )  /\  ran  ( F  oF  .+  G )  e. 
Fin  /\  A. z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) ( M `  ( `' ( F  oF  .+  G ) " { z } ) )  e.  ( 0 [,) +oo ) ) ) )
281175, 143, 275, 280mpbir3and 1180 1  |-  ( ph  ->  ( F  oF  .+  G )  e. 
dom  ( Ksitg M
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   _Vcvv 3095    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3770   {csn 4014   <.cop 4020   U.cuni 4234   U_ciun 4315  Disj wdisj 4407   class class class wbr 4437    X. cxp 4987   `'ccnv 4988   dom cdm 4989   ran crn 4990   "cima 4992   Fun wfun 5572    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281    oFcof 6523   omcom 6685   1stc1st 6783   2ndc2nd 6784    ^m cmap 7422    ~<_ cdom 7516    ~< csdm 7517   Fincfn 7518   RRcr 9494   0cc0 9495   +oocpnf 9628    <_ cle 9632   [,)cico 11540   sum_csu 13487   Basecbs 14509  Scalarcsca 14577   .scvsca 14578   TopOpenctopn 14696   0gc0g 14714   Topctop 19267   TopSpctps 19270   Clsdccld 19390   Frect1 19681  RRHomcrrh 27847  Σ*cesum 27913  sigAlgebracsiga 27980  sigaGencsigagen 28011  measurescmeas 28039  MblFnMcmbfm 28094  sitgcsitg 28144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-ac2 8846  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-disj 4408  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-acn 8326  df-ac 8500  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-ioc 11543  df-ico 11544  df-icc 11545  df-fz 11682  df-fzo 11804  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-fac 12333  df-bc 12360  df-hash 12385  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-sum 13488  df-ef 13681  df-sin 13683  df-cos 13684  df-pi 13686  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-hom 14598  df-cco 14599  df-rest 14697  df-topn 14698  df-0g 14716  df-gsum 14717  df-topgen 14718  df-pt 14719  df-prds 14722  df-ordt 14775  df-xrs 14776  df-qtop 14781  df-imas 14782  df-xps 14784  df-mre 14860  df-mrc 14861  df-acs 14863  df-ps 15704  df-tsr 15705  df-plusf 15745  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15840  df-submnd 15841  df-grp 15931  df-minusg 15932  df-sbg 15933  df-mulg 15934  df-subg 16072  df-cntz 16229  df-cmn 16674  df-abl 16675  df-mgp 17016  df-ur 17028  df-ring 17074  df-cring 17075  df-subrg 17301  df-abv 17340  df-lmod 17388  df-scaf 17389  df-sra 17692  df-rgmod 17693  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-fbas 18290  df-fg 18291  df-cnfld 18295  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cld 19393  df-ntr 19394  df-cls 19395  df-nei 19472  df-lp 19510  df-perf 19511  df-cn 19601  df-cnp 19602  df-t1 19688  df-haus 19689  df-tx 19936  df-hmeo 20129  df-fil 20220  df-fm 20312  df-flim 20313  df-flf 20314  df-tmd 20444  df-tgp 20445  df-tsms 20498  df-trg 20535  df-xms 20696  df-ms 20697  df-tms 20698  df-nm 20976  df-ngp 20977  df-nrg 20979  df-nlm 20980  df-ii 21254  df-cncf 21255  df-limc 22143  df-dv 22144  df-log 22816  df-esum 27914  df-siga 27981  df-sigagen 28012  df-meas 28040  df-mbfm 28095  df-sitg 28145
This theorem is referenced by: (None)
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