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Theorem sibfof 28788
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 19731 . . . . . . . . . . 11  |-  ( W  e.  TopSp  ->  B  =  U. J )
62, 5syl 17 . . . . . . . . . 10  |-  ( ph  ->  B  =  U. J
)
76sqxpeqd 4849 . . . . . . . . 9  |-  ( ph  ->  ( B  X.  B
)  =  ( U. J  X.  U. J ) )
87feq2d 5701 . . . . . . . 8  |-  ( ph  ->  (  .+  : ( B  X.  B ) --> C  <->  .+  : ( U. J  X.  U. J ) --> C ) )
91, 8mpbid 210 . . . . . . 7  |-  ( ph  ->  .+  : ( U. J  X.  U. J ) --> C )
109fovrnda 6427 . . . . . 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 28784 . . . . . 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 28784 . . . . . 6  |-  ( ph  ->  G : U. dom  M --> U. J )
21 dmexg 6715 . . . . . . 7  |-  ( M  e.  U. ran measures  ->  dom  M  e.  _V )
22 uniexg 6579 . . . . . . 7  |-  ( dom 
M  e.  _V  ->  U.
dom  M  e.  _V )
2316, 21, 223syl 18 . . . . . 6  |-  ( ph  ->  U. dom  M  e. 
_V )
24 inidm 3648 . . . . . 6  |-  ( U. dom  M  i^i  U. dom  M )  =  U. dom  M
2510, 18, 20, 23, 23, 24off 6536 . . . . 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 2402 . . . . . . . . 9  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
2927, 28tpsuni 19731 . . . . . . . 8  |-  ( K  e.  TopSp  ->  C  =  U. ( TopOpen `  K )
)
3026, 29syl 17 . . . . . . 7  |-  ( ph  ->  C  =  U. ( TopOpen
`  K ) )
31 fvex 5859 . . . . . . . 8  |-  ( TopOpen `  K )  e.  _V
32 unisg 28591 . . . . . . . 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 2461 . . . . . 6  |-  ( ph  ->  C  =  U. (sigaGen `  ( TopOpen `  K )
) )
3534feq3d 5702 . . . . 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 28590 . . . . . 6  |-  ( ph  ->  (sigaGen `  ( TopOpen `  K
) )  e.  U. ran sigAlgebra )
39 uniexg 6579 . . . . . 6  |-  ( (sigaGen `  ( TopOpen `  K )
)  e.  U. ran sigAlgebra  ->  U. (sigaGen `  ( TopOpen `  K
) )  e.  _V )
4038, 39syl 17 . . . . 5  |-  ( ph  ->  U. (sigaGen `  ( TopOpen
`  K ) )  e.  _V )
4140, 23elmapd 7471 . . . 4  |-  ( ph  ->  ( ( F  oF  .+  G )  e.  ( U. (sigaGen `  ( TopOpen
`  K ) )  ^m  U. dom  M
)  <->  ( F  oF  .+  G ) : U. dom  M --> U. (sigaGen `  ( TopOpen `  K )
) ) )
4236, 41mpbird 232 . . 3  |-  ( ph  ->  ( F  oF  .+  G )  e.  ( U. (sigaGen `  ( TopOpen
`  K ) )  ^m  U. dom  M
) )
43 inundif 3850 . . . . . . 7  |-  ( ( b  i^i  ran  ( F  oF  .+  G
) )  u.  (
b  \  ran  ( F  oF  .+  G
) ) )  =  b
4443imaeq2i 5155 . . . . . 6  |-  ( `' ( F  oF  .+  G ) "
( ( b  i^i 
ran  ( F  oF  .+  G ) )  u.  ( b  \  ran  ( F  oF  .+  G ) ) ) )  =  ( `' ( F  oF  .+  G ) "
b )
45 ffun 5716 . . . . . . . 8  |-  ( ( F  oF  .+  G ) : U. dom  M --> C  ->  Fun  ( F  oF  .+  G ) )
46 unpreima 5991 . . . . . . . 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
) ) ) ) )
4725, 45, 463syl 18 . . . . . . 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
) ) ) ) )
4847adantr 463 . . . . . 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
) ) ) ) )
4944, 48syl5eqr 2457 . . . . 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
) ) ) ) )
50 dmmeas 28649 . . . . . . . 8  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
5116, 50syl 17 . . . . . . 7  |-  ( ph  ->  dom  M  e.  U. ran sigAlgebra )
5251adantr 463 . . . . . 6  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  ->  dom  M  e.  U. ran sigAlgebra )
53 imaiun 6138 . . . . . . . 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 } )
54 iunid 4326 . . . . . . . . 9  |-  U_ z  e.  ( b  i^i  ran  ( F  oF  .+  G ) ) { z }  =  ( b  i^i  ran  ( F  oF  .+  G
) )
5554imaeq2i 5155 . . . . . . . 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 ) ) )
5653, 55eqtr3i 2433 . . . . . . 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
) ) )
57 inss2 3660 . . . . . . . . . 10  |-  ( b  i^i  ran  ( F  oF  .+  G ) )  C_  ran  ( F  oF  .+  G
)
586feq3d 5702 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( F : U. dom  M --> B  <->  F : U. dom  M --> U. J
) )
5918, 58mpbird 232 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : U. dom  M --> B )
606feq3d 5702 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G : U. dom  M --> B  <->  G : U. dom  M --> U. J
) )
6120, 60mpbird 232 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : U. dom  M --> B )
62 ffn 5714 . . . . . . . . . . . . . . 15  |-  (  .+  : ( B  X.  B ) --> C  ->  .+  Fn  ( B  X.  B ) )
631, 62syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  .+  Fn  ( B  X.  B ) )
6459, 61, 23, 63ofpreima2 27951 . . . . . . . . . . . . 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 ) } ) ) )
6564adantr 463 . . . . . . . . . . . 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
) } ) ) )
6651adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  dom  M  e.  U.
ran sigAlgebra )
6751ad2antrr 724 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  dom  M  e.  U.
ran sigAlgebra )
68 simpll 752 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ph )
69 inss1 3659 . . . . . . . . . . . . . . . . . 18  |-  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  C_  ( `'  .+  " { z } )
70 cnvimass 5177 . . . . . . . . . . . . . . . . . . . 20  |-  ( `' 
.+  " { z } )  C_  dom  .+
71 fdm 5718 . . . . . . . . . . . . . . . . . . . . 21  |-  (  .+  : ( B  X.  B ) --> C  ->  dom  .+  =  ( B  X.  B ) )
721, 71syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  dom  .+  =  ( B  X.  B ) )
7370, 72syl5sseq 3490 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( `'  .+  " {
z } )  C_  ( B  X.  B
) )
7473adantr 463 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( `'  .+  " { z } ) 
C_  ( B  X.  B ) )
7569, 74syl5ss 3453 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( ( `' 
.+  " { z } )  i^i  ( ran 
F  X.  ran  G
) )  C_  ( B  X.  B ) )
7675sselda 3442 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  p  e.  ( B  X.  B ) )
7751adantr 463 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  dom  M  e.  U. ran sigAlgebra )
78 sibfof.4 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  J  e.  Fre )
7978sgsiga 28590 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
8011, 79syl5eqel 2494 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
8180adantr 463 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  S  e.  U. ran sigAlgebra )
823, 4, 11, 12, 13, 14, 15, 16, 17sibfmbl 28783 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
8382adantr 463 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  F  e.  ( dom  MMblFnM S
) )
844tpstop 19732 . . . . . . . . . . . . . . . . . . . . 21  |-  ( W  e.  TopSp  ->  J  e.  Top )
85 cldssbrsiga 28635 . . . . . . . . . . . . . . . . . . . . 21  |-  ( J  e.  Top  ->  ( Clsd `  J )  C_  (sigaGen `  J ) )
862, 84, 853syl 18 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( Clsd `  J
)  C_  (sigaGen `  J
) )
8786adantr 463 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( Clsd `  J )  C_  (sigaGen `  J ) )
8878adantr 463 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  J  e.  Fre )
89 xp1st 6814 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p  e.  ( B  X.  B )  ->  ( 1st `  p )  e.  B )
9089adantl 464 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 1st `  p )  e.  B )
916adantr 463 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  B  =  U. J )
9290, 91eleqtrd 2492 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 1st `  p )  e. 
U. J )
93 eqid 2402 . . . . . . . . . . . . . . . . . . . . 21  |-  U. J  =  U. J
9493t1sncld 20120 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( J  e.  Fre  /\  ( 1st `  p )  e.  U. J )  ->  { ( 1st `  p ) }  e.  ( Clsd `  J )
)
9588, 92, 94syl2anc 659 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 1st `  p ) }  e.  ( Clsd `  J ) )
9687, 95sseldd 3443 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 1st `  p ) }  e.  (sigaGen `  J
) )
9796, 11syl6eleqr 2501 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 1st `  p ) }  e.  S )
9877, 81, 83, 97mbfmcnvima 28705 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( `' F " { ( 1st `  p ) } )  e.  dom  M )
9968, 76, 98syl2anc 659 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( `' F " { ( 1st `  p
) } )  e. 
dom  M )
1003, 4, 11, 12, 13, 14, 15, 16, 19sibfmbl 28783 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  G  e.  ( dom 
MMblFnM S ) )
101100adantr 463 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  G  e.  ( dom  MMblFnM S
) )
102 xp2nd 6815 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p  e.  ( B  X.  B )  ->  ( 2nd `  p )  e.  B )
103102adantl 464 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 2nd `  p )  e.  B )
104103, 91eleqtrd 2492 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 2nd `  p )  e. 
U. J )
10593t1sncld 20120 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( J  e.  Fre  /\  ( 2nd `  p )  e.  U. J )  ->  { ( 2nd `  p ) }  e.  ( Clsd `  J )
)
10688, 104, 105syl2anc 659 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 2nd `  p ) }  e.  ( Clsd `  J ) )
10787, 106sseldd 3443 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 2nd `  p ) }  e.  (sigaGen `  J
) )
108107, 11syl6eleqr 2501 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 2nd `  p ) }  e.  S )
10977, 81, 101, 108mbfmcnvima 28705 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( `' G " { ( 2nd `  p ) } )  e.  dom  M )
11068, 76, 109syl2anc 659 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( `' G " { ( 2nd `  p
) } )  e. 
dom  M )
111 inelsiga 28583 . . . . . . . . . . . . . . 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 )
11267, 99, 110, 111syl3anc 1230 . . . . . . . . . . . . . 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 )
113112ralrimiva 2818 . . . . . . . . . . . . 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 )
1143, 4, 11, 12, 13, 14, 15, 16, 17sibfrn 28785 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ran  F  e.  Fin )
1153, 4, 11, 12, 13, 14, 15, 16, 19sibfrn 28785 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ran  G  e.  Fin )
116 xpfi 7825 . . . . . . . . . . . . . . . . 17  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
117114, 115, 116syl2anc 659 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
118 inss2 3660 . . . . . . . . . . . . . . . 16  |-  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  C_  ( ran  F  X.  ran  G
)
119 ssdomg 7599 . . . . . . . . . . . . . . . 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 ) ) )
120117, 118, 119mpisyl 21 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  ~<_  ( ran  F  X.  ran  G ) )
121 isfinite 8102 . . . . . . . . . . . . . . . . 17  |-  ( ( ran  F  X.  ran  G )  e.  Fin  <->  ( ran  F  X.  ran  G ) 
~<  om )
122121biimpi 194 . . . . . . . . . . . . . . . 16  |-  ( ( ran  F  X.  ran  G )  e.  Fin  ->  ( ran  F  X.  ran  G )  ~<  om )
123 sdomdom 7581 . . . . . . . . . . . . . . . 16  |-  ( ( ran  F  X.  ran  G )  ~<  om  ->  ( ran  F  X.  ran  G )  ~<_  om )
124117, 122, 1233syl 18 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ran  F  X.  ran  G )  ~<_  om )
125 domtr 7606 . . . . . . . . . . . . . . 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 )
126120, 124, 125syl2anc 659 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  ~<_  om )
127126adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( ( `' 
.+  " { z } )  i^i  ( ran 
F  X.  ran  G
) )  ~<_  om )
128 nfcv 2564 . . . . . . . . . . . . . 14  |-  F/_ p
( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )
129128sigaclcuni 28566 . . . . . . . . . . . . 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 )
13066, 113, 127, 129syl3anc 1230 . . . . . . . . . . . 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 )
13165, 130eqeltrd 2490 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( `' ( F  oF  .+  G ) " {
z } )  e. 
dom  M )
132131ralrimiva 2818 . . . . . . . . . 10  |-  ( ph  ->  A. z  e.  ran  ( F  oF  .+  G ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
133 ssralv 3503 . . . . . . . . . 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 ) )
13457, 132, 133mpsyl 62 . . . . . . . . 9  |-  ( ph  ->  A. z  e.  ( b  i^i  ran  ( F  oF  .+  G
) ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
135134adantr 463 . . . . . . . 8  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  ->  A. z  e.  (
b  i^i  ran  ( F  oF  .+  G
) ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
136 ffun 5716 . . . . . . . . . . . . . 14  |-  (  .+  : ( B  X.  B ) --> C  ->  Fun  .+  )
1371, 136syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  Fun  .+  )
138 imafi 7847 . . . . . . . . . . . . 13  |-  ( ( Fun  .+  /\  ( ran  F  X.  ran  G
)  e.  Fin )  ->  (  .+  " ( ran  F  X.  ran  G
) )  e.  Fin )
139137, 117, 138syl2anc 659 . . . . . . . . . . . 12  |-  ( ph  ->  (  .+  " ( ran  F  X.  ran  G
) )  e.  Fin )
14018, 20, 9, 23ofrn2 27923 . . . . . . . . . . . 12  |-  ( ph  ->  ran  ( F  oF  .+  G )  C_  (  .+  " ( ran 
F  X.  ran  G
) ) )
141 ssfi 7775 . . . . . . . . . . . 12  |-  ( ( (  .+  " ( ran  F  X.  ran  G
) )  e.  Fin  /\ 
ran  ( F  oF  .+  G )  C_  (  .+  " ( ran 
F  X.  ran  G
) ) )  ->  ran  ( F  oF  .+  G )  e. 
Fin )
142139, 140, 141syl2anc 659 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( F  oF  .+  G )  e. 
Fin )
143 ssdomg 7599 . . . . . . . . . . 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 ) ) )
144142, 57, 143mpisyl 21 . . . . . . . . . 10  |-  ( ph  ->  ( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  ran  ( F  oF  .+  G ) )
145 isfinite 8102 . . . . . . . . . . . 12  |-  ( ran  ( F  oF  .+  G )  e. 
Fin 
<->  ran  ( F  oF  .+  G )  ~<  om )
146142, 145sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( F  oF  .+  G )  ~<  om )
147 sdomdom 7581 . . . . . . . . . . 11  |-  ( ran  ( F  oF  .+  G )  ~<  om  ->  ran  ( F  oF  .+  G )  ~<_  om )
148146, 147syl 17 . . . . . . . . . 10  |-  ( ph  ->  ran  ( F  oF  .+  G )  ~<_  om )
149 domtr 7606 . . . . . . . . . 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 )
150144, 148, 149syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  ( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  om )
151150adantr 463 . . . . . . . 8  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  om )
152 nfcv 2564 . . . . . . . . 9  |-  F/_ z
( b  i^i  ran  ( F  oF  .+  G ) )
153152sigaclcuni 28566 . . . . . . . 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 )
15452, 135, 151, 153syl3anc 1230 . . . . . . 7  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  ->  U_ z  e.  (
b  i^i  ran  ( F  oF  .+  G
) ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
15556, 154syl5eqelr 2495 . . . . . 6  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " ( b  i^i  ran  ( F  oF  .+  G ) ) )  e.  dom  M )
156 difpreima 5993 . . . . . . . . . 10  |-  ( Fun  ( F  oF  .+  G )  -> 
( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  =  ( ( `' ( F  oF  .+  G
) " b ) 
\  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) ) )
15725, 45, 1563syl 18 . . . . . . . . 9  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  =  ( ( `' ( F  oF  .+  G
) " b ) 
\  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) ) )
158 cnvimarndm 5178 . . . . . . . . . . 11  |-  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) )  =  dom  ( F  oF  .+  G
)
159158difeq2i 3558 . . . . . . . . . 10  |-  ( ( `' ( F  oF  .+  G ) "
b )  \  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) )  =  ( ( `' ( F  oF  .+  G ) "
b )  \  dom  ( F  oF  .+  G ) )
160 cnvimass 5177 . . . . . . . . . . 11  |-  ( `' ( F  oF  .+  G ) "
b )  C_  dom  ( F  oF  .+  G )
161 ssdif0 3828 . . . . . . . . . . 11  |-  ( ( `' ( F  oF  .+  G ) "
b )  C_  dom  ( F  oF  .+  G )  <->  ( ( `' ( F  oF  .+  G ) "
b )  \  dom  ( F  oF  .+  G ) )  =  (/) )
162160, 161mpbi 208 . . . . . . . . . 10  |-  ( ( `' ( F  oF  .+  G ) "
b )  \  dom  ( F  oF  .+  G ) )  =  (/)
163159, 162eqtri 2431 . . . . . . . . 9  |-  ( ( `' ( F  oF  .+  G ) "
b )  \  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) )  =  (/)
164157, 163syl6eq 2459 . . . . . . . 8  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  =  (/) )
165 0elsiga 28562 . . . . . . . . 9  |-  ( dom 
M  e.  U. ran sigAlgebra  ->  (/)  e.  dom  M )
16616, 50, 1653syl 18 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  dom  M )
167164, 166eqeltrd 2490 . . . . . . 7  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  e.  dom  M )
168167adantr 463 . . . . . 6  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  e.  dom  M )
169 unelsiga 28582 . . . . . 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 )
17052, 155, 168, 169syl3anc 1230 . . . . 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 )
17149, 170eqeltrd 2490 . . . 4  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " b )  e.  dom  M )
172171ralrimiva 2818 . . 3  |-  ( ph  ->  A. b  e.  (sigaGen `  ( TopOpen `  K )
) ( `' ( F  oF  .+  G ) " b
)  e.  dom  M
)
17351, 38ismbfm 28700 . . 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 ) ) )
17442, 172, 173mpbir2and 923 . 2  |-  ( ph  ->  ( F  oF  .+  G )  e.  ( dom  MMblFnM (sigaGen `  ( TopOpen `  K )
) ) )
17564adantr 463 . . . . . . 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
) } ) ) )
176175fveq2d 5853 . . . . . 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
) } ) ) ) )
177 measbasedom 28650 . . . . . . . . 9  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )
17816, 177sylib 196 . . . . . . . 8  |-  ( ph  ->  M  e.  (measures `  dom  M ) )
179178adantr 463 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  M  e.  (measures `  dom  M ) )
180 eldifi 3565 . . . . . . . 8  |-  ( z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } )  ->  z  e.  ran  ( F  oF  .+  G ) )
181180, 113sylan2 472 . . . . . . 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 )
182126adantr 463 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  ~<_  om )
183 sneq 3982 . . . . . . . . . . 11  |-  ( x  =  ( 1st `  p
)  ->  { x }  =  { ( 1st `  p ) } )
184183imaeq2d 5157 . . . . . . . . . 10  |-  ( x  =  ( 1st `  p
)  ->  ( `' F " { x }
)  =  ( `' F " { ( 1st `  p ) } ) )
185 sneq 3982 . . . . . . . . . . 11  |-  ( y  =  ( 2nd `  p
)  ->  { y }  =  { ( 2nd `  p ) } )
186185imaeq2d 5157 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  p
)  ->  ( `' G " { y } )  =  ( `' G " { ( 2nd `  p ) } ) )
187 ffun 5716 . . . . . . . . . . . 12  |-  ( F : U. dom  M --> U. J  ->  Fun  F
)
18818, 187syl 17 . . . . . . . . . . 11  |-  ( ph  ->  Fun  F )
189 sndisj 4387 . . . . . . . . . . 11  |- Disj  x  e. 
ran  F { x }
190 disjpreima 27876 . . . . . . . . . . 11  |-  ( ( Fun  F  /\ Disj  x  e. 
ran  F { x } )  -> Disj  x  e. 
ran  F ( `' F " { x } ) )
191188, 189, 190sylancl 660 . . . . . . . . . 10  |-  ( ph  -> Disj  x  e.  ran  F ( `' F " { x } ) )
192 ffun 5716 . . . . . . . . . . . 12  |-  ( G : U. dom  M --> U. J  ->  Fun  G
)
19320, 192syl 17 . . . . . . . . . . 11  |-  ( ph  ->  Fun  G )
194 sndisj 4387 . . . . . . . . . . 11  |- Disj  y  e. 
ran  G { y }
195 disjpreima 27876 . . . . . . . . . . 11  |-  ( ( Fun  G  /\ Disj  y  e. 
ran  G { y } )  -> Disj  y  e. 
ran  G ( `' G " { y } ) )
196193, 194, 195sylancl 660 . . . . . . . . . 10  |-  ( ph  -> Disj  y  e.  ran  G ( `' G " { y } ) )
197184, 186, 191, 196disjxpin 27880 . . . . . . . . 9  |-  ( ph  -> Disj  p  e.  ( ran  F  X.  ran  G ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )
198 disjss1 4372 . . . . . . . . 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 ) } ) ) ) )
199118, 197, 198mpsyl 62 . . . . . . . 8  |-  ( ph  -> Disj  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
200199adantr 463 . . . . . . 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
) } ) ) )
201 measvuni 28662 . . . . . . 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 ) } ) ) ) )
202179, 181, 182, 200, 201syl112anc 1234 . . . . . 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 ) } ) ) ) )
203 ssfi 7775 . . . . . . . . 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 )
204117, 118, 203sylancl 660 . . . . . . . 8  |-  ( ph  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  e.  Fin )
205204adantr 463 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  e.  Fin )
206 simpll 752 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ph )
207 simpr 459 . . . . . . . . . 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 ) ) )
208118, 207sseldi 3440 . . . . . . . . 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 ) )
209 xp1st 6814 . . . . . . . . 9  |-  ( p  e.  ( ran  F  X.  ran  G )  -> 
( 1st `  p
)  e.  ran  F
)
210208, 209syl 17 . . . . . . . 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 )
211 xp2nd 6815 . . . . . . . . 9  |-  ( p  e.  ( ran  F  X.  ran  G )  -> 
( 2nd `  p
)  e.  ran  G
)
212208, 211syl 17 . . . . . . . 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 )
213 oveq12 6287 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  .0.  /\  y  =  .0.  )  ->  ( x  .+  y
)  =  (  .0.  .+  .0.  ) )
214 sibfof.5 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  (  .0.  .+  .0.  )  =  ( 0g `  K ) )
215213, 214sylan9eqr 2465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  =  .0.  /\  y  =  .0.  ) )  -> 
( x  .+  y
)  =  ( 0g
`  K ) )
216215ex 432 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( x  =  .0.  /\  y  =  .0.  )  ->  (
x  .+  y )  =  ( 0g `  K ) ) )
217216necon3ad 2613 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( x  .+  y )  =/=  ( 0g `  K )  ->  -.  ( x  =  .0. 
/\  y  =  .0.  ) ) )
218 neorian 2730 . . . . . . . . . . . . 13  |-  ( ( x  =/=  .0.  \/  y  =/=  .0.  )  <->  -.  (
x  =  .0.  /\  y  =  .0.  )
)
219217, 218syl6ibr 227 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( x  .+  y )  =/=  ( 0g `  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) ) )
220219adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( x  .+  y )  =/=  ( 0g `  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) ) )
221220ralrimivva 2825 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  B  A. y  e.  B  ( ( x  .+  y )  =/=  ( 0g `  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) ) )
222206, 221syl 17 . . . . . . . . 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.  ) ) )
22369a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  C_  ( `'  .+  " { z } ) )
224223sselda 3442 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  p  e.  ( `'  .+  " {
z } ) )
225 fniniseg 5986 . . . . . . . . . . . . 13  |-  (  .+  Fn  ( B  X.  B
)  ->  ( p  e.  ( `'  .+  " {
z } )  <->  ( p  e.  ( B  X.  B
)  /\  (  .+  `  p )  =  z ) ) )
226206, 63, 2253syl 18 . . . . . . . . . . . 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 ) ) )
227224, 226mpbid 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 ) )
228 simpr 459 . . . . . . . . . . . 12  |-  ( ( p  e.  ( B  X.  B )  /\  (  .+  `  p )  =  z )  -> 
(  .+  `  p )  =  z )
229 1st2nd2 6821 . . . . . . . . . . . . . . 15  |-  ( p  e.  ( B  X.  B )  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
230229fveq2d 5853 . . . . . . . . . . . . . 14  |-  ( p  e.  ( B  X.  B )  ->  (  .+  `  p )  =  (  .+  `  <. ( 1st `  p ) ,  ( 2nd `  p
) >. ) )
231 df-ov 6281 . . . . . . . . . . . . . 14  |-  ( ( 1st `  p ) 
.+  ( 2nd `  p
) )  =  ( 
.+  `  <. ( 1st `  p ) ,  ( 2nd `  p )
>. )
232230, 231syl6eqr 2461 . . . . . . . . . . . . 13  |-  ( p  e.  ( B  X.  B )  ->  (  .+  `  p )  =  ( ( 1st `  p
)  .+  ( 2nd `  p ) ) )
233232adantr 463 . . . . . . . . . . . 12  |-  ( ( p  e.  ( B  X.  B )  /\  (  .+  `  p )  =  z )  -> 
(  .+  `  p )  =  ( ( 1st `  p )  .+  ( 2nd `  p ) ) )
234228, 233eqtr3d 2445 . . . . . . . . . . 11  |-  ( ( p  e.  ( B  X.  B )  /\  (  .+  `  p )  =  z )  -> 
z  =  ( ( 1st `  p ) 
.+  ( 2nd `  p
) ) )
235227, 234syl 17 . . . . . . . . . 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 ) ) )
236 simplr 754 . . . . . . . . . . . 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 ) } ) )
237236eldifbd 3427 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  -.  z  e.  { ( 0g `  K ) } )
238 elsn 3986 . . . . . . . . . . . 12  |-  ( z  e.  { ( 0g
`  K ) }  <-> 
z  =  ( 0g
`  K ) )
239238necon3bbii 2664 . . . . . . . . . . 11  |-  ( -.  z  e.  { ( 0g `  K ) }  <->  z  =/=  ( 0g `  K ) )
240237, 239sylib 196 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  z  =/=  ( 0g `  K ) )
241235, 240eqnetrrd 2697 . . . . . . . . 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 ) )
242180, 76sylanl2 649 . . . . . . . . . . 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
) )
243242, 89syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( 1st `  p )  e.  B
)
244242, 102syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( 2nd `  p )  e.  B
)
245 oveq1 6285 . . . . . . . . . . . . 13  |-  ( x  =  ( 1st `  p
)  ->  ( x  .+  y )  =  ( ( 1st `  p
)  .+  y )
)
246245neeq1d 2680 . . . . . . . . . . . 12  |-  ( x  =  ( 1st `  p
)  ->  ( (
x  .+  y )  =/=  ( 0g `  K
)  <->  ( ( 1st `  p )  .+  y
)  =/=  ( 0g
`  K ) ) )
247 neeq1 2684 . . . . . . . . . . . . 13  |-  ( x  =  ( 1st `  p
)  ->  ( x  =/=  .0.  <->  ( 1st `  p
)  =/=  .0.  )
)
248247orbi1d 701 . . . . . . . . . . . 12  |-  ( x  =  ( 1st `  p
)  ->  ( (
x  =/=  .0.  \/  y  =/=  .0.  )  <->  ( ( 1st `  p )  =/= 
.0.  \/  y  =/=  .0.  ) ) )
249246, 248imbi12d 318 . . . . . . . . . . 11  |-  ( x  =  ( 1st `  p
)  ->  ( (
( x  .+  y
)  =/=  ( 0g
`  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) )  <->  ( (
( 1st `  p
)  .+  y )  =/=  ( 0g `  K
)  ->  ( ( 1st `  p )  =/= 
.0.  \/  y  =/=  .0.  ) ) ) )
250 oveq2 6286 . . . . . . . . . . . . 13  |-  ( y  =  ( 2nd `  p
)  ->  ( ( 1st `  p )  .+  y )  =  ( ( 1st `  p
)  .+  ( 2nd `  p ) ) )
251250neeq1d 2680 . . . . . . . . . . . 12  |-  ( y  =  ( 2nd `  p
)  ->  ( (
( 1st `  p
)  .+  y )  =/=  ( 0g `  K
)  <->  ( ( 1st `  p )  .+  ( 2nd `  p ) )  =/=  ( 0g `  K ) ) )
252 neeq1 2684 . . . . . . . . . . . . 13  |-  ( y  =  ( 2nd `  p
)  ->  ( y  =/=  .0.  <->  ( 2nd `  p
)  =/=  .0.  )
)
253252orbi2d 700 . . . . . . . . . . . 12  |-  ( y  =  ( 2nd `  p
)  ->  ( (
( 1st `  p
)  =/=  .0.  \/  y  =/=  .0.  )  <->  ( ( 1st `  p )  =/= 
.0.  \/  ( 2nd `  p )  =/=  .0.  ) ) )
254251, 253imbi12d 318 . . . . . . . . . . 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.  ) ) ) )
255249, 254rspc2v 3169 . . . . . . . . . 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.  ) ) ) )
256243, 244, 255syl2anc 659 . . . . . . . . 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.  ) ) ) )
257222, 241, 256mp2d 43 . . . . . . . 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.  ) )
2583, 4, 11, 12, 13, 14, 15, 16, 17, 19, 2, 78sibfinima 28787 . . . . . . . 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 ) )
259206, 210, 212, 257, 258syl31anc 1233 . . . . . . 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 ) )
260205, 259esumpfinval 28522 . . . . . 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 ) } ) ) ) )
261176, 202, 2603eqtrd 2447 . . . . 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 ) } ) ) ) )
262 rge0ssre 11682 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR
263262, 259sseldi 3440 . . . . . 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 )
264205, 263fsumrecl 13705 . . . . 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 )
265261, 264eqeltrd 2490 . . . 4  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( M `  ( `' ( F  oF  .+  G
) " { z } ) )  e.  RR )
266179adantr 463 . . . . . . 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 ) )
267180, 112sylanl2 649 . . . . . . 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 )
268 measge0 28655 . . . . . . 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
) } ) ) ) )
269266, 267, 268syl2anc 659 . . . . . 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 ) } ) ) ) )
270205, 263, 269fsumge0 13760 . . . . 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 ) } ) ) ) )
271270, 261breqtrrd 4421 . . . 4  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  0  <_  ( M `  ( `' ( F  oF  .+  G ) " { z } ) ) )
272 elrege0 11681 . . . 4  |-  ( ( M `  ( `' ( F  oF  .+  G ) " { z } ) )  e.  ( 0 [,) +oo )  <->  ( ( M `  ( `' ( F  oF  .+  G ) " {
z } ) )  e.  RR  /\  0  <_  ( M `  ( `' ( F  oF  .+  G ) " { z } ) ) ) )
273265, 271, 272sylanbrc 662 . . 3  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( M `  ( `' ( F  oF  .+  G
) " { z } ) )  e.  ( 0 [,) +oo ) )
274273ralrimiva 2818 . 2  |-  ( ph  ->  A. z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) ( M `  ( `' ( F  oF  .+  G ) " { z } ) )  e.  ( 0 [,) +oo ) )
275 eqid 2402 . . 3  |-  (sigaGen `  ( TopOpen
`  K ) )  =  (sigaGen `  ( TopOpen
`  K ) )
276 eqid 2402 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
277 eqid 2402 . . 3  |-  ( .s
`  K )  =  ( .s `  K
)
278 eqid 2402 . . 3  |-  (RRHom `  (Scalar `  K ) )  =  (RRHom `  (Scalar `  K ) )
27927, 28, 275, 276, 277, 278, 26, 16issibf 28781 . 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 ) ) ) )
280174, 142, 274, 279mpbir3and 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 366    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   _Vcvv 3059    \ cdif 3411    u. cun 3412    i^i cin 3413    C_ wss 3414   (/)c0 3738   {csn 3972   <.cop 3978   U.cuni 4191   U_ciun 4271  Disj wdisj 4366   class class class wbr 4395    X. cxp 4821   `'ccnv 4822   dom cdm 4823   ran crn 4824   "cima 4826   Fun wfun 5563    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278    oFcof 6519   omcom 6683   1stc1st 6782   2ndc2nd 6783    ^m cmap 7457    ~<_ cdom 7552    ~< csdm 7553   Fincfn 7554   RRcr 9521   0cc0 9522   +oocpnf 9655    <_ cle 9659   [,)cico 11584   sum_csu 13657   Basecbs 14841  Scalarcsca 14912   .scvsca 14913   TopOpenctopn 15036   0gc0g 15054   Topctop 19686   TopSpctps 19689   Clsdccld 19809   Frect1 20101  RRHomcrrh 28426  Σ*cesum 28474  sigAlgebracsiga 28555  sigaGencsigagen 28586  measurescmeas 28643  MblFnMcmbfm 28698  sitgcsitg 28777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-ac2 8875  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-disj 4367  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-acn 8355  df-ac 8529  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-ioc 11587  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-fac 12398  df-bc 12425  df-hash 12453  df-shft 13049  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-limsup 13443  df-clim 13460  df-rlim 13461  df-sum 13658  df-ef 14012  df-sin 14014  df-cos 14015  df-pi 14017  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-ordt 15115  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-ps 16154  df-tsr 16155  df-plusf 16195  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-mhm 16290  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-mulg 16384  df-subg 16522  df-cntz 16679  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-cring 17521  df-subrg 17747  df-abv 17786  df-lmod 17834  df-scaf 17835  df-sra 18138  df-rgmod 18139  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-fbas 18736  df-fg 18737  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-lp 19930  df-perf 19931  df-cn 20021  df-cnp 20022  df-t1 20108  df-haus 20109  df-tx 20355  df-hmeo 20548  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-tmd 20863  df-tgp 20864  df-tsms 20917  df-trg 20954  df-xms 21115  df-ms 21116  df-tms 21117  df-nm 21395  df-ngp 21396  df-nrg 21398  df-nlm 21399  df-ii 21673  df-cncf 21674  df-limc 22562  df-dv 22563  df-log 23236  df-esum 28475  df-siga 28556  df-sigagen 28587  df-meas 28644  df-mbfm 28699  df-sitg 28778
This theorem is referenced by: (None)
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