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Theorem sibfof 29175
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 19945 . . . . . . . . . . 11  |-  ( W  e.  TopSp  ->  B  =  U. J )
62, 5syl 17 . . . . . . . . . 10  |-  ( ph  ->  B  =  U. J
)
76sqxpeqd 4877 . . . . . . . . 9  |-  ( ph  ->  ( B  X.  B
)  =  ( U. J  X.  U. J ) )
87feq2d 5731 . . . . . . . 8  |-  ( ph  ->  (  .+  : ( B  X.  B ) --> C  <->  .+  : ( U. J  X.  U. J ) --> C ) )
91, 8mpbid 214 . . . . . . 7  |-  ( ph  ->  .+  : ( U. J  X.  U. J ) --> C )
109fovrnda 6452 . . . . . 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 29171 . . . . . 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 29171 . . . . . 6  |-  ( ph  ->  G : U. dom  M --> U. J )
21 dmexg 6736 . . . . . . 7  |-  ( M  e.  U. ran measures  ->  dom  M  e.  _V )
22 uniexg 6600 . . . . . . 7  |-  ( dom 
M  e.  _V  ->  U.
dom  M  e.  _V )
2316, 21, 223syl 18 . . . . . 6  |-  ( ph  ->  U. dom  M  e. 
_V )
24 inidm 3672 . . . . . 6  |-  ( U. dom  M  i^i  U. dom  M )  =  U. dom  M
2510, 18, 20, 23, 23, 24off 6558 . . . . 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 2423 . . . . . . . . 9  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
2927, 28tpsuni 19945 . . . . . . . 8  |-  ( K  e.  TopSp  ->  C  =  U. ( TopOpen `  K )
)
3026, 29syl 17 . . . . . . 7  |-  ( ph  ->  C  =  U. ( TopOpen
`  K ) )
31 fvex 5889 . . . . . . . 8  |-  ( TopOpen `  K )  e.  _V
32 unisg 28967 . . . . . . . 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 2482 . . . . . 6  |-  ( ph  ->  C  =  U. (sigaGen `  ( TopOpen `  K )
) )
3534feq3d 5732 . . . . 5  |-  ( ph  ->  ( ( F  oF  .+  G ) : U. dom  M --> C  <->  ( F  oF  .+  G ) : U. dom  M --> U. (sigaGen `  ( TopOpen `  K
) ) ) )
3625, 35mpbid 214 . . . 4  |-  ( ph  ->  ( F  oF  .+  G ) : U. dom  M --> U. (sigaGen `  ( TopOpen `  K )
) )
3731a1i 11 . . . . . . 7  |-  ( ph  ->  ( TopOpen `  K )  e.  _V )
3837sgsiga 28966 . . . . . 6  |-  ( ph  ->  (sigaGen `  ( TopOpen `  K
) )  e.  U. ran sigAlgebra )
39 uniexg 6600 . . . . . 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 7492 . . . 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 236 . . 3  |-  ( ph  ->  ( F  oF  .+  G )  e.  ( U. (sigaGen `  ( TopOpen
`  K ) )  ^m  U. dom  M
) )
43 inundif 3874 . . . . . . 7  |-  ( ( b  i^i  ran  ( F  oF  .+  G
) )  u.  (
b  \  ran  ( F  oF  .+  G
) ) )  =  b
4443imaeq2i 5183 . . . . . 6  |-  ( `' ( F  oF  .+  G ) "
( ( b  i^i 
ran  ( F  oF  .+  G ) )  u.  ( b  \  ran  ( F  oF  .+  G ) ) ) )  =  ( `' ( F  oF  .+  G ) "
b )
45 ffun 5746 . . . . . . . 8  |-  ( ( F  oF  .+  G ) : U. dom  M --> C  ->  Fun  ( F  oF  .+  G ) )
46 unpreima 6019 . . . . . . . 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 467 . . . . . 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 2478 . . . . 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 29025 . . . . . . . 8  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
5116, 50syl 17 . . . . . . 7  |-  ( ph  ->  dom  M  e.  U. ran sigAlgebra )
5251adantr 467 . . . . . 6  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  ->  dom  M  e.  U. ran sigAlgebra )
53 imaiun 6163 . . . . . . . 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 4352 . . . . . . . . 9  |-  U_ z  e.  ( b  i^i  ran  ( F  oF  .+  G ) ) { z }  =  ( b  i^i  ran  ( F  oF  .+  G
) )
5554imaeq2i 5183 . . . . . . . 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 2454 . . . . . . 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 3684 . . . . . . . . . 10  |-  ( b  i^i  ran  ( F  oF  .+  G ) )  C_  ran  ( F  oF  .+  G
)
586feq3d 5732 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( F : U. dom  M --> B  <->  F : U. dom  M --> U. J
) )
5918, 58mpbird 236 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : U. dom  M --> B )
606feq3d 5732 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G : U. dom  M --> B  <->  G : U. dom  M --> U. J
) )
6120, 60mpbird 236 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : U. dom  M --> B )
62 ffn 5744 . . . . . . . . . . . . . . 15  |-  (  .+  : ( B  X.  B ) --> C  ->  .+  Fn  ( B  X.  B ) )
631, 62syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  .+  Fn  ( B  X.  B ) )
6459, 61, 23, 63ofpreima2 28265 . . . . . . . . . . . . 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 467 . . . . . . . . . . . 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 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  dom  M  e.  U.
ran sigAlgebra )
6751ad2antrr 731 . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ph )
69 inss1 3683 . . . . . . . . . . . . . . . . . 18  |-  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  C_  ( `'  .+  " { z } )
70 cnvimass 5205 . . . . . . . . . . . . . . . . . . . 20  |-  ( `' 
.+  " { z } )  C_  dom  .+
71 fdm 5748 . . . . . . . . . . . . . . . . . . . . 21  |-  (  .+  : ( B  X.  B ) --> C  ->  dom  .+  =  ( B  X.  B ) )
721, 71syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  dom  .+  =  ( B  X.  B ) )
7370, 72syl5sseq 3513 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( `'  .+  " {
z } )  C_  ( B  X.  B
) )
7473adantr 467 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( `'  .+  " { z } ) 
C_  ( B  X.  B ) )
7569, 74syl5ss 3476 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( ( `' 
.+  " { z } )  i^i  ( ran 
F  X.  ran  G
) )  C_  ( B  X.  B ) )
7675sselda 3465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  p  e.  ( B  X.  B ) )
7751adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  dom  M  e.  U. ran sigAlgebra )
78 sibfof.4 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  J  e.  Fre )
7978sgsiga 28966 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
8011, 79syl5eqel 2515 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
8180adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  S  e.  U. ran sigAlgebra )
823, 4, 11, 12, 13, 14, 15, 16, 17sibfmbl 29170 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
8382adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  F  e.  ( dom  MMblFnM S
) )
844tpstop 19946 . . . . . . . . . . . . . . . . . . . . 21  |-  ( W  e.  TopSp  ->  J  e.  Top )
85 cldssbrsiga 29011 . . . . . . . . . . . . . . . . . . . . 21  |-  ( J  e.  Top  ->  ( Clsd `  J )  C_  (sigaGen `  J ) )
862, 84, 853syl 18 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( Clsd `  J
)  C_  (sigaGen `  J
) )
8786adantr 467 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( Clsd `  J )  C_  (sigaGen `  J ) )
8878adantr 467 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  J  e.  Fre )
89 xp1st 6835 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p  e.  ( B  X.  B )  ->  ( 1st `  p )  e.  B )
9089adantl 468 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 1st `  p )  e.  B )
916adantr 467 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  B  =  U. J )
9290, 91eleqtrd 2513 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 1st `  p )  e. 
U. J )
93 eqid 2423 . . . . . . . . . . . . . . . . . . . . 21  |-  U. J  =  U. J
9493t1sncld 20334 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( J  e.  Fre  /\  ( 1st `  p )  e.  U. J )  ->  { ( 1st `  p ) }  e.  ( Clsd `  J )
)
9588, 92, 94syl2anc 666 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 1st `  p ) }  e.  ( Clsd `  J ) )
9687, 95sseldd 3466 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 1st `  p ) }  e.  (sigaGen `  J
) )
9796, 11syl6eleqr 2522 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 1st `  p ) }  e.  S )
9877, 81, 83, 97mbfmcnvima 29081 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( `' F " { ( 1st `  p ) } )  e.  dom  M )
9968, 76, 98syl2anc 666 . . . . . . . . . . . . . . 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 29170 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  G  e.  ( dom 
MMblFnM S ) )
101100adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  G  e.  ( dom  MMblFnM S
) )
102 xp2nd 6836 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p  e.  ( B  X.  B )  ->  ( 2nd `  p )  e.  B )
103102adantl 468 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 2nd `  p )  e.  B )
104103, 91eleqtrd 2513 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 2nd `  p )  e. 
U. J )
10593t1sncld 20334 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( J  e.  Fre  /\  ( 2nd `  p )  e.  U. J )  ->  { ( 2nd `  p ) }  e.  ( Clsd `  J )
)
10688, 104, 105syl2anc 666 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 2nd `  p ) }  e.  ( Clsd `  J ) )
10787, 106sseldd 3466 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 2nd `  p ) }  e.  (sigaGen `  J
) )
108107, 11syl6eleqr 2522 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 2nd `  p ) }  e.  S )
10977, 81, 101, 108mbfmcnvima 29081 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( `' G " { ( 2nd `  p ) } )  e.  dom  M )
11068, 76, 109syl2anc 666 . . . . . . . . . . . . . . 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 28959 . . . . . . . . . . . . . . 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 1265 . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . 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 29172 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ran  F  e.  Fin )
1153, 4, 11, 12, 13, 14, 15, 16, 19sibfrn 29172 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ran  G  e.  Fin )
116 xpfi 7846 . . . . . . . . . . . . . . . . 17  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
117114, 115, 116syl2anc 666 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
118 inss2 3684 . . . . . . . . . . . . . . . 16  |-  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  C_  ( ran  F  X.  ran  G
)
119 ssdomg 7620 . . . . . . . . . . . . . . . 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 22 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  ~<_  ( ran  F  X.  ran  G ) )
121 isfinite 8161 . . . . . . . . . . . . . . . . 17  |-  ( ( ran  F  X.  ran  G )  e.  Fin  <->  ( ran  F  X.  ran  G ) 
~<  om )
122121biimpi 198 . . . . . . . . . . . . . . . 16  |-  ( ( ran  F  X.  ran  G )  e.  Fin  ->  ( ran  F  X.  ran  G )  ~<  om )
123 sdomdom 7602 . . . . . . . . . . . . . . . 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 7627 . . . . . . . . . . . . . . 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 666 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  ~<_  om )
127126adantr 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( ( `' 
.+  " { z } )  i^i  ( ran 
F  X.  ran  G
) )  ~<_  om )
128 nfcv 2585 . . . . . . . . . . . . . 14  |-  F/_ p
( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )
129128sigaclcuni 28942 . . . . . . . . . . . . 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 1265 . . . . . . . . . . . 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 2511 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( `' ( F  oF  .+  G ) " {
z } )  e. 
dom  M )
132131ralrimiva 2840 . . . . . . . . . 10  |-  ( ph  ->  A. z  e.  ran  ( F  oF  .+  G ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
133 ssralv 3526 . . . . . . . . . 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 66 . . . . . . . . 9  |-  ( ph  ->  A. z  e.  ( b  i^i  ran  ( F  oF  .+  G
) ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
135134adantr 467 . . . . . . . 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 5746 . . . . . . . . . . . . . 14  |-  (  .+  : ( B  X.  B ) --> C  ->  Fun  .+  )
1371, 136syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  Fun  .+  )
138 imafi 7871 . . . . . . . . . . . . 13  |-  ( ( Fun  .+  /\  ( ran  F  X.  ran  G
)  e.  Fin )  ->  (  .+  " ( ran  F  X.  ran  G
) )  e.  Fin )
139137, 117, 138syl2anc 666 . . . . . . . . . . . 12  |-  ( ph  ->  (  .+  " ( ran  F  X.  ran  G
) )  e.  Fin )
14018, 20, 9, 23ofrn2 28237 . . . . . . . . . . . 12  |-  ( ph  ->  ran  ( F  oF  .+  G )  C_  (  .+  " ( ran 
F  X.  ran  G
) ) )
141 ssfi 7796 . . . . . . . . . . . 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 666 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( F  oF  .+  G )  e. 
Fin )
143 ssdomg 7620 . . . . . . . . . . 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 22 . . . . . . . . . 10  |-  ( ph  ->  ( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  ran  ( F  oF  .+  G ) )
145 isfinite 8161 . . . . . . . . . . . 12  |-  ( ran  ( F  oF  .+  G )  e. 
Fin 
<->  ran  ( F  oF  .+  G )  ~<  om )
146142, 145sylib 200 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( F  oF  .+  G )  ~<  om )
147 sdomdom 7602 . . . . . . . . . . 11  |-  ( ran  ( F  oF  .+  G )  ~<  om  ->  ran  ( F  oF  .+  G )  ~<_  om )
148146, 147syl 17 . . . . . . . . . 10  |-  ( ph  ->  ran  ( F  oF  .+  G )  ~<_  om )
149 domtr 7627 . . . . . . . . . 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 666 . . . . . . . . 9  |-  ( ph  ->  ( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  om )
151150adantr 467 . . . . . . . 8  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  om )
152 nfcv 2585 . . . . . . . . 9  |-  F/_ z
( b  i^i  ran  ( F  oF  .+  G ) )
153152sigaclcuni 28942 . . . . . . . 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 1265 . . . . . . 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 2516 . . . . . 6  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " ( b  i^i  ran  ( F  oF  .+  G ) ) )  e.  dom  M )
156 difpreima 6021 . . . . . . . . . 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 5206 . . . . . . . . . . 11  |-  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) )  =  dom  ( F  oF  .+  G
)
159158difeq2i 3581 . . . . . . . . . 10  |-  ( ( `' ( F  oF  .+  G ) "
b )  \  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) )  =  ( ( `' ( F  oF  .+  G ) "
b )  \  dom  ( F  oF  .+  G ) )
160 cnvimass 5205 . . . . . . . . . . 11  |-  ( `' ( F  oF  .+  G ) "
b )  C_  dom  ( F  oF  .+  G )
161 ssdif0 3852 . . . . . . . . . . 11  |-  ( ( `' ( F  oF  .+  G ) "
b )  C_  dom  ( F  oF  .+  G )  <->  ( ( `' ( F  oF  .+  G ) "
b )  \  dom  ( F  oF  .+  G ) )  =  (/) )
162160, 161mpbi 212 . . . . . . . . . 10  |-  ( ( `' ( F  oF  .+  G ) "
b )  \  dom  ( F  oF  .+  G ) )  =  (/)
163159, 162eqtri 2452 . . . . . . . . 9  |-  ( ( `' ( F  oF  .+  G ) "
b )  \  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) )  =  (/)
164157, 163syl6eq 2480 . . . . . . . 8  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  =  (/) )
165 0elsiga 28938 . . . . . . . . 9  |-  ( dom 
M  e.  U. ran sigAlgebra  ->  (/)  e.  dom  M )
16616, 50, 1653syl 18 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  dom  M )
167164, 166eqeltrd 2511 . . . . . . 7  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  e.  dom  M )
168167adantr 467 . . . . . 6  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  e.  dom  M )
169 unelsiga 28958 . . . . . 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 1265 . . . . 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 2511 . . . 4  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " b )  e.  dom  M )
172171ralrimiva 2840 . . 3  |-  ( ph  ->  A. b  e.  (sigaGen `  ( TopOpen `  K )
) ( `' ( F  oF  .+  G ) " b
)  e.  dom  M
)
17351, 38ismbfm 29076 . . 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 931 . 2  |-  ( ph  ->  ( F  oF  .+  G )  e.  ( dom  MMblFnM (sigaGen `  ( TopOpen `  K )
) ) )
17564adantr 467 . . . . . . 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 5883 . . . . . 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 29026 . . . . . . . . 9  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )
17816, 177sylib 200 . . . . . . . 8  |-  ( ph  ->  M  e.  (measures `  dom  M ) )
179178adantr 467 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  M  e.  (measures `  dom  M ) )
180 eldifi 3588 . . . . . . . 8  |-  ( z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } )  ->  z  e.  ran  ( F  oF  .+  G ) )
181180, 113sylan2 477 . . . . . . 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 467 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  ~<_  om )
183 sneq 4007 . . . . . . . . . . 11  |-  ( x  =  ( 1st `  p
)  ->  { x }  =  { ( 1st `  p ) } )
184183imaeq2d 5185 . . . . . . . . . 10  |-  ( x  =  ( 1st `  p
)  ->  ( `' F " { x }
)  =  ( `' F " { ( 1st `  p ) } ) )
185 sneq 4007 . . . . . . . . . . 11  |-  ( y  =  ( 2nd `  p
)  ->  { y }  =  { ( 2nd `  p ) } )
186185imaeq2d 5185 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  p
)  ->  ( `' G " { y } )  =  ( `' G " { ( 2nd `  p ) } ) )
187 ffun 5746 . . . . . . . . . . . 12  |-  ( F : U. dom  M --> U. J  ->  Fun  F
)
18818, 187syl 17 . . . . . . . . . . 11  |-  ( ph  ->  Fun  F )
189 sndisj 4413 . . . . . . . . . . 11  |- Disj  x  e. 
ran  F { x }
190 disjpreima 28190 . . . . . . . . . . 11  |-  ( ( Fun  F  /\ Disj  x  e. 
ran  F { x } )  -> Disj  x  e. 
ran  F ( `' F " { x } ) )
191188, 189, 190sylancl 667 . . . . . . . . . 10  |-  ( ph  -> Disj  x  e.  ran  F ( `' F " { x } ) )
192 ffun 5746 . . . . . . . . . . . 12  |-  ( G : U. dom  M --> U. J  ->  Fun  G
)
19320, 192syl 17 . . . . . . . . . . 11  |-  ( ph  ->  Fun  G )
194 sndisj 4413 . . . . . . . . . . 11  |- Disj  y  e. 
ran  G { y }
195 disjpreima 28190 . . . . . . . . . . 11  |-  ( ( Fun  G  /\ Disj  y  e. 
ran  G { y } )  -> Disj  y  e. 
ran  G ( `' G " { y } ) )
196193, 194, 195sylancl 667 . . . . . . . . . 10  |-  ( ph  -> Disj  y  e.  ran  G ( `' G " { y } ) )
197184, 186, 191, 196disjxpin 28194 . . . . . . . . 9  |-  ( ph  -> Disj  p  e.  ( ran  F  X.  ran  G ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )
198 disjss1 4398 . . . . . . . . 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 66 . . . . . . . 8  |-  ( ph  -> Disj  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
200199adantr 467 . . . . . . 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 29038 . . . . . . 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 1269 . . . . . 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 7796 . . . . . . . . 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 667 . . . . . . . 8  |-  ( ph  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  e.  Fin )
205204adantr 467 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  e.  Fin )
206 simpll 759 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ph )
207 simpr 463 . . . . . . . . . 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 3463 . . . . . . . . 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 6835 . . . . . . . . 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 6836 . . . . . . . . 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 6312 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  .0.  /\  y  =  .0.  )  ->  ( x  .+  y
)  =  (  .0.  .+  .0.  ) )
214 sibfof.5 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  (  .0.  .+  .0.  )  =  ( 0g `  K ) )
215213, 214sylan9eqr 2486 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  =  .0.  /\  y  =  .0.  ) )  -> 
( x  .+  y
)  =  ( 0g
`  K ) )
216215ex 436 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( x  =  .0.  /\  y  =  .0.  )  ->  (
x  .+  y )  =  ( 0g `  K ) ) )
217216necon3ad 2635 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( x  .+  y )  =/=  ( 0g `  K )  ->  -.  ( x  =  .0. 
/\  y  =  .0.  ) ) )
218 neorian 2752 . . . . . . . . . . . . 13  |-  ( ( x  =/=  .0.  \/  y  =/=  .0.  )  <->  -.  (
x  =  .0.  /\  y  =  .0.  )
)
219217, 218syl6ibr 231 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( x  .+  y )  =/=  ( 0g `  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) ) )
220219adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( x  .+  y )  =/=  ( 0g `  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) ) )
221220ralrimivva 2847 . . . . . . . . . 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 3465 . . . . . . . . . . . 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 6016 . . . . . . . . . . . . 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 214 . . . . . . . . . . 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 463 . . . . . . . . . . . 12  |-  ( ( p  e.  ( B  X.  B )  /\  (  .+  `  p )  =  z )  -> 
(  .+  `  p )  =  z )
229 1st2nd2 6842 . . . . . . . . . . . . . . 15  |-  ( p  e.  ( B  X.  B )  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
230229fveq2d 5883 . . . . . . . . . . . . . 14  |-  ( p  e.  ( B  X.  B )  ->  (  .+  `  p )  =  (  .+  `  <. ( 1st `  p ) ,  ( 2nd `  p
) >. ) )
231 df-ov 6306 . . . . . . . . . . . . . 14  |-  ( ( 1st `  p ) 
.+  ( 2nd `  p
) )  =  ( 
.+  `  <. ( 1st `  p ) ,  ( 2nd `  p )
>. )
232230, 231syl6eqr 2482 . . . . . . . . . . . . 13  |-  ( p  e.  ( B  X.  B )  ->  (  .+  `  p )  =  ( ( 1st `  p
)  .+  ( 2nd `  p ) ) )
233232adantr 467 . . . . . . . . . . . 12  |-  ( ( p  e.  ( B  X.  B )  /\  (  .+  `  p )  =  z )  -> 
(  .+  `  p )  =  ( ( 1st `  p )  .+  ( 2nd `  p ) ) )
234228, 233eqtr3d 2466 . . . . . . . . . . 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 761 . . . . . . . . . . . 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 3450 . . . . . . . . . . 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 4011 . . . . . . . . . . . 12  |-  ( z  e.  { ( 0g
`  K ) }  <-> 
z  =  ( 0g
`  K ) )
239238necon3bbii 2686 . . . . . . . . . . 11  |-  ( -.  z  e.  { ( 0g `  K ) }  <->  z  =/=  ( 0g `  K ) )
240237, 239sylib 200 . . . . . . . . . 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 2719 . . . . . . . . 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 656 . . . . . . . . . . 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 6310 . . . . . . . . . . . . 13  |-  ( x  =  ( 1st `  p
)  ->  ( x  .+  y )  =  ( ( 1st `  p
)  .+  y )
)
246245neeq1d 2702 . . . . . . . . . . . 12  |-  ( x  =  ( 1st `  p
)  ->  ( (
x  .+  y )  =/=  ( 0g `  K
)  <->  ( ( 1st `  p )  .+  y
)  =/=  ( 0g
`  K ) ) )
247 neeq1 2706 . . . . . . . . . . . . 13  |-  ( x  =  ( 1st `  p
)  ->  ( x  =/=  .0.  <->  ( 1st `  p
)  =/=  .0.  )
)
248247orbi1d 708 . . . . . . . . . . . 12  |-  ( x  =  ( 1st `  p
)  ->  ( (
x  =/=  .0.  \/  y  =/=  .0.  )  <->  ( ( 1st `  p )  =/= 
.0.  \/  y  =/=  .0.  ) ) )
249246, 248imbi12d 322 . . . . . . . . . . 11  |-  ( x  =  ( 1st `  p
)  ->  ( (
( x  .+  y
)  =/=  ( 0g
`  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) )  <->  ( (
( 1st `  p
)  .+  y )  =/=  ( 0g `  K
)  ->  ( ( 1st `  p )  =/= 
.0.  \/  y  =/=  .0.  ) ) ) )
250 oveq2 6311 . . . . . . . . . . . . 13  |-  ( y  =  ( 2nd `  p
)  ->  ( ( 1st `  p )  .+  y )  =  ( ( 1st `  p
)  .+  ( 2nd `  p ) ) )
251250neeq1d 2702 . . . . . . . . . . . 12  |-  ( y  =  ( 2nd `  p
)  ->  ( (
( 1st `  p
)  .+  y )  =/=  ( 0g `  K
)  <->  ( ( 1st `  p )  .+  ( 2nd `  p ) )  =/=  ( 0g `  K ) ) )
252 neeq1 2706 . . . . . . . . . . . . 13  |-  ( y  =  ( 2nd `  p
)  ->  ( y  =/=  .0.  <->  ( 2nd `  p
)  =/=  .0.  )
)
253252orbi2d 707 . . . . . . . . . . . 12  |-  ( y  =  ( 2nd `  p
)  ->  ( (
( 1st `  p
)  =/=  .0.  \/  y  =/=  .0.  )  <->  ( ( 1st `  p )  =/= 
.0.  \/  ( 2nd `  p )  =/=  .0.  ) ) )
254251, 253imbi12d 322 . . . . . . . . . . 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 3192 . . . . . . . . . 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 666 . . . . . . . . 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 47 . . . . . . . 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 29174 . . . . . . . 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 1268 . . . . . . 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 28898 . . . . . 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 2468 . . . . 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 11742 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR
263262, 259sseldi 3463 . . . . . 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 13793 . . . . 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 2511 . . . 4  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( M `  ( `' ( F  oF  .+  G
) " { z } ) )  e.  RR )
266179adantr 467 . . . . . . 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 656 . . . . . . 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 29031 . . . . . . 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 666 . . . . . 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 13848 . . . . 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 4448 . . . 4  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  0  <_  ( M `  ( `' ( F  oF  .+  G ) " { z } ) ) )
272 elrege0 11740 . . . 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 669 . . 3  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( M `  ( `' ( F  oF  .+  G
) " { z } ) )  e.  ( 0 [,) +oo ) )
274273ralrimiva 2840 . 2  |-  ( ph  ->  A. z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) ( M `  ( `' ( F  oF  .+  G ) " { z } ) )  e.  ( 0 [,) +oo ) )
275 eqid 2423 . . 3  |-  (sigaGen `  ( TopOpen
`  K ) )  =  (sigaGen `  ( TopOpen
`  K ) )
276 eqid 2423 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
277 eqid 2423 . . 3  |-  ( .s
`  K )  =  ( .s `  K
)
278 eqid 2423 . . 3  |-  (RRHom `  (Scalar `  K ) )  =  (RRHom `  (Scalar `  K ) )
27927, 28, 275, 276, 277, 278, 26, 16issibf 29168 . 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 1189 1  |-  ( ph  ->  ( F  oF  .+  G )  e. 
dom  ( Ksitg M
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1438    e. wcel 1869    =/= wne 2619   A.wral 2776   _Vcvv 3082    \ cdif 3434    u. cun 3435    i^i cin 3436    C_ wss 3437   (/)c0 3762   {csn 3997   <.cop 4003   U.cuni 4217   U_ciun 4297  Disj wdisj 4392   class class class wbr 4421    X. cxp 4849   `'ccnv 4850   dom cdm 4851   ran crn 4852   "cima 4854   Fun wfun 5593    Fn wfn 5594   -->wf 5595   ` cfv 5599  (class class class)co 6303    oFcof 6541   omcom 6704   1stc1st 6803   2ndc2nd 6804    ^m cmap 7478    ~<_ cdom 7573    ~< csdm 7574   Fincfn 7575   RRcr 9540   0cc0 9541   +oocpnf 9674    <_ cle 9678   [,)cico 11639   sum_csu 13745   Basecbs 15114  Scalarcsca 15186   .scvsca 15187   TopOpenctopn 15313   0gc0g 15331   Topctop 19909   TopSpctps 19911   Clsdccld 20023   Frect1 20315  RRHomcrrh 28799  Σ*cesum 28850  sigAlgebracsiga 28931  sigaGencsigagen 28962  measurescmeas 29019  MblFnMcmbfm 29074  sitgcsitg 29164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-ac2 8895  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619  ax-addf 9620  ax-mulf 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-disj 4393  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-ixp 7529  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-fi 7929  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-acn 8379  df-ac 8549  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-q 11267  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-ioo 11641  df-ioc 11642  df-ico 11643  df-icc 11644  df-fz 11787  df-fzo 11918  df-fl 12029  df-mod 12098  df-seq 12215  df-exp 12274  df-fac 12461  df-bc 12489  df-hash 12517  df-shft 13124  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117  df-pi 14119  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-ordt 15392  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-ps 16439  df-tsr 16440  df-plusf 16480  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-mhm 16575  df-submnd 16576  df-grp 16666  df-minusg 16667  df-sbg 16668  df-mulg 16669  df-subg 16807  df-cntz 16964  df-cmn 17425  df-abl 17426  df-mgp 17717  df-ur 17729  df-ring 17775  df-cring 17776  df-subrg 17999  df-abv 18038  df-lmod 18086  df-scaf 18087  df-sra 18388  df-rgmod 18389  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19913  df-bases 19914  df-topon 19915  df-topsp 19916  df-cld 20026  df-ntr 20027  df-cls 20028  df-nei 20106  df-lp 20144  df-perf 20145  df-cn 20235  df-cnp 20236  df-t1 20322  df-haus 20323  df-tx 20569  df-hmeo 20762  df-fil 20853  df-fm 20945  df-flim 20946  df-flf 20947  df-tmd 21079  df-tgp 21080  df-tsms 21133  df-trg 21166  df-xms 21327  df-ms 21328  df-tms 21329  df-nm 21589  df-ngp 21590  df-nrg 21592  df-nlm 21593  df-ii 21901  df-cncf 21902  df-limc 22813  df-dv 22814  df-log 23498  df-esum 28851  df-siga 28932  df-sigagen 28963  df-meas 29020  df-mbfm 29075  df-sitg 29165
This theorem is referenced by:  sitmcl  29186
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