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Theorem sibfof 27950
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 . . . . . . . . . 10  |-  ( ph  ->  .+  : ( B  X.  B ) --> C )
2 sibfof.0 . . . . . . . . . . . . 13  |-  ( ph  ->  W  e.  TopSp )
3 sitgval.b . . . . . . . . . . . . . 14  |-  B  =  ( Base `  W
)
4 sitgval.j . . . . . . . . . . . . . 14  |-  J  =  ( TopOpen `  W )
53, 4tpsuni 19234 . . . . . . . . . . . . 13  |-  ( W  e.  TopSp  ->  B  =  U. J )
62, 5syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  B  =  U. J
)
76, 6xpeq12d 5024 . . . . . . . . . . 11  |-  ( ph  ->  ( B  X.  B
)  =  ( U. J  X.  U. J ) )
87feq2d 5718 . . . . . . . . . 10  |-  ( ph  ->  (  .+  : ( B  X.  B ) --> C  <->  .+  : ( U. J  X.  U. J ) --> C ) )
91, 8mpbid 210 . . . . . . . . 9  |-  ( ph  ->  .+  : ( U. J  X.  U. J ) --> C )
109fovrnda 6430 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  U. J  /\  x  e.  U. J ) )  ->  ( z  .+  x )  e.  C
)
11 sitgval.s . . . . . . . . 9  |-  S  =  (sigaGen `  J )
12 sitgval.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  W )
13 sitgval.x . . . . . . . . 9  |-  .x.  =  ( .s `  W )
14 sitgval.h . . . . . . . . 9  |-  H  =  (RRHom `  (Scalar `  W
) )
15 sitgval.1 . . . . . . . . 9  |-  ( ph  ->  W  e.  V )
16 sitgval.2 . . . . . . . . 9  |-  ( ph  ->  M  e.  U. ran measures )
17 sibfmbl.1 . . . . . . . . 9  |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )
183, 4, 11, 12, 13, 14, 15, 16, 17sibff 27946 . . . . . . . 8  |-  ( ph  ->  F : U. dom  M --> U. J )
19 sibfof.2 . . . . . . . . 9  |-  ( ph  ->  G  e.  dom  ( Wsitg M ) )
203, 4, 11, 12, 13, 14, 15, 16, 19sibff 27946 . . . . . . . 8  |-  ( ph  ->  G : U. dom  M --> U. J )
21 dmexg 6715 . . . . . . . . 9  |-  ( M  e.  U. ran measures  ->  dom  M  e.  _V )
22 uniexg 6581 . . . . . . . . 9  |-  ( dom 
M  e.  _V  ->  U.
dom  M  e.  _V )
2316, 21, 223syl 20 . . . . . . . 8  |-  ( ph  ->  U. dom  M  e. 
_V )
24 inidm 3707 . . . . . . . 8  |-  ( U. dom  M  i^i  U. dom  M )  =  U. dom  M
2510, 18, 20, 23, 23, 24off 6538 . . . . . . 7  |-  ( ph  ->  ( F  oF  .+  G ) : U. dom  M --> C )
26 sibfof.3 . . . . . . . . . 10  |-  ( ph  ->  K  e.  TopSp )
27 sibfof.c . . . . . . . . . . 11  |-  C  =  ( Base `  K
)
28 eqid 2467 . . . . . . . . . . 11  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
2927, 28tpsuni 19234 . . . . . . . . . 10  |-  ( K  e.  TopSp  ->  C  =  U. ( TopOpen `  K )
)
3026, 29syl 16 . . . . . . . . 9  |-  ( ph  ->  C  =  U. ( TopOpen
`  K ) )
31 eqid 2467 . . . . . . . . . . 11  |-  (sigaGen `  ( TopOpen
`  K ) )  =  (sigaGen `  ( TopOpen
`  K ) )
3231unieqi 4254 . . . . . . . . . 10  |-  U. (sigaGen `  ( TopOpen `  K )
)  =  U. (sigaGen `  ( TopOpen `  K )
)
33 fvex 5876 . . . . . . . . . . 11  |-  ( TopOpen `  K )  e.  _V
34 unisg 27811 . . . . . . . . . . 11  |-  ( (
TopOpen `  K )  e. 
_V  ->  U. (sigaGen `  ( TopOpen
`  K ) )  =  U. ( TopOpen `  K ) )
3533, 34ax-mp 5 . . . . . . . . . 10  |-  U. (sigaGen `  ( TopOpen `  K )
)  =  U. ( TopOpen
`  K )
3632, 35eqtri 2496 . . . . . . . . 9  |-  U. (sigaGen `  ( TopOpen `  K )
)  =  U. ( TopOpen
`  K )
3730, 36syl6eqr 2526 . . . . . . . 8  |-  ( ph  ->  C  =  U. (sigaGen `  ( TopOpen `  K )
) )
38 feq3 5715 . . . . . . . 8  |-  ( C  =  U. (sigaGen `  ( TopOpen
`  K ) )  ->  ( ( F  oF  .+  G
) : U. dom  M --> C  <->  ( F  oF  .+  G ) : U. dom  M --> U. (sigaGen `  ( TopOpen `  K )
) ) )
3937, 38syl 16 . . . . . . 7  |-  ( ph  ->  ( ( F  oF  .+  G ) : U. dom  M --> C  <->  ( F  oF  .+  G ) : U. dom  M --> U. (sigaGen `  ( TopOpen `  K
) ) ) )
4025, 39mpbid 210 . . . . . 6  |-  ( ph  ->  ( F  oF  .+  G ) : U. dom  M --> U. (sigaGen `  ( TopOpen `  K )
) )
4133a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( TopOpen `  K )  e.  _V )
4241sgsiga 27810 . . . . . . . . 9  |-  ( ph  ->  (sigaGen `  ( TopOpen `  K
) )  e.  U. ran sigAlgebra )
4331, 42syl5eqel 2559 . . . . . . . 8  |-  ( ph  ->  (sigaGen `  ( TopOpen `  K
) )  e.  U. ran sigAlgebra )
44 uniexg 6581 . . . . . . . 8  |-  ( (sigaGen `  ( TopOpen `  K )
)  e.  U. ran sigAlgebra  ->  U. (sigaGen `  ( TopOpen `  K
) )  e.  _V )
4543, 44syl 16 . . . . . . 7  |-  ( ph  ->  U. (sigaGen `  ( TopOpen
`  K ) )  e.  _V )
46 elmapg 7433 . . . . . . 7  |-  ( ( U. (sigaGen `  ( TopOpen
`  K ) )  e.  _V  /\  U. dom  M  e.  _V )  ->  ( ( F  oF  .+  G )  e.  ( U. (sigaGen `  ( TopOpen
`  K ) )  ^m  U. dom  M
)  <->  ( F  oF  .+  G ) : U. dom  M --> U. (sigaGen `  ( TopOpen `  K )
) ) )
4745, 23, 46syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( F  oF  .+  G )  e.  ( U. (sigaGen `  ( TopOpen
`  K ) )  ^m  U. dom  M
)  <->  ( F  oF  .+  G ) : U. dom  M --> U. (sigaGen `  ( TopOpen `  K )
) ) )
4840, 47mpbird 232 . . . . 5  |-  ( ph  ->  ( F  oF  .+  G )  e.  ( U. (sigaGen `  ( TopOpen
`  K ) )  ^m  U. dom  M
) )
49 inundif 3905 . . . . . . . . 9  |-  ( ( b  i^i  ran  ( F  oF  .+  G
) )  u.  (
b  \  ran  ( F  oF  .+  G
) ) )  =  b
5049imaeq2i 5335 . . . . . . . 8  |-  ( `' ( F  oF  .+  G ) "
( ( b  i^i 
ran  ( F  oF  .+  G ) )  u.  ( b  \  ran  ( F  oF  .+  G ) ) ) )  =  ( `' ( F  oF  .+  G ) "
b )
51 ffun 5733 . . . . . . . . . 10  |-  ( ( F  oF  .+  G ) : U. dom  M --> C  ->  Fun  ( F  oF  .+  G ) )
52 unpreima 6007 . . . . . . . . . 10  |-  ( 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
) ) ) ) )
5325, 51, 523syl 20 . . . . . . . . 9  |-  ( 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
) ) ) ) )
5453adantr 465 . . . . . . . 8  |-  ( (
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
) ) ) ) )
5550, 54syl5eqr 2522 . . . . . . 7  |-  ( (
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
) ) ) ) )
56 dmmeas 27840 . . . . . . . . . 10  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
5716, 56syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  M  e.  U. ran sigAlgebra )
5857adantr 465 . . . . . . . 8  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  ->  dom  M  e.  U. ran sigAlgebra )
59 imaiun 6145 . . . . . . . . . 10  |-  ( `' ( 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 } )
60 iunid 4380 . . . . . . . . . . 11  |-  U_ z  e.  ( b  i^i  ran  ( F  oF  .+  G ) ) { z }  =  ( b  i^i  ran  ( F  oF  .+  G
) )
6160imaeq2i 5335 . . . . . . . . . 10  |-  ( `' ( F  oF  .+  G ) " U_ z  e.  (
b  i^i  ran  ( F  oF  .+  G
) ) { z } )  =  ( `' ( F  oF  .+  G ) "
( b  i^i  ran  ( F  oF  .+  G ) ) )
6259, 61eqtr3i 2498 . . . . . . . . 9  |-  U_ z  e.  ( b  i^i  ran  ( F  oF  .+  G ) ) ( `' ( F  oF  .+  G ) " { z } )  =  ( `' ( F  oF  .+  G ) " (
b  i^i  ran  ( F  oF  .+  G
) ) )
63 inss2 3719 . . . . . . . . . . . 12  |-  ( b  i^i  ran  ( F  oF  .+  G ) )  C_  ran  ( F  oF  .+  G
)
64 feq3 5715 . . . . . . . . . . . . . . . . . 18  |-  ( B  =  U. J  -> 
( F : U. dom  M --> B  <->  F : U. dom  M --> U. J
) )
656, 64syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( F : U. dom  M --> B  <->  F : U. dom  M --> U. J
) )
6618, 65mpbird 232 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F : U. dom  M --> B )
67 feq3 5715 . . . . . . . . . . . . . . . . . 18  |-  ( B  =  U. J  -> 
( G : U. dom  M --> B  <->  G : U. dom  M --> U. J
) )
686, 67syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( G : U. dom  M --> B  <->  G : U. dom  M --> U. J
) )
6920, 68mpbird 232 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G : U. dom  M --> B )
70 ffn 5731 . . . . . . . . . . . . . . . . 17  |-  (  .+  : ( B  X.  B ) --> C  ->  .+  Fn  ( B  X.  B ) )
711, 70syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  .+  Fn  ( B  X.  B ) )
7266, 69, 23, 71ofpreima2 27208 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " { z } )  =  U_ p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
7372adantr 465 . . . . . . . . . . . . . 14  |-  ( (
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
) } ) ) )
7457adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  dom  M  e.  U.
ran sigAlgebra )
7557ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  dom  M  e.  U.
ran sigAlgebra )
76 simpll 753 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ph )
77 inss1 3718 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  C_  ( `'  .+  " { z } )
78 cnvimass 5357 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( `' 
.+  " { z } )  C_  dom  .+
79 fdm 5735 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (  .+  : ( B  X.  B ) --> C  ->  dom  .+  =  ( B  X.  B ) )
801, 79syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  dom  .+  =  ( B  X.  B ) )
8178, 80syl5sseq 3552 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( `'  .+  " {
z } )  C_  ( B  X.  B
) )
8281adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( `'  .+  " { z } ) 
C_  ( B  X.  B ) )
8377, 82syl5ss 3515 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( ( `' 
.+  " { z } )  i^i  ( ran 
F  X.  ran  G
) )  C_  ( B  X.  B ) )
8483sselda 3504 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  p  e.  ( B  X.  B ) )
8557adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  dom  M  e.  U. ran sigAlgebra )
86 fvex 5876 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( TopOpen `  W )  e.  _V
874, 86eqeltri 2551 . . . . . . . . . . . . . . . . . . . . . . 23  |-  J  e. 
_V
8887a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  J  e.  _V )
8988sgsiga 27810 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
9011, 89syl5eqel 2559 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
9190adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  S  e.  U. ran sigAlgebra )
923, 4, 11, 12, 13, 14, 15, 16, 17sibfmbl 27945 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
9392adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  F  e.  ( dom  MMblFnM S
) )
944tpstop 19235 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( W  e.  TopSp  ->  J  e.  Top )
95 cldssbrsiga 27826 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( J  e.  Top  ->  ( Clsd `  J )  C_  (sigaGen `  J ) )
962, 94, 953syl 20 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( Clsd `  J
)  C_  (sigaGen `  J
) )
9796adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( Clsd `  J )  C_  (sigaGen `  J ) )
98 sibfof.4 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  J  e.  Fre )
9998adantr 465 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  J  e.  Fre )
100 xp1st 6814 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( p  e.  ( B  X.  B )  ->  ( 1st `  p )  e.  B )
101100adantl 466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 1st `  p )  e.  B )
1026adantr 465 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  B  =  U. J )
103101, 102eleqtrd 2557 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 1st `  p )  e. 
U. J )
104 eqid 2467 . . . . . . . . . . . . . . . . . . . . . . 23  |-  U. J  =  U. J
105104t1sncld 19621 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( J  e.  Fre  /\  ( 1st `  p )  e.  U. J )  ->  { ( 1st `  p ) }  e.  ( Clsd `  J )
)
10699, 103, 105syl2anc 661 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 1st `  p ) }  e.  ( Clsd `  J ) )
10797, 106sseldd 3505 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 1st `  p ) }  e.  (sigaGen `  J
) )
108107, 11syl6eleqr 2566 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 1st `  p ) }  e.  S )
10985, 91, 93, 108mbfmcnvima 27896 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( `' F " { ( 1st `  p ) } )  e.  dom  M )
11076, 84, 109syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( `' F " { ( 1st `  p
) } )  e. 
dom  M )
1113, 4, 11, 12, 13, 14, 15, 16, 19sibfmbl 27945 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  G  e.  ( dom 
MMblFnM S ) )
112111adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  G  e.  ( dom  MMblFnM S
) )
113 xp2nd 6815 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( p  e.  ( B  X.  B )  ->  ( 2nd `  p )  e.  B )
114113adantl 466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 2nd `  p )  e.  B )
115114, 102eleqtrd 2557 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 2nd `  p )  e. 
U. J )
116104t1sncld 19621 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( J  e.  Fre  /\  ( 2nd `  p )  e.  U. J )  ->  { ( 2nd `  p ) }  e.  ( Clsd `  J )
)
11799, 115, 116syl2anc 661 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 2nd `  p ) }  e.  ( Clsd `  J ) )
11897, 117sseldd 3505 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 2nd `  p ) }  e.  (sigaGen `  J
) )
119118, 11syl6eleqr 2566 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 2nd `  p ) }  e.  S )
12085, 91, 112, 119mbfmcnvima 27896 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( `' G " { ( 2nd `  p ) } )  e.  dom  M )
12176, 84, 120syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( `' G " { ( 2nd `  p
) } )  e. 
dom  M )
122 inelsiga 27803 . . . . . . . . . . . . . . . . 17  |-  ( ( 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 )
12375, 110, 121, 122syl3anc 1228 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 )
124123ralrimiva 2878 . . . . . . . . . . . . . . 15  |-  ( (
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 )
125 inss2 3719 . . . . . . . . . . . . . . . . . 18  |-  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  C_  ( ran  F  X.  ran  G
)
1263, 4, 11, 12, 13, 14, 15, 16, 17sibfrn 27947 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ran  F  e.  Fin )
1273, 4, 11, 12, 13, 14, 15, 16, 19sibfrn 27947 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ran  G  e.  Fin )
128 xpfi 7791 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
129126, 127, 128syl2anc 661 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
130 ssdomg 7561 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 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 ) ) )
131129, 130syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( ( `' 
.+  " { 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
) ) )
132125, 131mpi 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  ~<_  ( ran  F  X.  ran  G ) )
133 isfinite 8069 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ran  F  X.  ran  G )  e.  Fin  <->  ( ran  F  X.  ran  G ) 
~<  om )
134133biimpi 194 . . . . . . . . . . . . . . . . . 18  |-  ( ( ran  F  X.  ran  G )  e.  Fin  ->  ( ran  F  X.  ran  G )  ~<  om )
135 sdomdom 7543 . . . . . . . . . . . . . . . . . 18  |-  ( ( ran  F  X.  ran  G )  ~<  om  ->  ( ran  F  X.  ran  G )  ~<_  om )
136129, 134, 1353syl 20 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ran  F  X.  ran  G )  ~<_  om )
137 domtr 7568 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( `'  .+  " { 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 )
138132, 136, 137syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  ~<_  om )
139138adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( ( `' 
.+  " { z } )  i^i  ( ran 
F  X.  ran  G
) )  ~<_  om )
140 nfcv 2629 . . . . . . . . . . . . . . . 16  |-  F/_ p
( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )
141140sigaclcuni 27786 . . . . . . . . . . . . . . 15  |-  ( ( 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 )
14274, 124, 139, 141syl3anc 1228 . . . . . . . . . . . . . 14  |-  ( (
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 )
14373, 142eqeltrd 2555 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( `' ( F  oF  .+  G ) " {
z } )  e. 
dom  M )
144143ralrimiva 2878 . . . . . . . . . . . 12  |-  ( ph  ->  A. z  e.  ran  ( F  oF  .+  G ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
145 ssralv 3564 . . . . . . . . . . . 12  |-  ( ( 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 ) )
14663, 144, 145mpsyl 63 . . . . . . . . . . 11  |-  ( ph  ->  A. z  e.  ( b  i^i  ran  ( F  oF  .+  G
) ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
147146adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  ->  A. z  e.  (
b  i^i  ran  ( F  oF  .+  G
) ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
148 ffun 5733 . . . . . . . . . . . . . . . . 17  |-  (  .+  : ( B  X.  B ) --> C  ->  Fun  .+  )
1491, 148syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  Fun  .+  )
150 imafi 7813 . . . . . . . . . . . . . . . 16  |-  ( ( Fun  .+  /\  ( ran  F  X.  ran  G
)  e.  Fin )  ->  (  .+  " ( ran  F  X.  ran  G
) )  e.  Fin )
151149, 129, 150syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (  .+  " ( ran  F  X.  ran  G
) )  e.  Fin )
15218, 20, 9, 23ofrn2 27181 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  ( F  oF  .+  G )  C_  (  .+  " ( ran 
F  X.  ran  G
) ) )
153 ssfi 7740 . . . . . . . . . . . . . . 15  |-  ( ( (  .+  " ( ran  F  X.  ran  G
) )  e.  Fin  /\ 
ran  ( F  oF  .+  G )  C_  (  .+  " ( ran 
F  X.  ran  G
) ) )  ->  ran  ( F  oF  .+  G )  e. 
Fin )
154151, 152, 153syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  ( F  oF  .+  G )  e. 
Fin )
155 ssdomg 7561 . . . . . . . . . . . . . 14  |-  ( 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 ) ) )
156154, 155syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( b  i^i 
ran  ( F  oF  .+  G ) ) 
C_  ran  ( F  oF  .+  G )  ->  ( b  i^i 
ran  ( F  oF  .+  G ) )  ~<_  ran  ( F  oF  .+  G ) ) )
15763, 156mpi 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  ran  ( F  oF  .+  G ) )
158 isfinite 8069 . . . . . . . . . . . . . 14  |-  ( ran  ( F  oF  .+  G )  e. 
Fin 
<->  ran  ( F  oF  .+  G )  ~<  om )
159154, 158sylib 196 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  ( F  oF  .+  G )  ~<  om )
160 sdomdom 7543 . . . . . . . . . . . . 13  |-  ( ran  ( F  oF  .+  G )  ~<  om  ->  ran  ( F  oF  .+  G )  ~<_  om )
161159, 160syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ran  ( F  oF  .+  G )  ~<_  om )
162 domtr 7568 . . . . . . . . . . . 12  |-  ( ( ( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  ran  ( F  oF  .+  G )  /\  ran  ( F  oF  .+  G )  ~<_  om )  ->  ( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  om )
163157, 161, 162syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  om )
164163adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  om )
165 nfcv 2629 . . . . . . . . . . 11  |-  F/_ z
( b  i^i  ran  ( F  oF  .+  G ) )
166165sigaclcuni 27786 . . . . . . . . . 10  |-  ( ( 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 )
16758, 147, 164, 166syl3anc 1228 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  ->  U_ z  e.  (
b  i^i  ran  ( F  oF  .+  G
) ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
16862, 167syl5eqelr 2560 . . . . . . . 8  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " ( b  i^i  ran  ( F  oF  .+  G ) ) )  e.  dom  M )
169 difpreima 6009 . . . . . . . . . . . 12  |-  ( Fun  ( F  oF  .+  G )  -> 
( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  =  ( ( `' ( F  oF  .+  G
) " b ) 
\  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) ) )
17025, 51, 1693syl 20 . . . . . . . . . . 11  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  =  ( ( `' ( F  oF  .+  G
) " b ) 
\  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) ) )
171 cnvimarndm 5358 . . . . . . . . . . . . 13  |-  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) )  =  dom  ( F  oF  .+  G
)
172171difeq2i 3619 . . . . . . . . . . . 12  |-  ( ( `' ( F  oF  .+  G ) "
b )  \  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) )  =  ( ( `' ( F  oF  .+  G ) "
b )  \  dom  ( F  oF  .+  G ) )
173 cnvimass 5357 . . . . . . . . . . . . 13  |-  ( `' ( F  oF  .+  G ) "
b )  C_  dom  ( F  oF  .+  G )
174 ssdif0 3885 . . . . . . . . . . . . 13  |-  ( ( `' ( F  oF  .+  G ) "
b )  C_  dom  ( F  oF  .+  G )  <->  ( ( `' ( F  oF  .+  G ) "
b )  \  dom  ( F  oF  .+  G ) )  =  (/) )
175173, 174mpbi 208 . . . . . . . . . . . 12  |-  ( ( `' ( F  oF  .+  G ) "
b )  \  dom  ( F  oF  .+  G ) )  =  (/)
176172, 175eqtri 2496 . . . . . . . . . . 11  |-  ( ( `' ( F  oF  .+  G ) "
b )  \  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) )  =  (/)
177170, 176syl6eq 2524 . . . . . . . . . 10  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  =  (/) )
178 0elsiga 27782 . . . . . . . . . . 11  |-  ( dom 
M  e.  U. ran sigAlgebra  ->  (/)  e.  dom  M )
17957, 178syl 16 . . . . . . . . . 10  |-  ( ph  -> 
(/)  e.  dom  M )
180177, 179eqeltrd 2555 . . . . . . . . 9  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  e.  dom  M )
181180adantr 465 . . . . . . . 8  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  e.  dom  M )
182 unelsiga 27802 . . . . . . . 8  |-  ( ( 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 )
18358, 168, 181, 182syl3anc 1228 . . . . . . 7  |-  ( (
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 )
18455, 183eqeltrd 2555 . . . . . 6  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " b )  e.  dom  M )
185184ralrimiva 2878 . . . . 5  |-  ( ph  ->  A. b  e.  (sigaGen `  ( TopOpen `  K )
) ( `' ( F  oF  .+  G ) " b
)  e.  dom  M
)
18648, 185jca 532 . . . 4  |-  ( ph  ->  ( ( 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 ) )
18757, 43ismbfm 27891 . . . 4  |-  ( 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 ) ) )
188186, 187mpbird 232 . . 3  |-  ( ph  ->  ( F  oF  .+  G )  e.  ( dom  MMblFnM (sigaGen `  ( TopOpen `  K )
) ) )
18972adantr 465 . . . . . . . 8  |-  ( (
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
) } ) ) )
190189fveq2d 5870 . . . . . . 7  |-  ( (
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
) } ) ) ) )
191 measbasedom 27841 . . . . . . . . . 10  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )
19216, 191sylib 196 . . . . . . . . 9  |-  ( ph  ->  M  e.  (measures `  dom  M ) )
193192adantr 465 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  M  e.  (measures `  dom  M ) )
194 difss 3631 . . . . . . . . . 10  |-  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) 
C_  ran  ( F  oF  .+  G )
195194sseli 3500 . . . . . . . . 9  |-  ( z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } )  ->  z  e.  ran  ( F  oF  .+  G ) )
196195, 124sylan2 474 . . . . . . . 8  |-  ( (
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 )
197138adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  ~<_  om )
198 ffun 5733 . . . . . . . . . . . . . 14  |-  ( F : U. dom  M --> U. J  ->  Fun  F
)
19918, 198syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  Fun  F )
200 sndisj 4439 . . . . . . . . . . . . 13  |- Disj  x  e. 
ran  F { x }
201 disjpreima 27146 . . . . . . . . . . . . 13  |-  ( ( Fun  F  /\ Disj  x  e. 
ran  F { x } )  -> Disj  x  e. 
ran  F ( `' F " { x } ) )
202199, 200, 201sylancl 662 . . . . . . . . . . . 12  |-  ( ph  -> Disj  x  e.  ran  F ( `' F " { x } ) )
203 ffun 5733 . . . . . . . . . . . . . 14  |-  ( G : U. dom  M --> U. J  ->  Fun  G
)
20420, 203syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  Fun  G )
205 sndisj 4439 . . . . . . . . . . . . 13  |- Disj  y  e. 
ran  G { y }
206 disjpreima 27146 . . . . . . . . . . . . 13  |-  ( ( Fun  G  /\ Disj  y  e. 
ran  G { y } )  -> Disj  y  e. 
ran  G ( `' G " { y } ) )
207204, 205, 206sylancl 662 . . . . . . . . . . . 12  |-  ( ph  -> Disj  y  e.  ran  G ( `' G " { y } ) )
208 sneq 4037 . . . . . . . . . . . . . 14  |-  ( x  =  ( 1st `  p
)  ->  { x }  =  { ( 1st `  p ) } )
209208imaeq2d 5337 . . . . . . . . . . . . 13  |-  ( x  =  ( 1st `  p
)  ->  ( `' F " { x }
)  =  ( `' F " { ( 1st `  p ) } ) )
210 sneq 4037 . . . . . . . . . . . . . 14  |-  ( y  =  ( 2nd `  p
)  ->  { y }  =  { ( 2nd `  p ) } )
211210imaeq2d 5337 . . . . . . . . . . . . 13  |-  ( y  =  ( 2nd `  p
)  ->  ( `' G " { y } )  =  ( `' G " { ( 2nd `  p ) } ) )
212 simpl 457 . . . . . . . . . . . . 13  |-  ( (Disj  x  e.  ran  F ( `' F " { x } )  /\ Disj  y  e. 
ran  G ( `' G " { y } ) )  -> Disj  x  e.  ran  F ( `' F " { x } ) )
213 simpr 461 . . . . . . . . . . . . 13  |-  ( (Disj  x  e.  ran  F ( `' F " { x } )  /\ Disj  y  e. 
ran  G ( `' G " { y } ) )  -> Disj  y  e.  ran  G ( `' G " { y } ) )
214209, 211, 212, 213disjxpin 27148 . . . . . . . . . . . 12  |-  ( (Disj  x  e.  ran  F ( `' F " { x } )  /\ Disj  y  e. 
ran  G ( `' G " { y } ) )  -> Disj  p  e.  ( ran  F  X.  ran  G ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )
215202, 207, 214syl2anc 661 . . . . . . . . . . 11  |-  ( ph  -> Disj  p  e.  ( ran  F  X.  ran  G ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )
216 disjss1 4423 . . . . . . . . . . 11  |-  ( ( ( `'  .+  " {
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 ) } ) ) ) )
217125, 215, 216mpsyl 63 . . . . . . . . . 10  |-  ( ph  -> Disj  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
218217adantr 465 . . . . . . . . 9  |-  ( (
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
) } ) ) )
219197, 218jca 532 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( (
( `'  .+  " {
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 ) } ) ) ) )
220 measvuni 27853 . . . . . . . 8  |-  ( ( 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 ) } ) ) ) )
221193, 196, 219, 220syl3anc 1228 . . . . . . 7  |-  ( (
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 ) } ) ) ) )
222 ssfi 7740 . . . . . . . . . 10  |-  ( ( ( 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 )
223129, 125, 222sylancl 662 . . . . . . . . 9  |-  ( ph  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  e.  Fin )
224223adantr 465 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  e.  Fin )
225 rge0ssre 11628 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
226195, 76sylanl2 651 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ph )
227 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( 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 ) ) )
228125, 227sseldi 3502 . . . . . . . . . . . 12  |-  ( ( ( 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 ) )
229 xp1st 6814 . . . . . . . . . . . 12  |-  ( p  e.  ( ran  F  X.  ran  G )  -> 
( 1st `  p
)  e.  ran  F
)
230228, 229syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( 1st `  p )  e.  ran  F )
231 xp2nd 6815 . . . . . . . . . . . 12  |-  ( p  e.  ( ran  F  X.  ran  G )  -> 
( 2nd `  p
)  e.  ran  G
)
232228, 231syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( 2nd `  p )  e.  ran  G )
233 oveq12 6293 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =  .0.  /\  y  =  .0.  )  ->  ( x  .+  y
)  =  (  .0.  .+  .0.  ) )
234233adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( x  =  .0.  /\  y  =  .0.  ) )  -> 
( x  .+  y
)  =  (  .0.  .+  .0.  ) )
235 sibfof.5 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  (  .0.  .+  .0.  )  =  ( 0g `  K ) )
236235adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( x  =  .0.  /\  y  =  .0.  ) )  -> 
(  .0.  .+  .0.  )  =  ( 0g `  K ) )
237234, 236eqtrd 2508 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( x  =  .0.  /\  y  =  .0.  ) )  -> 
( x  .+  y
)  =  ( 0g
`  K ) )
238237ex 434 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( x  =  .0.  /\  y  =  .0.  )  ->  (
x  .+  y )  =  ( 0g `  K ) ) )
239238necon3ad 2677 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( x  .+  y )  =/=  ( 0g `  K )  ->  -.  ( x  =  .0. 
/\  y  =  .0.  ) ) )
240 oran 496 . . . . . . . . . . . . . . . . 17  |-  ( ( x  =/=  .0.  \/  y  =/=  .0.  )  <->  -.  ( -.  x  =/=  .0.  /\ 
-.  y  =/=  .0.  ) )
241 nne 2668 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  x  =/=  .0.  <->  x  =  .0.  )
242 nne 2668 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  y  =/=  .0.  <->  y  =  .0.  )
243241, 242anbi12i 697 . . . . . . . . . . . . . . . . . 18  |-  ( ( -.  x  =/=  .0.  /\ 
-.  y  =/=  .0.  ) 
<->  ( x  =  .0. 
/\  y  =  .0.  ) )
244243notbii 296 . . . . . . . . . . . . . . . . 17  |-  ( -.  ( -.  x  =/= 
.0.  /\  -.  y  =/=  .0.  )  <->  -.  (
x  =  .0.  /\  y  =  .0.  )
)
245240, 244bitri 249 . . . . . . . . . . . . . . . 16  |-  ( ( x  =/=  .0.  \/  y  =/=  .0.  )  <->  -.  (
x  =  .0.  /\  y  =  .0.  )
)
246239, 245syl6ibr 227 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( x  .+  y )  =/=  ( 0g `  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) ) )
247246adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( x  .+  y )  =/=  ( 0g `  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) ) )
248247ralrimivva 2885 . . . . . . . . . . . . 13  |-  ( ph  ->  A. x  e.  B  A. y  e.  B  ( ( x  .+  y )  =/=  ( 0g `  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) ) )
249226, 248syl 16 . . . . . . . . . . . 12  |-  ( ( ( 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.  ) ) )
25077a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  C_  ( `'  .+  " { z } ) )
251250sselda 3504 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  p  e.  ( `'  .+  " {
z } ) )
252226, 71syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  .+  Fn  ( B  X.  B ) )
253 fniniseg 6002 . . . . . . . . . . . . . . . 16  |-  (  .+  Fn  ( B  X.  B
)  ->  ( p  e.  ( `'  .+  " {
z } )  <->  ( p  e.  ( B  X.  B
)  /\  (  .+  `  p )  =  z ) ) )
254252, 253syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 ) ) )
255251, 254mpbid 210 . . . . . . . . . . . . . 14  |-  ( ( ( 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 ) )
256 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  ( B  X.  B )  /\  (  .+  `  p )  =  z )  -> 
(  .+  `  p )  =  z )
257 1st2nd2 6821 . . . . . . . . . . . . . . . . . 18  |-  ( p  e.  ( B  X.  B )  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
258257fveq2d 5870 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  ( B  X.  B )  ->  (  .+  `  p )  =  (  .+  `  <. ( 1st `  p ) ,  ( 2nd `  p
) >. ) )
259 df-ov 6287 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st `  p ) 
.+  ( 2nd `  p
) )  =  ( 
.+  `  <. ( 1st `  p ) ,  ( 2nd `  p )
>. )
260258, 259syl6eqr 2526 . . . . . . . . . . . . . . . 16  |-  ( p  e.  ( B  X.  B )  ->  (  .+  `  p )  =  ( ( 1st `  p
)  .+  ( 2nd `  p ) ) )
261260adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  ( B  X.  B )  /\  (  .+  `  p )  =  z )  -> 
(  .+  `  p )  =  ( ( 1st `  p )  .+  ( 2nd `  p ) ) )
262256, 261eqtr3d 2510 . . . . . . . . . . . . . 14  |-  ( ( p  e.  ( B  X.  B )  /\  (  .+  `  p )  =  z )  -> 
z  =  ( ( 1st `  p ) 
.+  ( 2nd `  p
) ) )
263255, 262syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  z  =  ( ( 1st `  p
)  .+  ( 2nd `  p ) ) )
264 simplr 754 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 ) } ) )
265264eldifbd 3489 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  -.  z  e.  { ( 0g `  K ) } )
266 elsn 4041 . . . . . . . . . . . . . . 15  |-  ( z  e.  { ( 0g
`  K ) }  <-> 
z  =  ( 0g
`  K ) )
267266necon3bbii 2728 . . . . . . . . . . . . . 14  |-  ( -.  z  e.  { ( 0g `  K ) }  <->  z  =/=  ( 0g `  K ) )
268265, 267sylib 196 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  z  =/=  ( 0g `  K ) )
269263, 268eqnetrrd 2761 . . . . . . . . . . . 12  |-  ( ( ( 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 ) )
270195, 84sylanl2 651 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  p  e.  ( B  X.  B
) )
271270, 100syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( 1st `  p )  e.  B
)
272270, 113syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ( 2nd `  p )  e.  B
)
273 oveq1 6291 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( 1st `  p
)  ->  ( x  .+  y )  =  ( ( 1st `  p
)  .+  y )
)
274273neeq1d 2744 . . . . . . . . . . . . . . 15  |-  ( x  =  ( 1st `  p
)  ->  ( (
x  .+  y )  =/=  ( 0g `  K
)  <->  ( ( 1st `  p )  .+  y
)  =/=  ( 0g
`  K ) ) )
275 neeq1 2748 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( 1st `  p
)  ->  ( x  =/=  .0.  <->  ( 1st `  p
)  =/=  .0.  )
)
276275orbi1d 702 . . . . . . . . . . . . . . 15  |-  ( x  =  ( 1st `  p
)  ->  ( (
x  =/=  .0.  \/  y  =/=  .0.  )  <->  ( ( 1st `  p )  =/= 
.0.  \/  y  =/=  .0.  ) ) )
277274, 276imbi12d 320 . . . . . . . . . . . . . 14  |-  ( x  =  ( 1st `  p
)  ->  ( (
( x  .+  y
)  =/=  ( 0g
`  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) )  <->  ( (
( 1st `  p
)  .+  y )  =/=  ( 0g `  K
)  ->  ( ( 1st `  p )  =/= 
.0.  \/  y  =/=  .0.  ) ) ) )
278 oveq2 6292 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( 2nd `  p
)  ->  ( ( 1st `  p )  .+  y )  =  ( ( 1st `  p
)  .+  ( 2nd `  p ) ) )
279278neeq1d 2744 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 2nd `  p
)  ->  ( (
( 1st `  p
)  .+  y )  =/=  ( 0g `  K
)  <->  ( ( 1st `  p )  .+  ( 2nd `  p ) )  =/=  ( 0g `  K ) ) )
280 neeq1 2748 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( 2nd `  p
)  ->  ( y  =/=  .0.  <->  ( 2nd `  p
)  =/=  .0.  )
)
281280orbi2d 701 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 2nd `  p
)  ->  ( (
( 1st `  p
)  =/=  .0.  \/  y  =/=  .0.  )  <->  ( ( 1st `  p )  =/= 
.0.  \/  ( 2nd `  p )  =/=  .0.  ) ) )
282279, 281imbi12d 320 . . . . . . . . . . . . . 14  |-  ( 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.  ) ) ) )
283277, 282rspc2v 3223 . . . . . . . . . . . . 13  |-  ( ( ( 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.  ) ) ) )
284271, 272, 283syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( 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.  ) ) ) )
285249, 269, 284mp2d 45 . . . . . . . . . . 11  |-  ( ( ( 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.  ) )
2863, 4, 11, 12, 13, 14, 15, 16, 17, 19, 2, 98sibfinima 27949 . . . . . . . . . . 11  |-  ( ( ( 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 ) )
287226, 230, 232, 285, 286syl31anc 1231 . . . . . . . . . 10  |-  ( ( ( 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 ) )
288225, 287sseldi 3502 . . . . . . . . 9  |-  ( ( ( 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 )
289193adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  M  e.  (measures `  dom  M ) )
290195, 123sylanl2 651 . . . . . . . . . 10  |-  ( ( ( 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 )
291 measge0 27846 . . . . . . . . . 10  |-  ( ( 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
) } ) ) ) )
292289, 290, 291syl2anc 661 . . . . . . . . 9  |-  ( ( ( 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 ) } ) ) ) )
293 elrege0 11627 . . . . . . . . 9  |-  ( ( M `  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )  e.  ( 0 [,) +oo )  <->  ( ( M `
 ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )  e.  RR  /\  0  <_  ( M `  (
( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) ) ) )
294288, 292, 293sylanbrc 664 . . . . . . . 8  |-  ( ( ( 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 ) )
295224, 294esumpfinval 27749 . . . . . . 7  |-  ( (
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 ) } ) ) ) )
296190, 221, 2953eqtrd 2512 . . . . . 6  |-  ( (
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 ) } ) ) ) )
297224, 288fsumrecl 13519 . . . . . 6  |-  ( (
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 )
298296, 297eqeltrd 2555 . . . . 5  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( M `  ( `' ( F  oF  .+  G
) " { z } ) )  e.  RR )
299224, 288, 292fsumge0 13572 . . . . . 6  |-  ( (
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 ) } ) ) ) )
300299, 296breqtrrd 4473 . . . . 5  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  0  <_  ( M `  ( `' ( F  oF  .+  G ) " { z } ) ) )
301 elrege0 11627 . . . . 5  |-  ( ( M `  ( `' ( F  oF  .+  G ) " { z } ) )  e.  ( 0 [,) +oo )  <->  ( ( M `  ( `' ( F  oF  .+  G ) " {
z } ) )  e.  RR  /\  0  <_  ( M `  ( `' ( F  oF  .+  G ) " { z } ) ) ) )
302298, 300, 301sylanbrc 664 . . . 4  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( M `  ( `' ( F  oF  .+  G
) " { z } ) )  e.  ( 0 [,) +oo ) )
303302ralrimiva 2878 . . 3  |-  ( ph  ->  A. z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) ( M `  ( `' ( F  oF  .+  G ) " { z } ) )  e.  ( 0 [,) +oo ) )
304188, 154, 3033jca 1176 . 2  |-  ( ph  ->  ( ( 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 ) ) )
305 eqid 2467 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
306 eqid 2467 . . 3  |-  ( .s
`  K )  =  ( .s `  K
)
307 eqid 2467 . . 3  |-  (RRHom `  (Scalar `  K ) )  =  (RRHom `  (Scalar `  K ) )
30827, 28, 31, 305, 306, 307, 26, 16issibf 27943 . 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 ) ) ) )
309304, 308mpbird 232 1  |-  ( ph  ->  ( F  oF  .+  G )  e. 
dom  ( Ksitg M
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   U.cuni 4245   U_ciun 4325  Disj wdisj 4417   class class class wbr 4447    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   Fun wfun 5582    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284    oFcof 6522   omcom 6684   1stc1st 6782   2ndc2nd 6783    ^m cmap 7420    ~<_ cdom 7514    ~< csdm 7515   Fincfn 7516   RRcr 9491   0cc0 9492   +oocpnf 9625    <_ cle 9629   [,)cico 11531   sum_csu 13471   Basecbs 14490  Scalarcsca 14558   .scvsca 14559   TopOpenctopn 14677   0gc0g 14695   Topctop 19189   TopSpctps 19192   Clsdccld 19311   Frect1 19602  RRHomcrrh 27638  Σ*cesum 27708  sigAlgebracsiga 27775  sigaGencsigagen 27806  measurescmeas 27834  MblFnMcmbfm 27889  sitgcsitg 27939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-ac2 8843  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-acn 8323  df-ac 8497  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-shft 12863  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-sum 13472  df-ef 13665  df-sin 13667  df-cos 13668  df-pi 13670  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-ordt 14756  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-ps 15687  df-tsr 15688  df-mnd 15732  df-plusf 15733  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-subg 16003  df-cntz 16160  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-cring 17003  df-subrg 17227  df-abv 17266  df-lmod 17314  df-scaf 17315  df-sra 17618  df-rgmod 17619  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cn 19522  df-cnp 19523  df-t1 19609  df-haus 19610  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-tmd 20334  df-tgp 20335  df-tsms 20388  df-trg 20425  df-xms 20586  df-ms 20587  df-tms 20588  df-nm 20866  df-ngp 20867  df-nrg 20869  df-nlm 20870  df-ii 21144  df-cncf 21145  df-limc 22033  df-dv 22034  df-log 22700  df-esum 27709  df-siga 27776  df-sigagen 27807  df-meas 27835  df-mbfm 27890  df-sitg 27940
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator