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Theorem sibfof 26640
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 18443 . . . . . . . . . . . . 13  |-  ( W  e.  TopSp  ->  B  =  U. J )
62, 5syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  B  =  U. J
)
76, 6xpeq12d 4861 . . . . . . . . . . 11  |-  ( ph  ->  ( B  X.  B
)  =  ( U. J  X.  U. J ) )
87feq2d 5544 . . . . . . . . . 10  |-  ( ph  ->  (  .+  : ( B  X.  B ) --> C  <->  .+  : ( U. J  X.  U. J ) --> C ) )
91, 8mpbid 210 . . . . . . . . 9  |-  ( ph  ->  .+  : ( U. J  X.  U. J ) --> C )
109fovrnda 6233 . . . . . . . 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 26636 . . . . . . . 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 26636 . . . . . . . 8  |-  ( ph  ->  G : U. dom  M --> U. J )
21 dmexg 6508 . . . . . . . . 9  |-  ( M  e.  U. ran measures  ->  dom  M  e.  _V )
22 uniexg 6376 . . . . . . . . 9  |-  ( dom 
M  e.  _V  ->  U.
dom  M  e.  _V )
2316, 21, 223syl 20 . . . . . . . 8  |-  ( ph  ->  U. dom  M  e. 
_V )
24 inidm 3556 . . . . . . . 8  |-  ( U. dom  M  i^i  U. dom  M )  =  U. dom  M
2510, 18, 20, 23, 23, 24off 6333 . . . . . . 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 2441 . . . . . . . . . . 11  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
2927, 28tpsuni 18443 . . . . . . . . . 10  |-  ( K  e.  TopSp  ->  C  =  U. ( TopOpen `  K )
)
3026, 29syl 16 . . . . . . . . 9  |-  ( ph  ->  C  =  U. ( TopOpen
`  K ) )
31 eqid 2441 . . . . . . . . . . 11  |-  (sigaGen `  ( TopOpen
`  K ) )  =  (sigaGen `  ( TopOpen
`  K ) )
3231unieqi 4097 . . . . . . . . . 10  |-  U. (sigaGen `  ( TopOpen `  K )
)  =  U. (sigaGen `  ( TopOpen `  K )
)
33 fvex 5698 . . . . . . . . . . 11  |-  ( TopOpen `  K )  e.  _V
34 unisg 26506 . . . . . . . . . . 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 2461 . . . . . . . . 9  |-  U. (sigaGen `  ( TopOpen `  K )
)  =  U. ( TopOpen
`  K )
3730, 36syl6eqr 2491 . . . . . . . 8  |-  ( ph  ->  C  =  U. (sigaGen `  ( TopOpen `  K )
) )
38 feq3 5541 . . . . . . . 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 26505 . . . . . . . . 9  |-  ( ph  ->  (sigaGen `  ( TopOpen `  K
) )  e.  U. ran sigAlgebra )
4331, 42syl5eqel 2525 . . . . . . . 8  |-  ( ph  ->  (sigaGen `  ( TopOpen `  K
) )  e.  U. ran sigAlgebra )
44 uniexg 6376 . . . . . . . 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 7223 . . . . . . 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 656 . . . . . 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 3754 . . . . . . . . 9  |-  ( ( b  i^i  ran  ( F  oF  .+  G
) )  u.  (
b  \  ran  ( F  oF  .+  G
) ) )  =  b
5049imaeq2i 5164 . . . . . . . 8  |-  ( `' ( F  oF  .+  G ) "
( ( b  i^i 
ran  ( F  oF  .+  G ) )  u.  ( b  \  ran  ( F  oF  .+  G ) ) ) )  =  ( `' ( F  oF  .+  G ) "
b )
51 ffun 5558 . . . . . . . . . 10  |-  ( ( F  oF  .+  G ) : U. dom  M --> C  ->  Fun  ( F  oF  .+  G ) )
52 unpreima 5826 . . . . . . . . . 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 462 . . . . . . . 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 2487 . . . . . . 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 26535 . . . . . . . . . 10  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
5716, 56syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  M  e.  U. ran sigAlgebra )
5857adantr 462 . . . . . . . 8  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  ->  dom  M  e.  U. ran sigAlgebra )
59 imaiun 5959 . . . . . . . . . 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 4222 . . . . . . . . . . 11  |-  U_ z  e.  ( b  i^i  ran  ( F  oF  .+  G ) ) { z }  =  ( b  i^i  ran  ( F  oF  .+  G
) )
6160imaeq2i 5164 . . . . . . . . . 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 2463 . . . . . . . . 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 3568 . . . . . . . . . . . 12  |-  ( b  i^i  ran  ( F  oF  .+  G ) )  C_  ran  ( F  oF  .+  G
)
64 feq3 5541 . . . . . . . . . . . . . . . . . 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 5541 . . . . . . . . . . . . . . . . . 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 5556 . . . . . . . . . . . . . . . . 17  |-  (  .+  : ( B  X.  B ) --> C  ->  .+  Fn  ( B  X.  B ) )
711, 70syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  .+  Fn  ( B  X.  B ) )
7266, 69, 23, 71ofpreima2 25904 . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  dom  M  e.  U.
ran sigAlgebra )
7557ad2antrr 720 . . . . . . . . . . . . . . . . 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 748 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ph )
77 inss1 3567 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  C_  ( `'  .+  " { z } )
78 cnvimass 5186 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( `' 
.+  " { z } )  C_  dom  .+
79 fdm 5560 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (  .+  : ( B  X.  B ) --> C  ->  dom  .+  =  ( B  X.  B ) )
801, 79syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  dom  .+  =  ( B  X.  B ) )
8178, 80syl5sseq 3401 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( `'  .+  " {
z } )  C_  ( B  X.  B
) )
8281adantr 462 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( `'  .+  " { z } ) 
C_  ( B  X.  B ) )
8377, 82syl5ss 3364 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( ( `' 
.+  " { z } )  i^i  ( ran 
F  X.  ran  G
) )  C_  ( B  X.  B ) )
8483sselda 3353 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  ran  ( F  oF  .+  G ) )  /\  p  e.  ( ( `'  .+  " {
z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  p  e.  ( B  X.  B ) )
8557adantr 462 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  dom  M  e.  U. ran sigAlgebra )
86 fvex 5698 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( TopOpen `  W )  e.  _V
874, 86eqeltri 2511 . . . . . . . . . . . . . . . . . . . . . . 23  |-  J  e. 
_V
8887a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  J  e.  _V )
8988sgsiga 26505 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
9011, 89syl5eqel 2525 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
9190adantr 462 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  S  e.  U. ran sigAlgebra )
923, 4, 11, 12, 13, 14, 15, 16, 17sibfmbl 26635 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
9392adantr 462 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  F  e.  ( dom  MMblFnM S
) )
944tpstop 18444 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( W  e.  TopSp  ->  J  e.  Top )
95 cldssbrsiga 26521 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( J  e.  Top  ->  ( Clsd `  J )  C_  (sigaGen `  J ) )
962, 94, 953syl 20 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( Clsd `  J
)  C_  (sigaGen `  J
) )
9796adantr 462 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( Clsd `  J )  C_  (sigaGen `  J ) )
98 sibfof.4 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  J  e.  Fre )
9998adantr 462 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  J  e.  Fre )
100 xp1st 6605 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( p  e.  ( B  X.  B )  ->  ( 1st `  p )  e.  B )
101100adantl 463 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 1st `  p )  e.  B )
1026adantr 462 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  B  =  U. J )
103101, 102eleqtrd 2517 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 1st `  p )  e. 
U. J )
104 eqid 2441 . . . . . . . . . . . . . . . . . . . . . . 23  |-  U. J  =  U. J
105104t1sncld 18830 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( J  e.  Fre  /\  ( 1st `  p )  e.  U. J )  ->  { ( 1st `  p ) }  e.  ( Clsd `  J )
)
10699, 103, 105syl2anc 656 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 1st `  p ) }  e.  ( Clsd `  J ) )
10797, 106sseldd 3354 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 1st `  p ) }  e.  (sigaGen `  J
) )
108107, 11syl6eleqr 2532 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 1st `  p ) }  e.  S )
10985, 91, 93, 108mbfmcnvima 26592 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( `' F " { ( 1st `  p ) } )  e.  dom  M )
11076, 84, 109syl2anc 656 . . . . . . . . . . . . . . . . 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 26635 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  G  e.  ( dom 
MMblFnM S ) )
112111adantr 462 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  G  e.  ( dom  MMblFnM S
) )
113 xp2nd 6606 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( p  e.  ( B  X.  B )  ->  ( 2nd `  p )  e.  B )
114113adantl 463 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 2nd `  p )  e.  B )
115114, 102eleqtrd 2517 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( 2nd `  p )  e. 
U. J )
116104t1sncld 18830 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( J  e.  Fre  /\  ( 2nd `  p )  e.  U. J )  ->  { ( 2nd `  p ) }  e.  ( Clsd `  J )
)
11799, 115, 116syl2anc 656 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 2nd `  p ) }  e.  ( Clsd `  J ) )
11897, 117sseldd 3354 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 2nd `  p ) }  e.  (sigaGen `  J
) )
119118, 11syl6eleqr 2532 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  { ( 2nd `  p ) }  e.  S )
12085, 91, 112, 119mbfmcnvima 26592 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  p  e.  ( B  X.  B
) )  ->  ( `' G " { ( 2nd `  p ) } )  e.  dom  M )
12176, 84, 120syl2anc 656 . . . . . . . . . . . . . . . . 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 26498 . . . . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . . . . 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 2797 . . . . . . . . . . . . . . 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 3568 . . . . . . . . . . . . . . . . . 18  |-  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  C_  ( ran  F  X.  ran  G
)
1263, 4, 11, 12, 13, 14, 15, 16, 17sibfrn 26637 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ran  F  e.  Fin )
1273, 4, 11, 12, 13, 14, 15, 16, 19sibfrn 26637 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ran  G  e.  Fin )
128 xpfi 7579 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
129126, 127, 128syl2anc 656 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
130 ssdomg 7351 . . . . . . . . . . . . . . . . . . 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 7854 . . . . . . . . . . . . . . . . . . 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 7333 . . . . . . . . . . . . . . . . . 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 7358 . . . . . . . . . . . . . . . . 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 656 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  ~<_  om )
139138adantr 462 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( ( `' 
.+  " { z } )  i^i  ( ran 
F  X.  ran  G
) )  ~<_  om )
140 nfcv 2577 . . . . . . . . . . . . . . . 16  |-  F/_ p
( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )
141140sigaclcuni 26481 . . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . . 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 2515 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ran  ( F  oF  .+  G ) )  ->  ( `' ( F  oF  .+  G ) " {
z } )  e. 
dom  M )
144143ralrimiva 2797 . . . . . . . . . . . 12  |-  ( ph  ->  A. z  e.  ran  ( F  oF  .+  G ) ( `' ( F  oF  .+  G ) " { z } )  e.  dom  M )
145 ssralv 3413 . . . . . . . . . . . 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 462 . . . . . . . . . 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 5558 . . . . . . . . . . . . . . . . 17  |-  (  .+  : ( B  X.  B ) --> C  ->  Fun  .+  )
1491, 148syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  Fun  .+  )
150 imafi 7600 . . . . . . . . . . . . . . . 16  |-  ( ( Fun  .+  /\  ( ran  F  X.  ran  G
)  e.  Fin )  ->  (  .+  " ( ran  F  X.  ran  G
) )  e.  Fin )
151149, 129, 150syl2anc 656 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (  .+  " ( ran  F  X.  ran  G
) )  e.  Fin )
15218, 20, 9, 23ofrn2 25877 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  ( F  oF  .+  G )  C_  (  .+  " ( ran 
F  X.  ran  G
) ) )
153 ssfi 7529 . . . . . . . . . . . . . . 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 656 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  ( F  oF  .+  G )  e. 
Fin )
155 ssdomg 7351 . . . . . . . . . . . . . 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 7854 . . . . . . . . . . . . . 14  |-  ( ran  ( F  oF  .+  G )  e. 
Fin 
<->  ran  ( F  oF  .+  G )  ~<  om )
159154, 158sylib 196 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  ( F  oF  .+  G )  ~<  om )
160 sdomdom 7333 . . . . . . . . . . . . 13  |-  ( ran  ( F  oF  .+  G )  ~<  om  ->  ran  ( F  oF  .+  G )  ~<_  om )
161159, 160syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ran  ( F  oF  .+  G )  ~<_  om )
162 domtr 7358 . . . . . . . . . . . 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 656 . . . . . . . . . . 11  |-  ( ph  ->  ( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  om )
164163adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( b  i^i  ran  ( F  oF  .+  G ) )  ~<_  om )
165 nfcv 2577 . . . . . . . . . . 11  |-  F/_ z
( b  i^i  ran  ( F  oF  .+  G ) )
166165sigaclcuni 26481 . . . . . . . . . 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 1213 . . . . . . . . 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 2526 . . . . . . . 8  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " ( b  i^i  ran  ( F  oF  .+  G ) ) )  e.  dom  M )
169 difpreima 5828 . . . . . . . . . . . 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 5187 . . . . . . . . . . . . 13  |-  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) )  =  dom  ( F  oF  .+  G
)
172171difeq2i 3468 . . . . . . . . . . . 12  |-  ( ( `' ( F  oF  .+  G ) "
b )  \  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) )  =  ( ( `' ( F  oF  .+  G ) "
b )  \  dom  ( F  oF  .+  G ) )
173 cnvimass 5186 . . . . . . . . . . . . 13  |-  ( `' ( F  oF  .+  G ) "
b )  C_  dom  ( F  oF  .+  G )
174 ssdif0 3734 . . . . . . . . . . . . 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 2461 . . . . . . . . . . 11  |-  ( ( `' ( F  oF  .+  G ) "
b )  \  ( `' ( F  oF  .+  G ) " ran  ( F  oF  .+  G ) ) )  =  (/)
177170, 176syl6eq 2489 . . . . . . . . . 10  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  =  (/) )
178 0elsiga 26477 . . . . . . . . . . 11  |-  ( dom 
M  e.  U. ran sigAlgebra  ->  (/)  e.  dom  M )
17957, 178syl 16 . . . . . . . . . 10  |-  ( ph  -> 
(/)  e.  dom  M )
180177, 179eqeltrd 2515 . . . . . . . . 9  |-  ( ph  ->  ( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  e.  dom  M )
181180adantr 462 . . . . . . . 8  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " ( b 
\  ran  ( F  oF  .+  G ) ) )  e.  dom  M )
182 unelsiga 26497 . . . . . . . 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 1213 . . . . . . 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 2515 . . . . . 6  |-  ( (
ph  /\  b  e.  (sigaGen `  ( TopOpen `  K
) ) )  -> 
( `' ( F  oF  .+  G
) " b )  e.  dom  M )
185184ralrimiva 2797 . . . . 5  |-  ( ph  ->  A. b  e.  (sigaGen `  ( TopOpen `  K )
) ( `' ( F  oF  .+  G ) " b
)  e.  dom  M
)
18648, 185jca 529 . . . 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 26587 . . . 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 462 . . . . . . . 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 5692 . . . . . . 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 26536 . . . . . . . . . 10  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )
19216, 191sylib 196 . . . . . . . . 9  |-  ( ph  ->  M  e.  (measures `  dom  M ) )
193192adantr 462 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  M  e.  (measures `  dom  M ) )
194 difss 3480 . . . . . . . . . 10  |-  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) 
C_  ran  ( F  oF  .+  G )
195194sseli 3349 . . . . . . . . 9  |-  ( z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } )  ->  z  e.  ran  ( F  oF  .+  G ) )
196195, 124sylan2 471 . . . . . . . 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 462 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  ~<_  om )
198 ffun 5558 . . . . . . . . . . . . . 14  |-  ( F : U. dom  M --> U. J  ->  Fun  F
)
19918, 198syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  Fun  F )
200 sndisj 4281 . . . . . . . . . . . . 13  |- Disj  x  e. 
ran  F { x }
201 disjpreima 25847 . . . . . . . . . . . . 13  |-  ( ( Fun  F  /\ Disj  x  e. 
ran  F { x } )  -> Disj  x  e. 
ran  F ( `' F " { x } ) )
202199, 200, 201sylancl 657 . . . . . . . . . . . 12  |-  ( ph  -> Disj  x  e.  ran  F ( `' F " { x } ) )
203 ffun 5558 . . . . . . . . . . . . . 14  |-  ( G : U. dom  M --> U. J  ->  Fun  G
)
20420, 203syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  Fun  G )
205 sndisj 4281 . . . . . . . . . . . . 13  |- Disj  y  e. 
ran  G { y }
206 disjpreima 25847 . . . . . . . . . . . . 13  |-  ( ( Fun  G  /\ Disj  y  e. 
ran  G { y } )  -> Disj  y  e. 
ran  G ( `' G " { y } ) )
207204, 205, 206sylancl 657 . . . . . . . . . . . 12  |-  ( ph  -> Disj  y  e.  ran  G ( `' G " { y } ) )
208 sneq 3884 . . . . . . . . . . . . . 14  |-  ( x  =  ( 1st `  p
)  ->  { x }  =  { ( 1st `  p ) } )
209208imaeq2d 5166 . . . . . . . . . . . . 13  |-  ( x  =  ( 1st `  p
)  ->  ( `' F " { x }
)  =  ( `' F " { ( 1st `  p ) } ) )
210 sneq 3884 . . . . . . . . . . . . . 14  |-  ( y  =  ( 2nd `  p
)  ->  { y }  =  { ( 2nd `  p ) } )
211210imaeq2d 5166 . . . . . . . . . . . . 13  |-  ( y  =  ( 2nd `  p
)  ->  ( `' G " { y } )  =  ( `' G " { ( 2nd `  p ) } ) )
212 simpl 454 . . . . . . . . . . . . 13  |-  ( (Disj  x  e.  ran  F ( `' F " { x } )  /\ Disj  y  e. 
ran  G ( `' G " { y } ) )  -> Disj  x  e.  ran  F ( `' F " { x } ) )
213 simpr 458 . . . . . . . . . . . . 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 25849 . . . . . . . . . . . 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 656 . . . . . . . . . . 11  |-  ( ph  -> Disj  p  e.  ( ran  F  X.  ran  G ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )
216 disjss1 4265 . . . . . . . . . . 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 462 . . . . . . . . 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 529 . . . . . . . 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 26548 . . . . . . . 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 1213 . . . . . . 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 7529 . . . . . . . . . 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 657 . . . . . . . . 9  |-  ( ph  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) )  e.  Fin )
224223adantr 462 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G
) )  e.  Fin )
225 rge0ssre 11389 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
226195, 76sylanl2 646 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  ( F  oF  .+  G
)  \  { ( 0g `  K ) } ) )  /\  p  e.  ( ( `'  .+  " { z } )  i^i  ( ran  F  X.  ran  G ) ) )  ->  ph )
227 simpr 458 . . . . . . . . . . . . 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 3351 . . . . . . . . . . . 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 6605 . . . . . . . . . . . 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 6606 . . . . . . . . . . . 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 6099 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =  .0.  /\  y  =  .0.  )  ->  ( x  .+  y
)  =  (  .0.  .+  .0.  ) )
234233adantl 463 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( x  =  .0.  /\  y  =  .0.  ) )  -> 
( x  .+  y
)  =  (  .0.  .+  .0.  ) )
235 sibfof.5 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  (  .0.  .+  .0.  )  =  ( 0g `  K ) )
236235adantr 462 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( x  =  .0.  /\  y  =  .0.  ) )  -> 
(  .0.  .+  .0.  )  =  ( 0g `  K ) )
237234, 236eqtrd 2473 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( x  =  .0.  /\  y  =  .0.  ) )  -> 
( x  .+  y
)  =  ( 0g
`  K ) )
238237ex 434 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( x  =  .0.  /\  y  =  .0.  )  ->  (
x  .+  y )  =  ( 0g `  K ) ) )
239238necon3ad 2642 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( x  .+  y )  =/=  ( 0g `  K )  ->  -.  ( x  =  .0. 
/\  y  =  .0.  ) ) )
240 oran 493 . . . . . . . . . . . . . . . . 17  |-  ( ( x  =/=  .0.  \/  y  =/=  .0.  )  <->  -.  ( -.  x  =/=  .0.  /\ 
-.  y  =/=  .0.  ) )
241 nne 2610 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  x  =/=  .0.  <->  x  =  .0.  )
242 nne 2610 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  y  =/=  .0.  <->  y  =  .0.  )
243241, 242anbi12i 692 . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( x  .+  y )  =/=  ( 0g `  K )  -> 
( x  =/=  .0.  \/  y  =/=  .0.  ) ) )
248247ralrimivva 2806 . . . . . . . . . . . . 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 3353 . . . . . . . . . . . . . . 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 5821 . . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  ( B  X.  B )  /\  (  .+  `  p )  =  z )  -> 
(  .+  `  p )  =  z )
257 1st2nd2 6612 . . . . . . . . . . . . . . . . . 18  |-  ( p  e.  ( B  X.  B )  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
258257fveq2d 5692 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  ( B  X.  B )  ->  (  .+  `  p )  =  (  .+  `  <. ( 1st `  p ) ,  ( 2nd `  p
) >. ) )
259 df-ov 6093 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st `  p ) 
.+  ( 2nd `  p
) )  =  ( 
.+  `  <. ( 1st `  p ) ,  ( 2nd `  p )
>. )
260258, 259syl6eqr 2491 . . . . . . . . . . . . . . . 16  |-  ( p  e.  ( B  X.  B )  ->  (  .+  `  p )  =  ( ( 1st `  p
)  .+  ( 2nd `  p ) ) )
261260adantr 462 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  ( B  X.  B )  /\  (  .+  `  p )  =  z )  -> 
(  .+  `  p )  =  ( ( 1st `  p )  .+  ( 2nd `  p ) ) )
262256, 261eqtr3d 2475 . . . . . . . . . . . . . 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 749 . . . . . . . . . . . . . . 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 3338 . . . . . . . . . . . . . 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 3888 . . . . . . . . . . . . . . 15  |-  ( z  e.  { ( 0g
`  K ) }  <-> 
z  =  ( 0g
`  K ) )
267266necon3bbii 2637 . . . . . . . . . . . . . 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 2626 . . . . . . . . . . . 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 646 . . . . . . . . . . . . . 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 6097 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( 1st `  p
)  ->  ( x  .+  y )  =  ( ( 1st `  p
)  .+  y )
)
274273neeq1d 2619 . . . . . . . . . . . . . . 15  |-  ( x  =  ( 1st `  p
)  ->  ( (
x  .+  y )  =/=  ( 0g `  K
)  <->  ( ( 1st `  p )  .+  y
)  =/=  ( 0g
`  K ) ) )
275 neeq1 2614 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( 1st `  p
)  ->  ( x  =/=  .0.  <->  ( 1st `  p
)  =/=  .0.  )
)
276275orbi1d 697 . . . . . . . . . . . . . . 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 6098 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( 2nd `  p
)  ->  ( ( 1st `  p )  .+  y )  =  ( ( 1st `  p
)  .+  ( 2nd `  p ) ) )
279278neeq1d 2619 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 2nd `  p
)  ->  ( (
( 1st `  p
)  .+  y )  =/=  ( 0g `  K
)  <->  ( ( 1st `  p )  .+  ( 2nd `  p ) )  =/=  ( 0g `  K ) ) )
280 neeq1 2614 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( 2nd `  p
)  ->  ( y  =/=  .0.  <->  ( 2nd `  p
)  =/=  .0.  )
)
281280orbi2d 696 . . . . . . . . . . . . . . 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 3076 . . . . . . . . . . . . 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 656 . . . . . . . . . . . 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 26639 . . . . . . . . . . 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 1216 . . . . . . . . . 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 3351 . . . . . . . . 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 462 . . . . . . . . . 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 646 . . . . . . . . . 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 26541 . . . . . . . . . 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 656 . . . . . . . . 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 11388 . . . . . . . . 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 659 . . . . . . . 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 26444 . . . . . . 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 2477 . . . . . 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 13207 . . . . . 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 2515 . . . . 5  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( M `  ( `' ( F  oF  .+  G
) " { z } ) )  e.  RR )
299224, 288, 292fsumge0 13254 . . . . . 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 4315 . . . . 5  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  0  <_  ( M `  ( `' ( F  oF  .+  G ) " { z } ) ) )
301 elrege0 11388 . . . . 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 659 . . . 4  |-  ( (
ph  /\  z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) )  ->  ( M `  ( `' ( F  oF  .+  G
) " { z } ) )  e.  ( 0 [,) +oo ) )
303302ralrimiva 2797 . . 3  |-  ( ph  ->  A. z  e.  ( ran  ( F  oF  .+  G )  \  { ( 0g `  K ) } ) ( M `  ( `' ( F  oF  .+  G ) " { z } ) )  e.  ( 0 [,) +oo ) )
304188, 154, 3033jca 1163 . 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 2441 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
306 eqid 2441 . . 3  |-  ( .s
`  K )  =  ( .s `  K
)
307 eqid 2441 . . 3  |-  (RRHom `  (Scalar `  K ) )  =  (RRHom `  (Scalar `  K ) )
30827, 28, 31, 305, 306, 307, 26, 16issibf 26633 . 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   _Vcvv 2970    \ cdif 3322    u. cun 3323    i^i cin 3324    C_ wss 3325   (/)c0 3634   {csn 3874   <.cop 3880   U.cuni 4088   U_ciun 4168  Disj wdisj 4259   class class class wbr 4289    X. cxp 4834   `'ccnv 4835   dom cdm 4836   ran crn 4837   "cima 4839   Fun wfun 5409    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317   omcom 6475   1stc1st 6574   2ndc2nd 6575    ^m cmap 7210    ~<_ cdom 7304    ~< csdm 7305   Fincfn 7306   RRcr 9277   0cc0 9278   +oocpnf 9411    <_ cle 9415   [,)cico 11298   sum_csu 13159   Basecbs 14170  Scalarcsca 14237   .scvsca 14238   TopOpenctopn 14356   0gc0g 14374   Topctop 18398   TopSpctps 18401   Clsdccld 18520   Frect1 18811  RRHomcrrh 26342  Σ*cesum 26403  sigAlgebracsiga 26470  sigaGencsigagen 26501  measurescmeas 26529  MblFnMcmbfm 26585  sitgcsitg 26629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-ac2 8628  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-disj 4260  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-acn 8108  df-ac 8282  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-ordt 14435  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-ps 15366  df-tsr 15367  df-mnd 15411  df-plusf 15412  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-cntz 15828  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-subrg 16843  df-abv 16882  df-lmod 16930  df-scaf 16931  df-sra 17231  df-rgmod 17232  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-fbas 17714  df-fg 17715  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cld 18523  df-ntr 18524  df-cls 18525  df-nei 18602  df-lp 18640  df-perf 18641  df-cn 18731  df-cnp 18732  df-t1 18818  df-haus 18819  df-tx 19035  df-hmeo 19228  df-fil 19319  df-fm 19411  df-flim 19412  df-flf 19413  df-tmd 19543  df-tgp 19544  df-tsms 19597  df-trg 19634  df-xms 19795  df-ms 19796  df-tms 19797  df-nm 20075  df-ngp 20076  df-nrg 20078  df-nlm 20079  df-ii 20353  df-cncf 20354  df-limc 21241  df-dv 21242  df-log 21951  df-esum 26404  df-siga 26471  df-sigagen 26502  df-meas 26530  df-mbfm 26586  df-sitg 26630
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator