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Theorem sibfinima 26747
Description: The measure of the intersection of any two preimages by simple functions is a real number. (Contributed by Thierry Arnoux, 21-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 ) )
sibfinima.g  |-  ( ph  ->  G  e.  dom  ( Wsitg M ) )
sibfinima.w  |-  ( ph  ->  W  e.  TopSp )
sibfinima.j  |-  ( ph  ->  J  e.  Fre )
Assertion
Ref Expression
sibfinima  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  ( X  =/=  .0.  \/  Y  =/=  .0.  ) )  ->  ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  ( 0 [,) +oo ) )

Proof of Theorem sibfinima
StepHypRef Expression
1 sitgval.2 . . . . . . . 8  |-  ( ph  ->  M  e.  U. ran measures )
2 measbasedom 26638 . . . . . . . 8  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )
31, 2sylib 196 . . . . . . 7  |-  ( ph  ->  M  e.  (measures `  dom  M ) )
433ad2ant1 1009 . . . . . 6  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  M  e.  (measures `  dom  M ) )
5 dmmeas 26637 . . . . . . . . 9  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
61, 5syl 16 . . . . . . . 8  |-  ( ph  ->  dom  M  e.  U. ran sigAlgebra )
763ad2ant1 1009 . . . . . . 7  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  dom  M  e.  U. ran sigAlgebra )
8 sitgval.s . . . . . . . . . 10  |-  S  =  (sigaGen `  J )
9 sibfinima.j . . . . . . . . . . 11  |-  ( ph  ->  J  e.  Fre )
109sgsiga 26607 . . . . . . . . . 10  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
118, 10syl5eqel 2527 . . . . . . . . 9  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
12113ad2ant1 1009 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  S  e.  U. ran sigAlgebra )
13 sitgval.b . . . . . . . . . 10  |-  B  =  ( Base `  W
)
14 sitgval.j . . . . . . . . . 10  |-  J  =  ( TopOpen `  W )
15 sitgval.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  W )
16 sitgval.x . . . . . . . . . 10  |-  .x.  =  ( .s `  W )
17 sitgval.h . . . . . . . . . 10  |-  H  =  (RRHom `  (Scalar `  W
) )
18 sitgval.1 . . . . . . . . . 10  |-  ( ph  ->  W  e.  V )
19 sibfmbl.1 . . . . . . . . . 10  |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )
2013, 14, 8, 15, 16, 17, 18, 1, 19sibfmbl 26743 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
21203ad2ant1 1009 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  F  e.  ( dom  MMblFnM
S ) )
22 sibfinima.w . . . . . . . . . . . 12  |-  ( ph  ->  W  e.  TopSp )
2314tpstop 18566 . . . . . . . . . . . 12  |-  ( W  e.  TopSp  ->  J  e.  Top )
24 cldssbrsiga 26623 . . . . . . . . . . . 12  |-  ( J  e.  Top  ->  ( Clsd `  J )  C_  (sigaGen `  J ) )
2522, 23, 243syl 20 . . . . . . . . . . 11  |-  ( ph  ->  ( Clsd `  J
)  C_  (sigaGen `  J
) )
2625, 8syl6sseqr 3424 . . . . . . . . . 10  |-  ( ph  ->  ( Clsd `  J
)  C_  S )
27263ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
( Clsd `  J )  C_  S )
2893ad2ant1 1009 . . . . . . . . . 10  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  J  e.  Fre )
2913, 14, 8, 15, 16, 17, 18, 1, 19sibff 26744 . . . . . . . . . . . . 13  |-  ( ph  ->  F : U. dom  M --> U. J )
30 frn 5586 . . . . . . . . . . . . 13  |-  ( F : U. dom  M --> U. J  ->  ran  F  C_ 
U. J )
3129, 30syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ran  F  C_  U. J
)
32313ad2ant1 1009 . . . . . . . . . . 11  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  ran  F  C_  U. J )
33 simp2 989 . . . . . . . . . . 11  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  X  e.  ran  F )
3432, 33sseldd 3378 . . . . . . . . . 10  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  X  e.  U. J )
35 eqid 2443 . . . . . . . . . . 11  |-  U. J  =  U. J
3635t1sncld 18952 . . . . . . . . . 10  |-  ( ( J  e.  Fre  /\  X  e.  U. J )  ->  { X }  e.  ( Clsd `  J
) )
3728, 34, 36syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  { X }  e.  (
Clsd `  J )
)
3827, 37sseldd 3378 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  { X }  e.  S
)
397, 12, 21, 38mbfmcnvima 26694 . . . . . . 7  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
( `' F " { X } )  e. 
dom  M )
40 sibfinima.g . . . . . . . . . 10  |-  ( ph  ->  G  e.  dom  ( Wsitg M ) )
4113, 14, 8, 15, 16, 17, 18, 1, 40sibfmbl 26743 . . . . . . . . 9  |-  ( ph  ->  G  e.  ( dom 
MMblFnM S ) )
42413ad2ant1 1009 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  G  e.  ( dom  MMblFnM
S ) )
4313, 14, 8, 15, 16, 17, 18, 1, 40sibff 26744 . . . . . . . . . . . . 13  |-  ( ph  ->  G : U. dom  M --> U. J )
44 frn 5586 . . . . . . . . . . . . 13  |-  ( G : U. dom  M --> U. J  ->  ran  G  C_ 
U. J )
4543, 44syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ran  G  C_  U. J
)
46453ad2ant1 1009 . . . . . . . . . . 11  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  ran  G  C_  U. J )
47 simp3 990 . . . . . . . . . . 11  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  Y  e.  ran  G )
4846, 47sseldd 3378 . . . . . . . . . 10  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  Y  e.  U. J )
4935t1sncld 18952 . . . . . . . . . 10  |-  ( ( J  e.  Fre  /\  Y  e.  U. J )  ->  { Y }  e.  ( Clsd `  J
) )
5028, 48, 49syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  { Y }  e.  (
Clsd `  J )
)
5127, 50sseldd 3378 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  { Y }  e.  S
)
527, 12, 42, 51mbfmcnvima 26694 . . . . . . 7  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
( `' G " { Y } )  e. 
dom  M )
53 inelsiga 26600 . . . . . . 7  |-  ( ( dom  M  e.  U. ran sigAlgebra  /\  ( `' F " { X } )  e. 
dom  M  /\  ( `' G " { Y } )  e.  dom  M )  ->  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) )  e. 
dom  M )
547, 39, 52, 53syl3anc 1218 . . . . . 6  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
( ( `' F " { X } )  i^i  ( `' G " { Y } ) )  e.  dom  M
)
55 measvxrge0 26641 . . . . . 6  |-  ( ( M  e.  (measures `  dom  M )  /\  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) )  e. 
dom  M )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  ( 0 [,] +oo ) )
564, 54, 55syl2anc 661 . . . . 5  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  ( 0 [,] +oo ) )
57 elxrge0 11415 . . . . . 6  |-  ( ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  ( 0 [,] +oo )  <->  ( ( M `
 ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  RR*  /\  0  <_  ( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) ) ) )
5857simplbi 460 . . . . 5  |-  ( ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  ( 0 [,] +oo )  ->  ( M `
 ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  RR* )
5956, 58syl 16 . . . 4  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  RR* )
6059adantr 465 . . 3  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  ( X  =/=  .0.  \/  Y  =/=  .0.  ) )  ->  ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  RR* )
61 0re 9407 . . . 4  |-  0  e.  RR
6261a1i 11 . . 3  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  ( X  =/=  .0.  \/  Y  =/=  .0.  ) )  ->  0  e.  RR )
6357simprbi 464 . . . . 5  |-  ( ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  ( 0 [,] +oo )  ->  0  <_ 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) ) )
6456, 63syl 16 . . . 4  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
0  <_  ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) ) )
6564adantr 465 . . 3  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  ( X  =/=  .0.  \/  Y  =/=  .0.  ) )  ->  0  <_  ( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) ) )
6659adantr 465 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  RR* )
674adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  ->  M  e.  (measures `  dom  M ) )
6839adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( `' F " { X } )  e. 
dom  M )
69 measvxrge0 26641 . . . . . . 7  |-  ( ( M  e.  (measures `  dom  M )  /\  ( `' F " { X } )  e.  dom  M )  ->  ( M `  ( `' F " { X } ) )  e.  ( 0 [,] +oo ) )
7067, 68, 69syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( M `  ( `' F " { X } ) )  e.  ( 0 [,] +oo ) )
71 elxrge0 11415 . . . . . . 7  |-  ( ( M `  ( `' F " { X } ) )  e.  ( 0 [,] +oo ) 
<->  ( ( M `  ( `' F " { X } ) )  e. 
RR*  /\  0  <_  ( M `  ( `' F " { X } ) ) ) )
7271simplbi 460 . . . . . 6  |-  ( ( M `  ( `' F " { X } ) )  e.  ( 0 [,] +oo )  ->  ( M `  ( `' F " { X } ) )  e. 
RR* )
7370, 72syl 16 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( M `  ( `' F " { X } ) )  e. 
RR* )
74 pnfxr 11113 . . . . . 6  |- +oo  e.  RR*
7574a1i 11 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> +oo  e.  RR* )
7654adantr 465 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( ( `' F " { X } )  i^i  ( `' G " { Y } ) )  e.  dom  M
)
77 inss1 3591 . . . . . . 7  |-  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) )  C_  ( `' F " { X } )
7877a1i 11 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( ( `' F " { X } )  i^i  ( `' G " { Y } ) )  C_  ( `' F " { X }
) )
7967, 76, 68, 78measssd 26651 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  <_  ( M `  ( `' F " { X } ) ) )
80 simpl1 991 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  ->  ph )
8133anim1i 568 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( X  e.  ran  F  /\  X  =/=  .0.  ) )
82 eldifsn 4021 . . . . . . . 8  |-  ( X  e.  ( ran  F  \  {  .0.  } )  <-> 
( X  e.  ran  F  /\  X  =/=  .0.  ) )
8381, 82sylibr 212 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  ->  X  e.  ( ran  F 
\  {  .0.  }
) )
8413, 14, 8, 15, 16, 17, 18, 1, 19sibfima 26746 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( ran  F  \  {  .0.  } ) )  -> 
( M `  ( `' F " { X } ) )  e.  ( 0 [,) +oo ) )
8580, 83, 84syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( M `  ( `' F " { X } ) )  e.  ( 0 [,) +oo ) )
86 elico2 11380 . . . . . . . 8  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
( M `  ( `' F " { X } ) )  e.  ( 0 [,) +oo ) 
<->  ( ( M `  ( `' F " { X } ) )  e.  RR  /\  0  <_ 
( M `  ( `' F " { X } ) )  /\  ( M `  ( `' F " { X } ) )  < +oo ) ) )
8761, 74, 86mp2an 672 . . . . . . 7  |-  ( ( M `  ( `' F " { X } ) )  e.  ( 0 [,) +oo ) 
<->  ( ( M `  ( `' F " { X } ) )  e.  RR  /\  0  <_ 
( M `  ( `' F " { X } ) )  /\  ( M `  ( `' F " { X } ) )  < +oo ) )
8887simp3bi 1005 . . . . . 6  |-  ( ( M `  ( `' F " { X } ) )  e.  ( 0 [,) +oo )  ->  ( M `  ( `' F " { X } ) )  < +oo )
8985, 88syl 16 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( M `  ( `' F " { X } ) )  < +oo )
9066, 73, 75, 79, 89xrlelttrd 11155 . . . 4  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  < +oo )
9159adantr 465 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  RR* )
924adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  ->  M  e.  (measures `  dom  M ) )
9352adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( `' G " { Y } )  e. 
dom  M )
94 measvxrge0 26641 . . . . . . 7  |-  ( ( M  e.  (measures `  dom  M )  /\  ( `' G " { Y } )  e.  dom  M )  ->  ( M `  ( `' G " { Y } ) )  e.  ( 0 [,] +oo ) )
9592, 93, 94syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( M `  ( `' G " { Y } ) )  e.  ( 0 [,] +oo ) )
96 elxrge0 11415 . . . . . . 7  |-  ( ( M `  ( `' G " { Y } ) )  e.  ( 0 [,] +oo ) 
<->  ( ( M `  ( `' G " { Y } ) )  e. 
RR*  /\  0  <_  ( M `  ( `' G " { Y } ) ) ) )
9796simplbi 460 . . . . . 6  |-  ( ( M `  ( `' G " { Y } ) )  e.  ( 0 [,] +oo )  ->  ( M `  ( `' G " { Y } ) )  e. 
RR* )
9895, 97syl 16 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( M `  ( `' G " { Y } ) )  e. 
RR* )
9974a1i 11 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> +oo  e.  RR* )
10054adantr 465 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( ( `' F " { X } )  i^i  ( `' G " { Y } ) )  e.  dom  M
)
101 inss2 3592 . . . . . . 7  |-  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) )  C_  ( `' G " { Y } )
102101a1i 11 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( ( `' F " { X } )  i^i  ( `' G " { Y } ) )  C_  ( `' G " { Y }
) )
10392, 100, 93, 102measssd 26651 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  <_  ( M `  ( `' G " { Y } ) ) )
104 simpl1 991 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  ->  ph )
10547anim1i 568 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( Y  e.  ran  G  /\  Y  =/=  .0.  ) )
106 eldifsn 4021 . . . . . . . 8  |-  ( Y  e.  ( ran  G  \  {  .0.  } )  <-> 
( Y  e.  ran  G  /\  Y  =/=  .0.  ) )
107105, 106sylibr 212 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  ->  Y  e.  ( ran  G 
\  {  .0.  }
) )
10813, 14, 8, 15, 16, 17, 18, 1, 40sibfima 26746 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( ran  G  \  {  .0.  } ) )  -> 
( M `  ( `' G " { Y } ) )  e.  ( 0 [,) +oo ) )
109104, 107, 108syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( M `  ( `' G " { Y } ) )  e.  ( 0 [,) +oo ) )
110 elico2 11380 . . . . . . . 8  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
( M `  ( `' G " { Y } ) )  e.  ( 0 [,) +oo ) 
<->  ( ( M `  ( `' G " { Y } ) )  e.  RR  /\  0  <_ 
( M `  ( `' G " { Y } ) )  /\  ( M `  ( `' G " { Y } ) )  < +oo ) ) )
11161, 74, 110mp2an 672 . . . . . . 7  |-  ( ( M `  ( `' G " { Y } ) )  e.  ( 0 [,) +oo ) 
<->  ( ( M `  ( `' G " { Y } ) )  e.  RR  /\  0  <_ 
( M `  ( `' G " { Y } ) )  /\  ( M `  ( `' G " { Y } ) )  < +oo ) )
112111simp3bi 1005 . . . . . 6  |-  ( ( M `  ( `' G " { Y } ) )  e.  ( 0 [,) +oo )  ->  ( M `  ( `' G " { Y } ) )  < +oo )
113109, 112syl 16 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( M `  ( `' G " { Y } ) )  < +oo )
11491, 98, 99, 103, 113xrlelttrd 11155 . . . 4  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  < +oo )
11590, 114jaodan 783 . . 3  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  ( X  =/=  .0.  \/  Y  =/=  .0.  ) )  ->  ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  < +oo )
116 xrre3 11164 . . 3  |-  ( ( ( ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  RR*  /\  0  e.  RR )  /\  ( 0  <_ 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  /\  ( M `
 ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  < +oo ) )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  RR )
11760, 62, 65, 115, 116syl22anc 1219 . 2  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  ( X  =/=  .0.  \/  Y  =/=  .0.  ) )  ->  ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  RR )
118 elico2 11380 . . 3  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  ( 0 [,) +oo )  <->  ( ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  RR  /\  0  <_  ( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  /\  ( M `
 ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  < +oo ) ) )
11961, 74, 118mp2an 672 . 2  |-  ( ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  ( 0 [,) +oo )  <->  ( ( M `
 ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  RR  /\  0  <_  ( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  /\  ( M `
 ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  < +oo ) )
120117, 65, 115, 119syl3anbrc 1172 1  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  ( X  =/=  .0.  \/  Y  =/=  .0.  ) )  ->  ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  ( 0 [,) +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620    \ cdif 3346    i^i cin 3348    C_ wss 3349   {csn 3898   U.cuni 4112   class class class wbr 4313   `'ccnv 4860   dom cdm 4861   ran crn 4862   "cima 4864   -->wf 5435   ` cfv 5439  (class class class)co 6112   RRcr 9302   0cc0 9303   +oocpnf 9436   RR*cxr 9438    < clt 9439    <_ cle 9440   [,)cico 11323   [,]cicc 11324   Basecbs 14195  Scalarcsca 14262   .scvsca 14263   TopOpenctopn 14381   0gc0g 14399   Topctop 18520   TopSpctps 18523   Clsdccld 18642   Frect1 18933  RRHomcrrh 26444  sigAlgebracsiga 26572  sigaGencsigagen 26603  measurescmeas 26631  MblFnMcmbfm 26687  sitgcsitg 26737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-ac2 8653  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-disj 4284  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-ixp 7285  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fsupp 7642  df-fi 7682  df-sup 7712  df-oi 7745  df-card 8130  df-acn 8133  df-ac 8307  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-ioo 11325  df-ioc 11326  df-ico 11327  df-icc 11328  df-fz 11459  df-fzo 11570  df-fl 11663  df-mod 11730  df-seq 11828  df-exp 11887  df-fac 12073  df-bc 12100  df-hash 12125  df-shft 12577  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-limsup 12970  df-clim 12987  df-rlim 12988  df-sum 13185  df-ef 13374  df-sin 13376  df-cos 13377  df-pi 13379  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-starv 14274  df-sca 14275  df-vsca 14276  df-ip 14277  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-hom 14283  df-cco 14284  df-rest 14382  df-topn 14383  df-0g 14401  df-gsum 14402  df-topgen 14403  df-pt 14404  df-prds 14407  df-ordt 14460  df-xrs 14461  df-qtop 14466  df-imas 14467  df-xps 14469  df-mre 14545  df-mrc 14546  df-acs 14548  df-ps 15391  df-tsr 15392  df-mnd 15436  df-plusf 15437  df-mhm 15485  df-submnd 15486  df-grp 15566  df-minusg 15567  df-sbg 15568  df-mulg 15569  df-subg 15699  df-cntz 15856  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-rng 16669  df-cring 16670  df-subrg 16885  df-abv 16924  df-lmod 16972  df-scaf 16973  df-sra 17275  df-rgmod 17276  df-psmet 17831  df-xmet 17832  df-met 17833  df-bl 17834  df-mopn 17835  df-fbas 17836  df-fg 17837  df-cnfld 17841  df-top 18525  df-bases 18527  df-topon 18528  df-topsp 18529  df-cld 18645  df-ntr 18646  df-cls 18647  df-nei 18724  df-lp 18762  df-perf 18763  df-cn 18853  df-cnp 18854  df-t1 18940  df-haus 18941  df-tx 19157  df-hmeo 19350  df-fil 19441  df-fm 19533  df-flim 19534  df-flf 19535  df-tmd 19665  df-tgp 19666  df-tsms 19719  df-trg 19756  df-xms 19917  df-ms 19918  df-tms 19919  df-nm 20197  df-ngp 20198  df-nrg 20200  df-nlm 20201  df-ii 20475  df-cncf 20476  df-limc 21363  df-dv 21364  df-log 22030  df-esum 26506  df-siga 26573  df-sigagen 26604  df-meas 26632  df-mbfm 26688  df-sitg 26738
This theorem is referenced by:  sibfof  26748
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