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Theorem sibfinima 28113
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 28005 . . . . . . . 8  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )
31, 2sylib 196 . . . . . . 7  |-  ( ph  ->  M  e.  (measures `  dom  M ) )
433ad2ant1 1017 . . . . . 6  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  M  e.  (measures `  dom  M ) )
5 dmmeas 28004 . . . . . . . . 9  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
61, 5syl 16 . . . . . . . 8  |-  ( ph  ->  dom  M  e.  U. ran sigAlgebra )
763ad2ant1 1017 . . . . . . 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 27974 . . . . . . . . . 10  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
118, 10syl5eqel 2559 . . . . . . . . 9  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
12113ad2ant1 1017 . . . . . . . 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 28109 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
21203ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  F  e.  ( dom  MMblFnM
S ) )
22 sibfinima.w . . . . . . . . . . . 12  |-  ( ph  ->  W  e.  TopSp )
2314tpstop 19307 . . . . . . . . . . . 12  |-  ( W  e.  TopSp  ->  J  e.  Top )
24 cldssbrsiga 27990 . . . . . . . . . . . 12  |-  ( J  e.  Top  ->  ( Clsd `  J )  C_  (sigaGen `  J ) )
2522, 23, 243syl 20 . . . . . . . . . . 11  |-  ( ph  ->  ( Clsd `  J
)  C_  (sigaGen `  J
) )
2625, 8syl6sseqr 3556 . . . . . . . . . 10  |-  ( ph  ->  ( Clsd `  J
)  C_  S )
27263ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
( Clsd `  J )  C_  S )
2893ad2ant1 1017 . . . . . . . . . 10  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  J  e.  Fre )
2913, 14, 8, 15, 16, 17, 18, 1, 19sibff 28110 . . . . . . . . . . . . 13  |-  ( ph  ->  F : U. dom  M --> U. J )
30 frn 5743 . . . . . . . . . . . . 13  |-  ( F : U. dom  M --> U. J  ->  ran  F  C_ 
U. J )
3129, 30syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ran  F  C_  U. J
)
32313ad2ant1 1017 . . . . . . . . . . 11  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  ran  F  C_  U. J )
33 simp2 997 . . . . . . . . . . 11  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  X  e.  ran  F )
3432, 33sseldd 3510 . . . . . . . . . 10  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  X  e.  U. J )
35 eqid 2467 . . . . . . . . . . 11  |-  U. J  =  U. J
3635t1sncld 19693 . . . . . . . . . 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 3510 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  { X }  e.  S
)
397, 12, 21, 38mbfmcnvima 28060 . . . . . . 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 28109 . . . . . . . . 9  |-  ( ph  ->  G  e.  ( dom 
MMblFnM S ) )
42413ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  G  e.  ( dom  MMblFnM
S ) )
4313, 14, 8, 15, 16, 17, 18, 1, 40sibff 28110 . . . . . . . . . . . . 13  |-  ( ph  ->  G : U. dom  M --> U. J )
44 frn 5743 . . . . . . . . . . . . 13  |-  ( G : U. dom  M --> U. J  ->  ran  G  C_ 
U. J )
4543, 44syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ran  G  C_  U. J
)
46453ad2ant1 1017 . . . . . . . . . . 11  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  ran  G  C_  U. J )
47 simp3 998 . . . . . . . . . . 11  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  Y  e.  ran  G )
4846, 47sseldd 3510 . . . . . . . . . 10  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  Y  e.  U. J )
4935t1sncld 19693 . . . . . . . . . 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 3510 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  { Y }  e.  S
)
527, 12, 42, 51mbfmcnvima 28060 . . . . . . 7  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
( `' G " { Y } )  e. 
dom  M )
53 inelsiga 27967 . . . . . . 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 1228 . . . . . 6  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
( ( `' F " { X } )  i^i  ( `' G " { Y } ) )  e.  dom  M
)
55 measvxrge0 28008 . . . . . 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 11641 . . . . . 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 9608 . . . 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 28008 . . . . . . 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 11641 . . . . . . 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 11333 . . . . . 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 3723 . . . . . . 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 28018 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  <_  ( M `  ( `' F " { X } ) ) )
80 simpl1 999 . . . . . . 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 4158 . . . . . . . 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 28112 . . . . . . 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 11600 . . . . . . . 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 1013 . . . . . 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 11375 . . . 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 28008 . . . . . . 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 11641 . . . . . . 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 3724 . . . . . . 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 28018 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  <_  ( M `  ( `' G " { Y } ) ) )
104 simpl1 999 . . . . . . 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 4158 . . . . . . . 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 28112 . . . . . . 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 11600 . . . . . . . 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 1013 . . . . . 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 11375 . . . 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 11384 . . 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 1229 . 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 11600 . . 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 1180 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3478    i^i cin 3480    C_ wss 3481   {csn 4033   U.cuni 4251   class class class wbr 4453   `'ccnv 5004   dom cdm 5005   ran crn 5006   "cima 5008   -->wf 5590   ` cfv 5594  (class class class)co 6295   RRcr 9503   0cc0 9504   +oocpnf 9637   RR*cxr 9639    < clt 9640    <_ cle 9641   [,)cico 11543   [,]cicc 11544   Basecbs 14506  Scalarcsca 14574   .scvsca 14575   TopOpenctopn 14693   0gc0g 14711   Topctop 19261   TopSpctps 19264   Clsdccld 19383   Frect1 19674  RRHomcrrh 27806  sigAlgebracsiga 27939  sigaGencsigagen 27970  measurescmeas 27998  MblFnMcmbfm 28053  sitgcsitg 28103
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-ac2 8855  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-acn 8335  df-ac 8509  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-sum 13488  df-ef 13681  df-sin 13683  df-cos 13684  df-pi 13686  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-starv 14586  df-sca 14587  df-vsca 14588  df-ip 14589  df-tset 14590  df-ple 14591  df-ds 14593  df-unif 14594  df-hom 14595  df-cco 14596  df-rest 14694  df-topn 14695  df-0g 14713  df-gsum 14714  df-topgen 14715  df-pt 14716  df-prds 14719  df-ordt 14772  df-xrs 14773  df-qtop 14778  df-imas 14779  df-xps 14781  df-mre 14857  df-mrc 14858  df-acs 14860  df-ps 15703  df-tsr 15704  df-plusf 15744  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-mhm 15838  df-submnd 15839  df-grp 15928  df-minusg 15929  df-sbg 15930  df-mulg 15931  df-subg 16069  df-cntz 16226  df-cmn 16671  df-abl 16672  df-mgp 17012  df-ur 17024  df-ring 17070  df-cring 17071  df-subrg 17296  df-abv 17335  df-lmod 17383  df-scaf 17384  df-sra 17687  df-rgmod 17688  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-fbas 18284  df-fg 18285  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cld 19386  df-ntr 19387  df-cls 19388  df-nei 19465  df-lp 19503  df-perf 19504  df-cn 19594  df-cnp 19595  df-t1 19681  df-haus 19682  df-tx 19929  df-hmeo 20122  df-fil 20213  df-fm 20305  df-flim 20306  df-flf 20307  df-tmd 20437  df-tgp 20438  df-tsms 20491  df-trg 20528  df-xms 20689  df-ms 20690  df-tms 20691  df-nm 20969  df-ngp 20970  df-nrg 20972  df-nlm 20973  df-ii 21247  df-cncf 21248  df-limc 22136  df-dv 22137  df-log 22808  df-esum 27873  df-siga 27940  df-sigagen 27971  df-meas 27999  df-mbfm 28054  df-sitg 28104
This theorem is referenced by:  sibfof  28114
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