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Theorem sibfinima 28773
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 28636 . . . . . . . 8  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )
31, 2sylib 196 . . . . . . 7  |-  ( ph  ->  M  e.  (measures `  dom  M ) )
433ad2ant1 1018 . . . . . 6  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  M  e.  (measures `  dom  M ) )
5 dmmeas 28635 . . . . . . . . 9  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
61, 5syl 17 . . . . . . . 8  |-  ( ph  ->  dom  M  e.  U. ran sigAlgebra )
763ad2ant1 1018 . . . . . . 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 28576 . . . . . . . . . 10  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
118, 10syl5eqel 2494 . . . . . . . . 9  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
12113ad2ant1 1018 . . . . . . . 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 28769 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
21203ad2ant1 1018 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  F  e.  ( dom  MMblFnM
S ) )
22 sibfinima.w . . . . . . . . . . . 12  |-  ( ph  ->  W  e.  TopSp )
2314tpstop 19730 . . . . . . . . . . . 12  |-  ( W  e.  TopSp  ->  J  e.  Top )
24 cldssbrsiga 28621 . . . . . . . . . . . 12  |-  ( J  e.  Top  ->  ( Clsd `  J )  C_  (sigaGen `  J ) )
2522, 23, 243syl 20 . . . . . . . . . . 11  |-  ( ph  ->  ( Clsd `  J
)  C_  (sigaGen `  J
) )
2625, 8syl6sseqr 3488 . . . . . . . . . 10  |-  ( ph  ->  ( Clsd `  J
)  C_  S )
27263ad2ant1 1018 . . . . . . . . 9  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
( Clsd `  J )  C_  S )
2893ad2ant1 1018 . . . . . . . . . 10  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  J  e.  Fre )
2913, 14, 8, 15, 16, 17, 18, 1, 19sibff 28770 . . . . . . . . . . . . 13  |-  ( ph  ->  F : U. dom  M --> U. J )
30 frn 5719 . . . . . . . . . . . . 13  |-  ( F : U. dom  M --> U. J  ->  ran  F  C_ 
U. J )
3129, 30syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ran  F  C_  U. J
)
32313ad2ant1 1018 . . . . . . . . . . 11  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  ran  F  C_  U. J )
33 simp2 998 . . . . . . . . . . 11  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  X  e.  ran  F )
3432, 33sseldd 3442 . . . . . . . . . 10  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  X  e.  U. J )
35 eqid 2402 . . . . . . . . . . 11  |-  U. J  =  U. J
3635t1sncld 20118 . . . . . . . . . 10  |-  ( ( J  e.  Fre  /\  X  e.  U. J )  ->  { X }  e.  ( Clsd `  J
) )
3728, 34, 36syl2anc 659 . . . . . . . . 9  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  { X }  e.  (
Clsd `  J )
)
3827, 37sseldd 3442 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  { X }  e.  S
)
397, 12, 21, 38mbfmcnvima 28691 . . . . . . 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 28769 . . . . . . . . 9  |-  ( ph  ->  G  e.  ( dom 
MMblFnM S ) )
42413ad2ant1 1018 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  G  e.  ( dom  MMblFnM
S ) )
4313, 14, 8, 15, 16, 17, 18, 1, 40sibff 28770 . . . . . . . . . . . . 13  |-  ( ph  ->  G : U. dom  M --> U. J )
44 frn 5719 . . . . . . . . . . . . 13  |-  ( G : U. dom  M --> U. J  ->  ran  G  C_ 
U. J )
4543, 44syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ran  G  C_  U. J
)
46453ad2ant1 1018 . . . . . . . . . . 11  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  ran  G  C_  U. J )
47 simp3 999 . . . . . . . . . . 11  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  Y  e.  ran  G )
4846, 47sseldd 3442 . . . . . . . . . 10  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  Y  e.  U. J )
4935t1sncld 20118 . . . . . . . . . 10  |-  ( ( J  e.  Fre  /\  Y  e.  U. J )  ->  { Y }  e.  ( Clsd `  J
) )
5028, 48, 49syl2anc 659 . . . . . . . . 9  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  { Y }  e.  (
Clsd `  J )
)
5127, 50sseldd 3442 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  ->  { Y }  e.  S
)
527, 12, 42, 51mbfmcnvima 28691 . . . . . . 7  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
( `' G " { Y } )  e. 
dom  M )
53 inelsiga 28569 . . . . . . 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 1230 . . . . . 6  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
( ( `' F " { X } )  i^i  ( `' G " { Y } ) )  e.  dom  M
)
55 measvxrge0 28639 . . . . . 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 659 . . . . 5  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  ( 0 [,] +oo ) )
57 elxrge0 11681 . . . . . 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 458 . . . . 5  |-  ( ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  ( 0 [,] +oo )  ->  ( M `
 ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  RR* )
5956, 58syl 17 . . . 4  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  RR* )
6059adantr 463 . . 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 9625 . . . 4  |-  0  e.  RR
6261a1i 11 . . 3  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  ( X  =/=  .0.  \/  Y  =/=  .0.  ) )  ->  0  e.  RR )
6357simprbi 462 . . . . 5  |-  ( ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  ( 0 [,] +oo )  ->  0  <_ 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) ) )
6456, 63syl 17 . . . 4  |-  ( (
ph  /\  X  e.  ran  F  /\  Y  e. 
ran  G )  -> 
0  <_  ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) ) )
6564adantr 463 . . 3  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  ( X  =/=  .0.  \/  Y  =/=  .0.  ) )  ->  0  <_  ( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) ) )
6659adantr 463 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  RR* )
674adantr 463 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  ->  M  e.  (measures `  dom  M ) )
6839adantr 463 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( `' F " { X } )  e. 
dom  M )
69 measvxrge0 28639 . . . . . . 7  |-  ( ( M  e.  (measures `  dom  M )  /\  ( `' F " { X } )  e.  dom  M )  ->  ( M `  ( `' F " { X } ) )  e.  ( 0 [,] +oo ) )
7067, 68, 69syl2anc 659 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( M `  ( `' F " { X } ) )  e.  ( 0 [,] +oo ) )
71 elxrge0 11681 . . . . . . 7  |-  ( ( M `  ( `' F " { X } ) )  e.  ( 0 [,] +oo ) 
<->  ( ( M `  ( `' F " { X } ) )  e. 
RR*  /\  0  <_  ( M `  ( `' F " { X } ) ) ) )
7271simplbi 458 . . . . . 6  |-  ( ( M `  ( `' F " { X } ) )  e.  ( 0 [,] +oo )  ->  ( M `  ( `' F " { X } ) )  e. 
RR* )
7370, 72syl 17 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( M `  ( `' F " { X } ) )  e. 
RR* )
74 pnfxr 11373 . . . . . 6  |- +oo  e.  RR*
7574a1i 11 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> +oo  e.  RR* )
7654adantr 463 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( ( `' F " { X } )  i^i  ( `' G " { Y } ) )  e.  dom  M
)
77 inss1 3658 . . . . . . 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 28649 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  <_  ( M `  ( `' F " { X } ) ) )
80 simpl1 1000 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  ->  ph )
8133anim1i 566 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( X  e.  ran  F  /\  X  =/=  .0.  ) )
82 eldifsn 4096 . . . . . . . 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 28772 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( ran  F  \  {  .0.  } ) )  -> 
( M `  ( `' F " { X } ) )  e.  ( 0 [,) +oo ) )
8580, 83, 84syl2anc 659 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( M `  ( `' F " { X } ) )  e.  ( 0 [,) +oo ) )
86 elico2 11640 . . . . . . . 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 670 . . . . . . 7  |-  ( ( M `  ( `' F " { X } ) )  e.  ( 0 [,) +oo ) 
<->  ( ( M `  ( `' F " { X } ) )  e.  RR  /\  0  <_ 
( M `  ( `' F " { X } ) )  /\  ( M `  ( `' F " { X } ) )  < +oo ) )
8887simp3bi 1014 . . . . . 6  |-  ( ( M `  ( `' F " { X } ) )  e.  ( 0 [,) +oo )  ->  ( M `  ( `' F " { X } ) )  < +oo )
8985, 88syl 17 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( M `  ( `' F " { X } ) )  < +oo )
9066, 73, 75, 79, 89xrlelttrd 11415 . . . 4  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  X  =/=  .0.  )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  < +oo )
9159adantr 463 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  e.  RR* )
924adantr 463 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  ->  M  e.  (measures `  dom  M ) )
9352adantr 463 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( `' G " { Y } )  e. 
dom  M )
94 measvxrge0 28639 . . . . . . 7  |-  ( ( M  e.  (measures `  dom  M )  /\  ( `' G " { Y } )  e.  dom  M )  ->  ( M `  ( `' G " { Y } ) )  e.  ( 0 [,] +oo ) )
9592, 93, 94syl2anc 659 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( M `  ( `' G " { Y } ) )  e.  ( 0 [,] +oo ) )
96 elxrge0 11681 . . . . . . 7  |-  ( ( M `  ( `' G " { Y } ) )  e.  ( 0 [,] +oo ) 
<->  ( ( M `  ( `' G " { Y } ) )  e. 
RR*  /\  0  <_  ( M `  ( `' G " { Y } ) ) ) )
9796simplbi 458 . . . . . 6  |-  ( ( M `  ( `' G " { Y } ) )  e.  ( 0 [,] +oo )  ->  ( M `  ( `' G " { Y } ) )  e. 
RR* )
9895, 97syl 17 . . . . 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 463 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( ( `' F " { X } )  i^i  ( `' G " { Y } ) )  e.  dom  M
)
101 inss2 3659 . . . . . . 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 28649 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  <_  ( M `  ( `' G " { Y } ) ) )
104 simpl1 1000 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  ->  ph )
10547anim1i 566 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( Y  e.  ran  G  /\  Y  =/=  .0.  ) )
106 eldifsn 4096 . . . . . . . 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 28772 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( ran  G  \  {  .0.  } ) )  -> 
( M `  ( `' G " { Y } ) )  e.  ( 0 [,) +oo ) )
109104, 107, 108syl2anc 659 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( M `  ( `' G " { Y } ) )  e.  ( 0 [,) +oo ) )
110 elico2 11640 . . . . . . . 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 670 . . . . . . 7  |-  ( ( M `  ( `' G " { Y } ) )  e.  ( 0 [,) +oo ) 
<->  ( ( M `  ( `' G " { Y } ) )  e.  RR  /\  0  <_ 
( M `  ( `' G " { Y } ) )  /\  ( M `  ( `' G " { Y } ) )  < +oo ) )
112111simp3bi 1014 . . . . . 6  |-  ( ( M `  ( `' G " { Y } ) )  e.  ( 0 [,) +oo )  ->  ( M `  ( `' G " { Y } ) )  < +oo )
113109, 112syl 17 . . . . 5  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( M `  ( `' G " { Y } ) )  < +oo )
11491, 98, 99, 103, 113xrlelttrd 11415 . . . 4  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  Y  =/=  .0.  )  -> 
( M `  (
( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  < +oo )
11590, 114jaodan 786 . . 3  |-  ( ( ( ph  /\  X  e.  ran  F  /\  Y  e.  ran  G )  /\  ( X  =/=  .0.  \/  Y  =/=  .0.  ) )  ->  ( M `  ( ( `' F " { X } )  i^i  ( `' G " { Y } ) ) )  < +oo )
116 xrre3 11424 . . 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 1231 . 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 11640 . . 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 670 . 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 1181 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 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598    \ cdif 3410    i^i cin 3412    C_ wss 3413   {csn 3971   U.cuni 4190   class class class wbr 4394   `'ccnv 4821   dom cdm 4822   ran crn 4823   "cima 4825   -->wf 5564   ` cfv 5568  (class class class)co 6277   RRcr 9520   0cc0 9521   +oocpnf 9654   RR*cxr 9656    < clt 9657    <_ cle 9658   [,)cico 11583   [,]cicc 11584   Basecbs 14839  Scalarcsca 14910   .scvsca 14911   TopOpenctopn 15034   0gc0g 15052   Topctop 19684   TopSpctps 19687   Clsdccld 19807   Frect1 20099  RRHomcrrh 28412  sigAlgebracsiga 28541  sigaGencsigagen 28572  measurescmeas 28629  MblFnMcmbfm 28684  sitgcsitg 28763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-ac2 8874  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599  ax-addf 9600  ax-mulf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-disj 4366  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-ixp 7507  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-fi 7904  df-sup 7934  df-oi 7968  df-card 8351  df-acn 8354  df-ac 8528  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-q 11227  df-rp 11265  df-xneg 11370  df-xadd 11371  df-xmul 11372  df-ioo 11585  df-ioc 11586  df-ico 11587  df-icc 11588  df-fz 11725  df-fzo 11853  df-fl 11964  df-mod 12033  df-seq 12150  df-exp 12209  df-fac 12396  df-bc 12423  df-hash 12451  df-shft 13047  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-limsup 13441  df-clim 13458  df-rlim 13459  df-sum 13656  df-ef 14010  df-sin 14012  df-cos 14013  df-pi 14015  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-starv 14922  df-sca 14923  df-vsca 14924  df-ip 14925  df-tset 14926  df-ple 14927  df-ds 14929  df-unif 14930  df-hom 14931  df-cco 14932  df-rest 15035  df-topn 15036  df-0g 15054  df-gsum 15055  df-topgen 15056  df-pt 15057  df-prds 15060  df-ordt 15113  df-xrs 15114  df-qtop 15119  df-imas 15120  df-xps 15122  df-mre 15198  df-mrc 15199  df-acs 15201  df-ps 16152  df-tsr 16153  df-plusf 16193  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-mhm 16288  df-submnd 16289  df-grp 16379  df-minusg 16380  df-sbg 16381  df-mulg 16382  df-subg 16520  df-cntz 16677  df-cmn 17122  df-abl 17123  df-mgp 17460  df-ur 17472  df-ring 17518  df-cring 17519  df-subrg 17745  df-abv 17784  df-lmod 17832  df-scaf 17833  df-sra 18136  df-rgmod 18137  df-psmet 18729  df-xmet 18730  df-met 18731  df-bl 18732  df-mopn 18733  df-fbas 18734  df-fg 18735  df-cnfld 18739  df-top 19689  df-bases 19691  df-topon 19692  df-topsp 19693  df-cld 19810  df-ntr 19811  df-cls 19812  df-nei 19890  df-lp 19928  df-perf 19929  df-cn 20019  df-cnp 20020  df-t1 20106  df-haus 20107  df-tx 20353  df-hmeo 20546  df-fil 20637  df-fm 20729  df-flim 20730  df-flf 20731  df-tmd 20861  df-tgp 20862  df-tsms 20915  df-trg 20952  df-xms 21113  df-ms 21114  df-tms 21115  df-nm 21393  df-ngp 21394  df-nrg 21396  df-nlm 21397  df-ii 21671  df-cncf 21672  df-limc 22560  df-dv 22561  df-log 23234  df-esum 28461  df-siga 28542  df-sigagen 28573  df-meas 28630  df-mbfm 28685  df-sitg 28764
This theorem is referenced by:  sibfof  28774
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