MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nmofval Unicode version

Theorem nmofval 18612
Description: Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
Hypotheses
Ref Expression
nmofval.1  |-  N  =  ( S normOp T )
nmofval.2  |-  V  =  ( Base `  S
)
nmofval.3  |-  L  =  ( norm `  S
)
nmofval.4  |-  M  =  ( norm `  T
)
Assertion
Ref Expression
nmofval  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  N  =  ( f  e.  ( S  GrpHom  T )  |->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `  x
) )  <_  (
r  x.  ( L `
 x ) ) } ,  RR* ,  `'  <  ) ) )
Distinct variable groups:    f, r, x, L    f, M, r, x    S, f, r, x    T, f, r, x    f, V, r, x    N, r, x
Allowed substitution hint:    N( f)

Proof of Theorem nmofval
Dummy variables  s 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmofval.1 . 2  |-  N  =  ( S normOp T )
2 oveq12 6022 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s  GrpHom  t )  =  ( S  GrpHom  T ) )
3 simpl 444 . . . . . . . . 9  |-  ( ( s  =  S  /\  t  =  T )  ->  s  =  S )
43fveq2d 5665 . . . . . . . 8  |-  ( ( s  =  S  /\  t  =  T )  ->  ( Base `  s
)  =  ( Base `  S ) )
5 nmofval.2 . . . . . . . 8  |-  V  =  ( Base `  S
)
64, 5syl6eqr 2430 . . . . . . 7  |-  ( ( s  =  S  /\  t  =  T )  ->  ( Base `  s
)  =  V )
7 simpr 448 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  t  =  T )  ->  t  =  T )
87fveq2d 5665 . . . . . . . . . 10  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  t
)  =  ( norm `  T ) )
9 nmofval.4 . . . . . . . . . 10  |-  M  =  ( norm `  T
)
108, 9syl6eqr 2430 . . . . . . . . 9  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  t
)  =  M )
1110fveq1d 5663 . . . . . . . 8  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( norm `  t
) `  ( f `  x ) )  =  ( M `  (
f `  x )
) )
123fveq2d 5665 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  s
)  =  ( norm `  S ) )
13 nmofval.3 . . . . . . . . . . 11  |-  L  =  ( norm `  S
)
1412, 13syl6eqr 2430 . . . . . . . . . 10  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  s
)  =  L )
1514fveq1d 5663 . . . . . . . . 9  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( norm `  s
) `  x )  =  ( L `  x ) )
1615oveq2d 6029 . . . . . . . 8  |-  ( ( s  =  S  /\  t  =  T )  ->  ( r  x.  (
( norm `  s ) `  x ) )  =  ( r  x.  ( L `  x )
) )
1711, 16breq12d 4159 . . . . . . 7  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( ( norm `  t ) `  (
f `  x )
)  <_  ( r  x.  ( ( norm `  s
) `  x )
)  <->  ( M `  ( f `  x
) )  <_  (
r  x.  ( L `
 x ) ) ) )
186, 17raleqbidv 2852 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( A. x  e.  ( Base `  s
) ( ( norm `  t ) `  (
f `  x )
)  <_  ( r  x.  ( ( norm `  s
) `  x )
)  <->  A. x  e.  V  ( M `  ( f `
 x ) )  <_  ( r  x.  ( L `  x
) ) ) )
1918rabbidv 2884 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  { r  e.  ( 0 [,)  +oo )  |  A. x  e.  (
Base `  s )
( ( norm `  t
) `  ( f `  x ) )  <_ 
( r  x.  (
( norm `  s ) `  x ) ) }  =  { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `  x
) )  <_  (
r  x.  ( L `
 x ) ) } )
2019supeq1d 7379 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  sup ( { r  e.  ( 0 [,) 
+oo )  |  A. x  e.  ( Base `  s ) ( (
norm `  t ) `  ( f `  x
) )  <_  (
r  x.  ( (
norm `  s ) `  x ) ) } ,  RR* ,  `'  <  )  =  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) } ,  RR* ,  `'  <  ) )
212, 20mpteq12dv 4221 . . 3  |-  ( ( s  =  S  /\  t  =  T )  ->  ( f  e.  ( s  GrpHom  t )  |->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  ( Base `  s
) ( ( norm `  t ) `  (
f `  x )
)  <_  ( r  x.  ( ( norm `  s
) `  x )
) } ,  RR* ,  `'  <  ) )  =  ( f  e.  ( S  GrpHom  T )  |->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `  x
) )  <_  (
r  x.  ( L `
 x ) ) } ,  RR* ,  `'  <  ) ) )
22 df-nmo 18606 . . 3  |-  normOp  =  ( s  e. NrmGrp ,  t  e. NrmGrp  |->  ( f  e.  ( s  GrpHom  t )  |->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  ( Base `  s
) ( ( norm `  t ) `  (
f `  x )
)  <_  ( r  x.  ( ( norm `  s
) `  x )
) } ,  RR* ,  `'  <  ) ) )
23 eqid 2380 . . . . 5  |-  ( f  e.  ( S  GrpHom  T )  |->  sup ( { r  e.  ( 0 [,) 
+oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) } ,  RR* ,  `'  <  ) )  =  ( f  e.  ( S  GrpHom  T )  |->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `  x
) )  <_  (
r  x.  ( L `
 x ) ) } ,  RR* ,  `'  <  ) )
24 ssrab2 3364 . . . . . . 7  |-  { r  e.  ( 0 [,) 
+oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) }  C_  (
0 [,)  +oo )
25 icossxr 10920 . . . . . . 7  |-  ( 0 [,)  +oo )  C_  RR*
2624, 25sstri 3293 . . . . . 6  |-  { r  e.  ( 0 [,) 
+oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) }  C_  RR*
27 infmxrcl 10820 . . . . . 6  |-  ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) }  C_  RR*  ->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `  x
) )  <_  (
r  x.  ( L `
 x ) ) } ,  RR* ,  `'  <  )  e.  RR* )
2826, 27mp1i 12 . . . . 5  |-  ( f  e.  ( S  GrpHom  T )  ->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `
 x ) )  <_  ( r  x.  ( L `  x
) ) } ,  RR* ,  `'  <  )  e.  RR* )
2923, 28fmpti 5824 . . . 4  |-  ( f  e.  ( S  GrpHom  T )  |->  sup ( { r  e.  ( 0 [,) 
+oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) } ,  RR* ,  `'  <  ) ) : ( S  GrpHom  T ) -->
RR*
30 ovex 6038 . . . 4  |-  ( S 
GrpHom  T )  e.  _V
31 xrex 10534 . . . 4  |-  RR*  e.  _V
32 fex2 5536 . . . 4  |-  ( ( ( f  e.  ( S  GrpHom  T )  |->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `  x
) )  <_  (
r  x.  ( L `
 x ) ) } ,  RR* ,  `'  <  ) ) : ( S  GrpHom  T ) --> RR* 
/\  ( S  GrpHom  T )  e.  _V  /\  RR* 
e.  _V )  ->  (
f  e.  ( S 
GrpHom  T )  |->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `
 x ) )  <_  ( r  x.  ( L `  x
) ) } ,  RR* ,  `'  <  )
)  e.  _V )
3329, 30, 31, 32mp3an 1279 . . 3  |-  ( f  e.  ( S  GrpHom  T )  |->  sup ( { r  e.  ( 0 [,) 
+oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) } ,  RR* ,  `'  <  ) )  e. 
_V
3421, 22, 33ovmpt2a 6136 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( S normOp T )  =  ( f  e.  ( S 
GrpHom  T )  |->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `
 x ) )  <_  ( r  x.  ( L `  x
) ) } ,  RR* ,  `'  <  )
) )
351, 34syl5eq 2424 1  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  N  =  ( f  e.  ( S  GrpHom  T )  |->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `  x
) )  <_  (
r  x.  ( L `
 x ) ) } ,  RR* ,  `'  <  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   {crab 2646   _Vcvv 2892    C_ wss 3256   class class class wbr 4146    e. cmpt 4200   `'ccnv 4810   -->wf 5383   ` cfv 5387  (class class class)co 6013   supcsup 7373   0cc0 8916    x. cmul 8921    +oocpnf 9043   RR*cxr 9045    < clt 9046    <_ cle 9047   [,)cico 10843   Basecbs 13389    GrpHom cghm 14923   normcnm 18488  NrmGrpcngp 18489   normOpcnmo 18603
This theorem is referenced by:  nmoval  18613  nmof  18617
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-ico 10847  df-nmo 18606
  Copyright terms: Public domain W3C validator