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

Theorem nmofval 18223
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 5867 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s  GrpHom  t )  =  ( S  GrpHom  T ) )
3 simpl 443 . . . . . . . . 9  |-  ( ( s  =  S  /\  t  =  T )  ->  s  =  S )
43fveq2d 5529 . . . . . . . 8  |-  ( ( s  =  S  /\  t  =  T )  ->  ( Base `  s
)  =  ( Base `  S ) )
5 nmofval.2 . . . . . . . 8  |-  V  =  ( Base `  S
)
64, 5syl6eqr 2333 . . . . . . 7  |-  ( ( s  =  S  /\  t  =  T )  ->  ( Base `  s
)  =  V )
7 simpr 447 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  t  =  T )  ->  t  =  T )
87fveq2d 5529 . . . . . . . . . 10  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  t
)  =  ( norm `  T ) )
9 nmofval.4 . . . . . . . . . 10  |-  M  =  ( norm `  T
)
108, 9syl6eqr 2333 . . . . . . . . 9  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  t
)  =  M )
1110fveq1d 5527 . . . . . . . 8  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( norm `  t
) `  ( f `  x ) )  =  ( M `  (
f `  x )
) )
123fveq2d 5529 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  s
)  =  ( norm `  S ) )
13 nmofval.3 . . . . . . . . . . 11  |-  L  =  ( norm `  S
)
1412, 13syl6eqr 2333 . . . . . . . . . 10  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  s
)  =  L )
1514fveq1d 5527 . . . . . . . . 9  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( norm `  s
) `  x )  =  ( L `  x ) )
1615oveq2d 5874 . . . . . . . 8  |-  ( ( s  =  S  /\  t  =  T )  ->  ( r  x.  (
( norm `  s ) `  x ) )  =  ( r  x.  ( L `  x )
) )
1711, 16breq12d 4036 . . . . . . 7  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( ( norm `  t ) `  (
f `  x )
)  <_  ( r  x.  ( ( norm `  s
) `  x )
)  <->  ( M `  ( f `  x
) )  <_  (
r  x.  ( L `
 x ) ) ) )
186, 17raleqbidv 2748 . . . . . 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 2780 . . . . 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 7199 . . . 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 4098 . . 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 18217 . . 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 2283 . . . . 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 3258 . . . . . . 7  |-  { r  e.  ( 0 [,) 
+oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) }  C_  (
0 [,)  +oo )
25 icossxr 10734 . . . . . . 7  |-  ( 0 [,)  +oo )  C_  RR*
2624, 25sstri 3188 . . . . . 6  |-  { r  e.  ( 0 [,) 
+oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) }  C_  RR*
27 infmxrcl 10635 . . . . . 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 11 . . . . 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 5683 . . . 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 5883 . . . 4  |-  ( S 
GrpHom  T )  e.  _V
31 xrex 10351 . . . 4  |-  RR*  e.  _V
32 fex2 5401 . . . 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 1277 . . 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 5978 . 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 2327 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 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   0cc0 8737    x. cmul 8742    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868   [,)cico 10658   Basecbs 13148    GrpHom cghm 14680   normcnm 18099  NrmGrpcngp 18100   normOpcnmo 18214
This theorem is referenced by:  nmoval  18224  nmof  18228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-ico 10662  df-nmo 18217
  Copyright terms: Public domain W3C validator