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Theorem nmofval 20305
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 6112 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s  GrpHom  t )  =  ( S  GrpHom  T ) )
3 simpl 457 . . . . . . . . 9  |-  ( ( s  =  S  /\  t  =  T )  ->  s  =  S )
43fveq2d 5707 . . . . . . . 8  |-  ( ( s  =  S  /\  t  =  T )  ->  ( Base `  s
)  =  ( Base `  S ) )
5 nmofval.2 . . . . . . . 8  |-  V  =  ( Base `  S
)
64, 5syl6eqr 2493 . . . . . . 7  |-  ( ( s  =  S  /\  t  =  T )  ->  ( Base `  s
)  =  V )
7 simpr 461 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  t  =  T )  ->  t  =  T )
87fveq2d 5707 . . . . . . . . . 10  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  t
)  =  ( norm `  T ) )
9 nmofval.4 . . . . . . . . . 10  |-  M  =  ( norm `  T
)
108, 9syl6eqr 2493 . . . . . . . . 9  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  t
)  =  M )
1110fveq1d 5705 . . . . . . . 8  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( norm `  t
) `  ( f `  x ) )  =  ( M `  (
f `  x )
) )
123fveq2d 5707 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  s
)  =  ( norm `  S ) )
13 nmofval.3 . . . . . . . . . . 11  |-  L  =  ( norm `  S
)
1412, 13syl6eqr 2493 . . . . . . . . . 10  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  s
)  =  L )
1514fveq1d 5705 . . . . . . . . 9  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( norm `  s
) `  x )  =  ( L `  x ) )
1615oveq2d 6119 . . . . . . . 8  |-  ( ( s  =  S  /\  t  =  T )  ->  ( r  x.  (
( norm `  s ) `  x ) )  =  ( r  x.  ( L `  x )
) )
1711, 16breq12d 4317 . . . . . . 7  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( ( norm `  t ) `  (
f `  x )
)  <_  ( r  x.  ( ( norm `  s
) `  x )
)  <->  ( M `  ( f `  x
) )  <_  (
r  x.  ( L `
 x ) ) ) )
186, 17raleqbidv 2943 . . . . . 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 2976 . . . . 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 7708 . . . 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 4382 . . 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 20299 . . 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 2443 . . . . 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 3449 . . . . . . 7  |-  { r  e.  ( 0 [,) +oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) }  C_  (
0 [,) +oo )
25 icossxr 11392 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR*
2624, 25sstri 3377 . . . . . 6  |-  { r  e.  ( 0 [,) +oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) }  C_  RR*
27 infmxrcl 11291 . . . . . 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 5878 . . . 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 6128 . . . 4  |-  ( S 
GrpHom  T )  e.  _V
31 xrex 11000 . . . 4  |-  RR*  e.  _V
32 fex2 6544 . . . 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 1314 . . 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 6233 . 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 2487 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2727   {crab 2731   _Vcvv 2984    C_ wss 3340   class class class wbr 4304    e. cmpt 4362   `'ccnv 4851   -->wf 5426   ` cfv 5430  (class class class)co 6103   supcsup 7702   0cc0 9294    x. cmul 9299   +oocpnf 9427   RR*cxr 9429    < clt 9430    <_ cle 9431   [,)cico 11314   Basecbs 14186    GrpHom cghm 15756   normcnm 20181  NrmGrpcngp 20182   normOpcnmo 20296
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-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-ico 11318  df-nmo 20299
This theorem is referenced by:  nmoval  20306  nmof  20310
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