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Theorem nmofval 18740
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 6082 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s  GrpHom  t )  =  ( S  GrpHom  T ) )
3 simpl 444 . . . . . . . . 9  |-  ( ( s  =  S  /\  t  =  T )  ->  s  =  S )
43fveq2d 5724 . . . . . . . 8  |-  ( ( s  =  S  /\  t  =  T )  ->  ( Base `  s
)  =  ( Base `  S ) )
5 nmofval.2 . . . . . . . 8  |-  V  =  ( Base `  S
)
64, 5syl6eqr 2485 . . . . . . 7  |-  ( ( s  =  S  /\  t  =  T )  ->  ( Base `  s
)  =  V )
7 simpr 448 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  t  =  T )  ->  t  =  T )
87fveq2d 5724 . . . . . . . . . 10  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  t
)  =  ( norm `  T ) )
9 nmofval.4 . . . . . . . . . 10  |-  M  =  ( norm `  T
)
108, 9syl6eqr 2485 . . . . . . . . 9  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  t
)  =  M )
1110fveq1d 5722 . . . . . . . 8  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( norm `  t
) `  ( f `  x ) )  =  ( M `  (
f `  x )
) )
123fveq2d 5724 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  s
)  =  ( norm `  S ) )
13 nmofval.3 . . . . . . . . . . 11  |-  L  =  ( norm `  S
)
1412, 13syl6eqr 2485 . . . . . . . . . 10  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  s
)  =  L )
1514fveq1d 5722 . . . . . . . . 9  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( norm `  s
) `  x )  =  ( L `  x ) )
1615oveq2d 6089 . . . . . . . 8  |-  ( ( s  =  S  /\  t  =  T )  ->  ( r  x.  (
( norm `  s ) `  x ) )  =  ( r  x.  ( L `  x )
) )
1711, 16breq12d 4217 . . . . . . 7  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( ( norm `  t ) `  (
f `  x )
)  <_  ( r  x.  ( ( norm `  s
) `  x )
)  <->  ( M `  ( f `  x
) )  <_  (
r  x.  ( L `
 x ) ) ) )
186, 17raleqbidv 2908 . . . . . 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 2940 . . . . 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 7443 . . . 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 4279 . . 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 18734 . . 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 2435 . . . . 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 3420 . . . . . . 7  |-  { r  e.  ( 0 [,) 
+oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) }  C_  (
0 [,)  +oo )
25 icossxr 10987 . . . . . . 7  |-  ( 0 [,)  +oo )  C_  RR*
2624, 25sstri 3349 . . . . . 6  |-  { r  e.  ( 0 [,) 
+oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) }  C_  RR*
27 infmxrcl 10887 . . . . . 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 5884 . . . 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 6098 . . . 4  |-  ( S 
GrpHom  T )  e.  _V
31 xrex 10601 . . . 4  |-  RR*  e.  _V
32 fex2 5595 . . . 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 6196 . 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 2479 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 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948    C_ wss 3312   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   -->wf 5442   ` cfv 5446  (class class class)co 6073   supcsup 7437   0cc0 8982    x. cmul 8987    +oocpnf 9109   RR*cxr 9111    < clt 9112    <_ cle 9113   [,)cico 10910   Basecbs 13461    GrpHom cghm 14995   normcnm 18616  NrmGrpcngp 18617   normOpcnmo 18731
This theorem is referenced by:  nmoval  18741  nmof  18745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-ico 10914  df-nmo 18734
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