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Theorem nmofval 21347
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 6305 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s  GrpHom  t )  =  ( S  GrpHom  T ) )
3 simpl 457 . . . . . . . . 9  |-  ( ( s  =  S  /\  t  =  T )  ->  s  =  S )
43fveq2d 5876 . . . . . . . 8  |-  ( ( s  =  S  /\  t  =  T )  ->  ( Base `  s
)  =  ( Base `  S ) )
5 nmofval.2 . . . . . . . 8  |-  V  =  ( Base `  S
)
64, 5syl6eqr 2516 . . . . . . 7  |-  ( ( s  =  S  /\  t  =  T )  ->  ( Base `  s
)  =  V )
7 simpr 461 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  t  =  T )  ->  t  =  T )
87fveq2d 5876 . . . . . . . . . 10  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  t
)  =  ( norm `  T ) )
9 nmofval.4 . . . . . . . . . 10  |-  M  =  ( norm `  T
)
108, 9syl6eqr 2516 . . . . . . . . 9  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  t
)  =  M )
1110fveq1d 5874 . . . . . . . 8  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( norm `  t
) `  ( f `  x ) )  =  ( M `  (
f `  x )
) )
123fveq2d 5876 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  s
)  =  ( norm `  S ) )
13 nmofval.3 . . . . . . . . . . 11  |-  L  =  ( norm `  S
)
1412, 13syl6eqr 2516 . . . . . . . . . 10  |-  ( ( s  =  S  /\  t  =  T )  ->  ( norm `  s
)  =  L )
1514fveq1d 5874 . . . . . . . . 9  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( norm `  s
) `  x )  =  ( L `  x ) )
1615oveq2d 6312 . . . . . . . 8  |-  ( ( s  =  S  /\  t  =  T )  ->  ( r  x.  (
( norm `  s ) `  x ) )  =  ( r  x.  ( L `  x )
) )
1711, 16breq12d 4469 . . . . . . 7  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( ( norm `  t ) `  (
f `  x )
)  <_  ( r  x.  ( ( norm `  s
) `  x )
)  <->  ( M `  ( f `  x
) )  <_  (
r  x.  ( L `
 x ) ) ) )
186, 17raleqbidv 3068 . . . . . 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 3101 . . . . 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 7923 . . . 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 4535 . . 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 21341 . . 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 2457 . . . . 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 3581 . . . . . . 7  |-  { r  e.  ( 0 [,) +oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) }  C_  (
0 [,) +oo )
25 icossxr 11634 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR*
2624, 25sstri 3508 . . . . . 6  |-  { r  e.  ( 0 [,) +oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) }  C_  RR*
27 infmxrcl 11533 . . . . . 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 6055 . . . 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 6324 . . . 4  |-  ( S 
GrpHom  T )  e.  _V
31 xrex 11242 . . . 4  |-  RR*  e.  _V
32 fex2 6754 . . . 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 1324 . . 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 6432 . 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 2510 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 1395    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109    C_ wss 3471   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   -->wf 5590   ` cfv 5594  (class class class)co 6296   supcsup 7918   0cc0 9509    x. cmul 9514   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646   [,)cico 11556   Basecbs 14644    GrpHom cghm 16391   normcnm 21223  NrmGrpcngp 21224   normOpcnmo 21338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-ico 11560  df-nmo 21341
This theorem is referenced by:  nmoval  21348  nmof  21352
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