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Theorem nmof 18753
Description: The operator norm is a function into the extended reals. (Contributed by Mario Carneiro, 18-Oct-2015.)
Hypothesis
Ref Expression
nmofval.1  |-  N  =  ( S normOp T )
Assertion
Ref Expression
nmof  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  N :
( S  GrpHom  T ) -->
RR* )

Proof of Theorem nmof
Dummy variables  f 
r  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3428 . . . . 5  |-  { r  e.  ( 0 [,) 
+oo )  |  A. x  e.  ( Base `  S ) ( (
norm `  T ) `  ( f `  x
) )  <_  (
r  x.  ( (
norm `  S ) `  x ) ) } 
C_  ( 0 [,) 
+oo )
2 icossxr 10995 . . . . 5  |-  ( 0 [,)  +oo )  C_  RR*
31, 2sstri 3357 . . . 4  |-  { r  e.  ( 0 [,) 
+oo )  |  A. x  e.  ( Base `  S ) ( (
norm `  T ) `  ( f `  x
) )  <_  (
r  x.  ( (
norm `  S ) `  x ) ) } 
C_  RR*
4 infmxrcl 10895 . . . 4  |-  ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  ( Base `  S ) ( (
norm `  T ) `  ( f `  x
) )  <_  (
r  x.  ( (
norm `  S ) `  x ) ) } 
C_  RR*  ->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  (
Base `  S )
( ( norm `  T
) `  ( f `  x ) )  <_ 
( r  x.  (
( norm `  S ) `  x ) ) } ,  RR* ,  `'  <  )  e.  RR* )
53, 4mp1i 12 . . 3  |-  ( ( ( 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* ,  `'  <  )  e.  RR* )
6 eqid 2436 . . 3  |-  ( 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.  ( Base `  S ) ( (
norm `  T ) `  ( f `  x
) )  <_  (
r  x.  ( (
norm `  S ) `  x ) ) } ,  RR* ,  `'  <  ) )
75, 6fmptd 5893 . 2  |-  ( ( 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* ,  `'  <  ) ) : ( S 
GrpHom  T ) --> RR* )
8 nmofval.1 . . . 4  |-  N  =  ( S normOp T )
9 eqid 2436 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2436 . . . 4  |-  ( norm `  S )  =  (
norm `  S )
11 eqid 2436 . . . 4  |-  ( norm `  T )  =  (
norm `  T )
128, 9, 10, 11nmofval 18748 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  N  =  ( 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* ,  `'  <  ) ) )
1312feq1d 5580 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N : ( S  GrpHom  T ) --> RR*  <->  ( 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* ,  `'  <  ) ) : ( S 
GrpHom  T ) --> RR* )
)
147, 13mpbird 224 1  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  N :
( S  GrpHom  T ) -->
RR* )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709    C_ wss 3320   class class class wbr 4212    e. cmpt 4266   `'ccnv 4877   -->wf 5450   ` cfv 5454  (class class class)co 6081   supcsup 7445   0cc0 8990    x. cmul 8995    +oocpnf 9117   RR*cxr 9119    < clt 9120    <_ cle 9121   [,)cico 10918   Basecbs 13469    GrpHom cghm 15003   normcnm 18624  NrmGrpcngp 18625   normOpcnmo 18739
This theorem is referenced by:  nmocl  18754  isnghm  18757
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-ico 10922  df-nmo 18742
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