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Theorem nmofOLD 21754
Description: The operator norm is a function into the extended reals. (Contributed by Mario Carneiro, 18-Oct-2015.) Obsolete version of nmof 21736 as of 26-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
nmofvalOLD.1  |-  N  =  ( S normOp T )
Assertion
Ref Expression
nmofOLD  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  N :
( S  GrpHom  T ) -->
RR* )

Proof of Theorem nmofOLD
Dummy variables  f 
r  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3516 . . . . 5  |-  { r  e.  ( 0 [,) +oo )  |  A. x  e.  ( Base `  S ) ( (
norm `  T ) `  ( f `  x
) )  <_  (
r  x.  ( (
norm `  S ) `  x ) ) } 
C_  ( 0 [,) +oo )
2 icossxr 11726 . . . . 5  |-  ( 0 [,) +oo )  C_  RR*
31, 2sstri 3443 . . . 4  |-  { r  e.  ( 0 [,) +oo )  |  A. x  e.  ( Base `  S ) ( (
norm `  T ) `  ( f `  x
) )  <_  (
r  x.  ( (
norm `  S ) `  x ) ) } 
C_  RR*
4 infmxrclOLD 11609 . . . 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 13 . . 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 2453 . . 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 6051 . 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 nmofvalOLD.1 . . . 4  |-  N  =  ( S normOp T )
9 eqid 2453 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2453 . . . 4  |-  ( norm `  S )  =  (
norm `  S )
11 eqid 2453 . . . 4  |-  ( norm `  T )  =  (
norm `  T )
128, 9, 10, 11nmofvalOLD 21750 . . 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 5719 . 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 236 1  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  N :
( S  GrpHom  T ) -->
RR* )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1446    e. wcel 1889   A.wral 2739   {crab 2743    C_ wss 3406   class class class wbr 4405    |-> cmpt 4464   `'ccnv 4836   -->wf 5581   ` cfv 5585  (class class class)co 6295   supcsup 7959   0cc0 9544    x. cmul 9549   +oocpnf 9677   RR*cxr 9679    < clt 9680    <_ cle 9681   [,)cico 11644   Basecbs 15133    GrpHom cghm 16892   normcnm 21603  NrmGrpcngp 21604   normOpcnmoold 21719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-sup 7961  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-ico 11648  df-nmoOLD 21724
This theorem is referenced by:  nmoclOLD  21755  isnghmOLD  21758
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