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Theorem nmoval 18276
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
nmoval  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( N `  F )  =  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( F `
 x ) )  <_  ( r  x.  ( L `  x
) ) } ,  RR* ,  `'  <  )
)
Distinct variable groups:    x, r, L    M, r, x    S, r, x    T, r, x    F, r, x    V, r, x    N, r, x

Proof of Theorem nmoval
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 nmofval.1 . . . . 5  |-  N  =  ( S normOp T )
2 nmofval.2 . . . . 5  |-  V  =  ( Base `  S
)
3 nmofval.3 . . . . 5  |-  L  =  ( norm `  S
)
4 nmofval.4 . . . . 5  |-  M  =  ( norm `  T
)
51, 2, 3, 4nmofval 18275 . . . 4  |-  ( ( 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* ,  `'  <  ) ) )
65fveq1d 5565 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  F )  =  ( ( f  e.  ( S  GrpHom  T )  |->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `  x
) )  <_  (
r  x.  ( L `
 x ) ) } ,  RR* ,  `'  <  ) ) `  F
) )
7 fveq1 5562 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
87fveq2d 5567 . . . . . . . 8  |-  ( f  =  F  ->  ( M `  ( f `  x ) )  =  ( M `  ( F `  x )
) )
98breq1d 4070 . . . . . . 7  |-  ( f  =  F  ->  (
( M `  (
f `  x )
)  <_  ( r  x.  ( L `  x
) )  <->  ( M `  ( F `  x
) )  <_  (
r  x.  ( L `
 x ) ) ) )
109ralbidv 2597 . . . . . 6  |-  ( f  =  F  ->  ( A. x  e.  V  ( M `  ( f `
 x ) )  <_  ( r  x.  ( L `  x
) )  <->  A. x  e.  V  ( M `  ( F `  x
) )  <_  (
r  x.  ( L `
 x ) ) ) )
1110rabbidv 2814 . . . . 5  |-  ( f  =  F  ->  { r  e.  ( 0 [,) 
+oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_ 
( r  x.  ( L `  x )
) }  =  {
r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( F `  x ) )  <_ 
( r  x.  ( L `  x )
) } )
1211supeq1d 7244 . . . 4  |-  ( f  =  F  ->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `
 x ) )  <_  ( r  x.  ( L `  x
) ) } ,  RR* ,  `'  <  )  =  sup ( { r  e.  ( 0 [,) 
+oo )  |  A. x  e.  V  ( M `  ( F `  x ) )  <_ 
( r  x.  ( L `  x )
) } ,  RR* ,  `'  <  ) )
13 eqid 2316 . . . 4  |-  ( 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* ,  `'  <  ) )
14 xrltso 10522 . . . . . 6  |-  <  Or  RR*
15 cnvso 5251 . . . . . 6  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
1614, 15mpbi 199 . . . . 5  |-  `'  <  Or 
RR*
1716supex 7259 . . . 4  |-  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( F `
 x ) )  <_  ( r  x.  ( L `  x
) ) } ,  RR* ,  `'  <  )  e.  _V
1812, 13, 17fvmpt 5640 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  ( (
f  e.  ( S 
GrpHom  T )  |->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `
 x ) )  <_  ( r  x.  ( L `  x
) ) } ,  RR* ,  `'  <  )
) `  F )  =  sup ( { r  e.  ( 0 [,) 
+oo )  |  A. x  e.  V  ( M `  ( F `  x ) )  <_ 
( r  x.  ( L `  x )
) } ,  RR* ,  `'  <  ) )
196, 18sylan9eq 2368 . 2  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  F  e.  ( S  GrpHom  T ) )  ->  ( N `  F )  =  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( F `
 x ) )  <_  ( r  x.  ( L `  x
) ) } ,  RR* ,  `'  <  )
)
20193impa 1146 1  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( N `  F )  =  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 358    /\ w3a 934    = wceq 1633    e. wcel 1701   A.wral 2577   {crab 2581   class class class wbr 4060    e. cmpt 4114    Or wor 4350   `'ccnv 4725   ` cfv 5292  (class class class)co 5900   supcsup 7238   0cc0 8782    x. cmul 8787    +oocpnf 8909   RR*cxr 8911    < clt 8912    <_ cle 8913   [,)cico 10705   Basecbs 13195    GrpHom cghm 14729   normcnm 18151  NrmGrpcngp 18152   normOpcnmo 18266
This theorem is referenced by:  nmogelb  18277  nmolb  18278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-ico 10709  df-nmo 18269
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