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Theorem nmoofval 27617
Description: The operator norm function. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoofval.1 |- X = ( BaseSet ` U )
nmoofval.2 |- Y = ( BaseSet ` W )
nmoofval.3 |- L = ( normCV ` U )
nmoofval.4 |- M = ( normCV ` W )
nmoofval.6 |- N = ( U normOpOLD W )
Assertion
Ref Expression
nmoofval |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> N = ( t e. ( Y ^m X ) |-> sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) } , RR* , < ) ) )
Distinct variable groups:   x, t, z, U   t, W, x, z   t, X, z   t, Y, x   t, L   t, M
Allowed substitution hints:   L( x, z)   M( x, z)   N( x, z, t)   X( x)   Y( z)
Proof of Theorem nmoofval
Dummy variables u w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmoofval.6 . 2 |- N = ( U normOpOLD W )
2 fveq2 6191 . . . . . 6 |- ( u = U -> ( BaseSet ` u ) = ( BaseSet ` U ) )
3 nmoofval.1 . . . . . 6 |- X = ( BaseSet ` U )
42, 3syl6eqr 2674 . . . . 5 |- ( u = U -> ( BaseSet ` u ) = X )
54oveq2d 6666 . . . 4 |- ( u = U -> ( ( BaseSet ` w ) ^m ( BaseSet ` u ) ) = ( ( BaseSet ` w ) ^m X ) )
6 fveq2 6191 . . . . . . . . . . 11 |- ( u = U -> ( normCV ` u ) = ( normCV ` U ) )
7 nmoofval.3 . . . . . . . . . . 11 |- L = ( normCV ` U )
86, 7syl6eqr 2674 . . . . . . . . . 10 |- ( u = U -> ( normCV ` u ) = L )
98fveq1d 6193 . . . . . . . . 9 |- ( u = U -> ( ( normCV ` u ) ` z ) = ( L ` z ) )
109breq1d 4663 . . . . . . . 8 |- ( u = U -> ( ( ( normCV ` u ) ` z ) <_ 1 <-> ( L ` z ) <_ 1 ) )
1110anbi1d 741 . . . . . . 7 |- ( u = U -> ( ( ( ( normCV ` u ) ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) <-> ( ( L ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) ) )
124, 11rexeqbidv 3153 . . . . . 6 |- ( u = U -> ( E. z e. ( BaseSet ` u ) ( ( ( normCV ` u ) ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) <-> E. z e. X ( ( L ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) ) )
1312abbidv 2741 . . . . 5 |- ( u = U -> { x | E. z e. ( BaseSet ` u ) ( ( ( normCV ` u ) ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) } = { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) } )
1413supeq1d 8352 . . . 4 |- ( u = U -> sup ( { x | E. z e. ( BaseSet ` u ) ( ( ( normCV ` u ) ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) } , RR* , < ) = sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) } , RR* , < ) )
155, 14mpteq12dv 4733 . . 3 |- ( u = U -> ( t e. ( ( BaseSet ` w ) ^m ( BaseSet ` u ) ) |-> sup ( { x | E. z e. ( BaseSet ` u ) ( ( ( normCV ` u ) ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) } , RR* , < ) ) = ( t e. ( ( BaseSet ` w ) ^m X ) |-> sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) } , RR* , < ) ) )
16 fveq2 6191 . . . . . 6 |- ( w = W -> ( BaseSet ` w ) = ( BaseSet ` W ) )
17 nmoofval.2 . . . . . 6 |- Y = ( BaseSet ` W )
1816, 17syl6eqr 2674 . . . . 5 |- ( w = W -> ( BaseSet ` w ) = Y )
1918oveq1d 6665 . . . 4 |- ( w = W -> ( ( BaseSet ` w ) ^m X ) = ( Y ^m X ) )
20 fveq2 6191 . . . . . . . . . . 11 |- ( w = W -> ( normCV ` w ) = ( normCV ` W ) )
21 nmoofval.4 . . . . . . . . . . 11 |- M = ( normCV ` W )
2220, 21syl6eqr 2674 . . . . . . . . . 10 |- ( w = W -> ( normCV ` w ) = M )
2322fveq1d 6193 . . . . . . . . 9 |- ( w = W -> ( ( normCV ` w ) ` ( t ` z ) ) = ( M ` ( t ` z ) ) )
2423eqeq2d 2632 . . . . . . . 8 |- ( w = W -> ( x = ( ( normCV ` w ) ` ( t ` z ) ) <-> x = ( M ` ( t ` z ) ) ) )
2524anbi2d 740 . . . . . . 7 |- ( w = W -> ( ( ( L ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) <-> ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) ) )
2625rexbidv 3052 . . . . . 6 |- ( w = W -> ( E. z e. X ( ( L ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) <-> E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) ) )
2726abbidv 2741 . . . . 5 |- ( w = W -> { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) } = { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) } )
2827supeq1d 8352 . . . 4 |- ( w = W -> sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) } , RR* , < ) = sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) } , RR* , < ) )
2919, 28mpteq12dv 4733 . . 3 |- ( w = W -> ( t e. ( ( BaseSet ` w ) ^m X ) |-> sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) } , RR* , < ) ) = ( t e. ( Y ^m X ) |-> sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) } , RR* , < ) ) )
30 df-nmoo 27600 . . 3 |- normOpOLD = ( u e. NrmCVec , w e. NrmCVec |-> ( t e. ( ( BaseSet ` w ) ^m ( BaseSet ` u ) ) |-> sup ( { x | E. z e. ( BaseSet ` u ) ( ( ( normCV ` u ) ` z ) <_ 1 /\ x = ( ( normCV ` w ) ` ( t ` z ) ) ) } , RR* , < ) ) )
31 ovex 6678 . . . 4 |- ( Y ^m X ) e. _V
3231mptex 6486 . . 3 |- ( t e. ( Y ^m X ) |-> sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) } , RR* , < ) ) e. _V
3315, 29, 30, 32ovmpt2 6796 . 2 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( U normOpOLD W ) = ( t e. ( Y ^m X ) |-> sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) } , RR* , < ) ) )
341, 33syl5eq 2668 1 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> N = ( t e. ( Y ^m X ) |-> sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) } , RR* , < ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   supcsup 8346   1c1 9937   RR*cxr 10073   < clt 10074   <_ cle 10075   NrmCVeccnv 27439   BaseSetcba 27441   normCVcnmcv 27445   normOpOLDcnmoo 27596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sup 8348  df-nmoo 27600
This theorem is referenced by:  nmooval  27618  hhnmoi  28760
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