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Theorem ipblnfi 21450
Description: A function  F generated by varying the first argument of an inner product (with its second argument a fixed vector  A) is a bounded linear functional, i.e. a bounded linear operator from the vector space to  CC. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ipblnfi.1  |-  X  =  ( BaseSet `  U )
ipblnfi.7  |-  P  =  ( .i OLD `  U
)
ipblnfi.9  |-  U  e.  CPreHil
OLD
ipblnfi.c  |-  C  = 
<. <.  +  ,  x.  >. ,  abs >.
ipblnfi.l  |-  B  =  ( U  BLnOp  C )
ipblnfi.f  |-  F  =  ( x  e.  X  |->  ( x P A ) )
Assertion
Ref Expression
ipblnfi  |-  ( A  e.  X  ->  F  e.  B )
Distinct variable groups:    x, A    x, U    x, X    x, P
Allowed substitution hints:    B( x)    C( x)    F( x)

Proof of Theorem ipblnfi
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ipblnfi.9 . . . . . . 7  |-  U  e.  CPreHil
OLD
21phnvi 21410 . . . . . 6  |-  U  e.  NrmCVec
3 ipblnfi.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
4 ipblnfi.7 . . . . . . 7  |-  P  =  ( .i OLD `  U
)
53, 4dipcl 21304 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  x  e.  X  /\  A  e.  X )  ->  (
x P A )  e.  CC )
62, 5mp3an1 1264 . . . . 5  |-  ( ( x  e.  X  /\  A  e.  X )  ->  ( x P A )  e.  CC )
76ancoms 439 . . . 4  |-  ( ( A  e.  X  /\  x  e.  X )  ->  ( x P A )  e.  CC )
8 ipblnfi.f . . . 4  |-  F  =  ( x  e.  X  |->  ( x P A ) )
97, 8fmptd 5700 . . 3  |-  ( A  e.  X  ->  F : X --> CC )
10 eqid 2296 . . . . . . . . . . 11  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
113, 10nvscl 21200 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  y  e.  CC  /\  z  e.  X )  ->  (
y ( .s OLD `  U ) z )  e.  X )
122, 11mp3an1 1264 . . . . . . . . 9  |-  ( ( y  e.  CC  /\  z  e.  X )  ->  ( y ( .s
OLD `  U )
z )  e.  X
)
1312ad2ant2lr 728 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( y ( .s
OLD `  U )
z )  e.  X
)
14 simprr 733 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  ->  w  e.  X )
15 simpll 730 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  ->  A  e.  X )
16 eqid 2296 . . . . . . . . . 10  |-  ( +v
`  U )  =  ( +v `  U
)
173, 16, 4dipdir 21436 . . . . . . . . 9  |-  ( ( U  e.  CPreHil OLD  /\  ( ( y ( .s OLD `  U
) z )  e.  X  /\  w  e.  X  /\  A  e.  X ) )  -> 
( ( ( y ( .s OLD `  U
) z ) ( +v `  U ) w ) P A )  =  ( ( ( y ( .s
OLD `  U )
z ) P A )  +  ( w P A ) ) )
181, 17mpan 651 . . . . . . . 8  |-  ( ( ( y ( .s
OLD `  U )
z )  e.  X  /\  w  e.  X  /\  A  e.  X
)  ->  ( (
( y ( .s
OLD `  U )
z ) ( +v
`  U ) w ) P A )  =  ( ( ( y ( .s OLD `  U ) z ) P A )  +  ( w P A ) ) )
1913, 14, 15, 18syl3anc 1182 . . . . . . 7  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( ( ( y ( .s OLD `  U
) z ) ( +v `  U ) w ) P A )  =  ( ( ( y ( .s
OLD `  U )
z ) P A )  +  ( w P A ) ) )
20 simplr 731 . . . . . . . . 9  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
y  e.  CC )
21 simprl 732 . . . . . . . . 9  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
z  e.  X )
223, 16, 10, 4, 1ipassi 21435 . . . . . . . . 9  |-  ( ( y  e.  CC  /\  z  e.  X  /\  A  e.  X )  ->  ( ( y ( .s OLD `  U
) z ) P A )  =  ( y  x.  ( z P A ) ) )
2320, 21, 15, 22syl3anc 1182 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( ( y ( .s OLD `  U
) z ) P A )  =  ( y  x.  ( z P A ) ) )
2423oveq1d 5889 . . . . . . 7  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( ( ( y ( .s OLD `  U
) z ) P A )  +  ( w P A ) )  =  ( ( y  x.  ( z P A ) )  +  ( w P A ) ) )
2519, 24eqtrd 2328 . . . . . 6  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( ( ( y ( .s OLD `  U
) z ) ( +v `  U ) w ) P A )  =  ( ( y  x.  ( z P A ) )  +  ( w P A ) ) )
2612adantll 694 . . . . . . . . 9  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  z  e.  X
)  ->  ( y
( .s OLD `  U
) z )  e.  X )
273, 16nvgcl 21192 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  (
y ( .s OLD `  U ) z )  e.  X  /\  w  e.  X )  ->  (
( y ( .s
OLD `  U )
z ) ( +v
`  U ) w )  e.  X )
282, 27mp3an1 1264 . . . . . . . . 9  |-  ( ( ( y ( .s
OLD `  U )
z )  e.  X  /\  w  e.  X
)  ->  ( (
y ( .s OLD `  U ) z ) ( +v `  U
) w )  e.  X )
2926, 28sylan 457 . . . . . . . 8  |-  ( ( ( ( A  e.  X  /\  y  e.  CC )  /\  z  e.  X )  /\  w  e.  X )  ->  (
( y ( .s
OLD `  U )
z ) ( +v
`  U ) w )  e.  X )
3029anasss 628 . . . . . . 7  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( ( y ( .s OLD `  U
) z ) ( +v `  U ) w )  e.  X
)
31 oveq1 5881 . . . . . . . 8  |-  ( x  =  ( ( y ( .s OLD `  U
) z ) ( +v `  U ) w )  ->  (
x P A )  =  ( ( ( y ( .s OLD `  U ) z ) ( +v `  U
) w ) P A ) )
32 ovex 5899 . . . . . . . 8  |-  ( ( ( y ( .s
OLD `  U )
z ) ( +v
`  U ) w ) P A )  e.  _V
3331, 8, 32fvmpt 5618 . . . . . . 7  |-  ( ( ( y ( .s
OLD `  U )
z ) ( +v
`  U ) w )  e.  X  -> 
( F `  (
( y ( .s
OLD `  U )
z ) ( +v
`  U ) w ) )  =  ( ( ( y ( .s OLD `  U
) z ) ( +v `  U ) w ) P A ) )
3430, 33syl 15 . . . . . 6  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( F `  (
( y ( .s
OLD `  U )
z ) ( +v
`  U ) w ) )  =  ( ( ( y ( .s OLD `  U
) z ) ( +v `  U ) w ) P A ) )
35 oveq1 5881 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x P A )  =  ( z P A ) )
36 ovex 5899 . . . . . . . . . 10  |-  ( z P A )  e. 
_V
3735, 8, 36fvmpt 5618 . . . . . . . . 9  |-  ( z  e.  X  ->  ( F `  z )  =  ( z P A ) )
3837ad2antrl 708 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( F `  z
)  =  ( z P A ) )
3938oveq2d 5890 . . . . . . 7  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( y  x.  ( F `  z )
)  =  ( y  x.  ( z P A ) ) )
40 oveq1 5881 . . . . . . . . 9  |-  ( x  =  w  ->  (
x P A )  =  ( w P A ) )
41 ovex 5899 . . . . . . . . 9  |-  ( w P A )  e. 
_V
4240, 8, 41fvmpt 5618 . . . . . . . 8  |-  ( w  e.  X  ->  ( F `  w )  =  ( w P A ) )
4342ad2antll 709 . . . . . . 7  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( F `  w
)  =  ( w P A ) )
4439, 43oveq12d 5892 . . . . . 6  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( ( y  x.  ( F `  z
) )  +  ( F `  w ) )  =  ( ( y  x.  ( z P A ) )  +  ( w P A ) ) )
4525, 34, 443eqtr4d 2338 . . . . 5  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( F `  (
( y ( .s
OLD `  U )
z ) ( +v
`  U ) w ) )  =  ( ( y  x.  ( F `  z )
)  +  ( F `
 w ) ) )
4645ralrimivva 2648 . . . 4  |-  ( ( A  e.  X  /\  y  e.  CC )  ->  A. z  e.  X  A. w  e.  X  ( F `  ( ( y ( .s OLD `  U ) z ) ( +v `  U
) w ) )  =  ( ( y  x.  ( F `  z ) )  +  ( F `  w
) ) )
4746ralrimiva 2639 . . 3  |-  ( A  e.  X  ->  A. y  e.  CC  A. z  e.  X  A. w  e.  X  ( F `  ( ( y ( .s OLD `  U
) z ) ( +v `  U ) w ) )  =  ( ( y  x.  ( F `  z
) )  +  ( F `  w ) ) )
48 ipblnfi.c . . . . 5  |-  C  = 
<. <.  +  ,  x.  >. ,  abs >.
4948cnnv 21261 . . . 4  |-  C  e.  NrmCVec
5048cnnvba 21263 . . . . 5  |-  CC  =  ( BaseSet `  C )
5148cnnvg 21262 . . . . 5  |-  +  =  ( +v `  C )
5248cnnvs 21265 . . . . 5  |-  x.  =  ( .s OLD `  C
)
53 eqid 2296 . . . . 5  |-  ( U 
LnOp  C )  =  ( U  LnOp  C )
543, 50, 16, 51, 10, 52, 53islno 21347 . . . 4  |-  ( ( U  e.  NrmCVec  /\  C  e.  NrmCVec )  ->  ( F  e.  ( U  LnOp  C )  <->  ( F : X --> CC  /\  A. y  e.  CC  A. z  e.  X  A. w  e.  X  ( F `  ( ( y ( .s OLD `  U
) z ) ( +v `  U ) w ) )  =  ( ( y  x.  ( F `  z
) )  +  ( F `  w ) ) ) ) )
552, 49, 54mp2an 653 . . 3  |-  ( F  e.  ( U  LnOp  C )  <->  ( F : X
--> CC  /\  A. y  e.  CC  A. z  e.  X  A. w  e.  X  ( F `  ( ( y ( .s OLD `  U
) z ) ( +v `  U ) w ) )  =  ( ( y  x.  ( F `  z
) )  +  ( F `  w ) ) ) )
569, 47, 55sylanbrc 645 . 2  |-  ( A  e.  X  ->  F  e.  ( U  LnOp  C
) )
57 eqid 2296 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
583, 57nvcl 21241 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( normCV `  U ) `  A )  e.  RR )
592, 58mpan 651 . 2  |-  ( A  e.  X  ->  (
( normCV `  U ) `  A )  e.  RR )
603, 57, 4, 1sii 21448 . . . . 5  |-  ( ( z  e.  X  /\  A  e.  X )  ->  ( abs `  (
z P A ) )  <_  ( (
( normCV `  U ) `  z )  x.  (
( normCV `  U ) `  A ) ) )
6160ancoms 439 . . . 4  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( abs `  (
z P A ) )  <_  ( (
( normCV `  U ) `  z )  x.  (
( normCV `  U ) `  A ) ) )
6237adantl 452 . . . . 5  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( F `  z
)  =  ( z P A ) )
6362fveq2d 5545 . . . 4  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( abs `  ( F `  z )
)  =  ( abs `  ( z P A ) ) )
6459recnd 8877 . . . . 5  |-  ( A  e.  X  ->  (
( normCV `  U ) `  A )  e.  CC )
653, 57nvcl 21241 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  z  e.  X )  ->  (
( normCV `  U ) `  z )  e.  RR )
662, 65mpan 651 . . . . . 6  |-  ( z  e.  X  ->  (
( normCV `  U ) `  z )  e.  RR )
6766recnd 8877 . . . . 5  |-  ( z  e.  X  ->  (
( normCV `  U ) `  z )  e.  CC )
68 mulcom 8839 . . . . 5  |-  ( ( ( ( normCV `  U
) `  A )  e.  CC  /\  ( (
normCV
`  U ) `  z )  e.  CC )  ->  ( ( (
normCV
`  U ) `  A )  x.  (
( normCV `  U ) `  z ) )  =  ( ( ( normCV `  U ) `  z
)  x.  ( (
normCV
`  U ) `  A ) ) )
6964, 67, 68syl2an 463 . . . 4  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( ( ( normCV `  U ) `  A
)  x.  ( (
normCV
`  U ) `  z ) )  =  ( ( ( normCV `  U ) `  z
)  x.  ( (
normCV
`  U ) `  A ) ) )
7061, 63, 693brtr4d 4069 . . 3  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( abs `  ( F `  z )
)  <_  ( (
( normCV `  U ) `  A )  x.  (
( normCV `  U ) `  z ) ) )
7170ralrimiva 2639 . 2  |-  ( A  e.  X  ->  A. z  e.  X  ( abs `  ( F `  z
) )  <_  (
( ( normCV `  U
) `  A )  x.  ( ( normCV `  U
) `  z )
) )
7248cnnvnm 21266 . . 3  |-  abs  =  ( normCV `  C )
73 ipblnfi.l . . 3  |-  B  =  ( U  BLnOp  C )
743, 57, 72, 53, 73, 2, 49blo3i 21396 . 2  |-  ( ( F  e.  ( U 
LnOp  C )  /\  (
( normCV `  U ) `  A )  e.  RR  /\ 
A. z  e.  X  ( abs `  ( F `
 z ) )  <_  ( ( (
normCV
`  U ) `  A )  x.  (
( normCV `  U ) `  z ) ) )  ->  F  e.  B
)
7556, 59, 71, 74syl3anc 1182 1  |-  ( A  e.  X  ->  F  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   <.cop 3656   class class class wbr 4039    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752    + caddc 8756    x. cmul 8758    <_ cle 8884   abscabs 11735   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s
OLDcns 21159   normCVcnmcv 21162   .i OLDcdip 21289    LnOp clno 21334    BLnOp cblo 21336   CPreHil OLDccphlo 21406
This theorem is referenced by:  htthlem  21513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-cn 16973  df-cnp 16974  df-t1 17058  df-haus 17059  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-vs 21171  df-nmcv 21172  df-ims 21173  df-dip 21290  df-lno 21338  df-nmoo 21339  df-blo 21340  df-0o 21341  df-ph 21407
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