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Theorem ipblnfi 22310
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 22270 . . . . . 6  |-  U  e.  NrmCVec
3 ipblnfi.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
4 ipblnfi.7 . . . . . . 7  |-  P  =  ( .i OLD `  U
)
53, 4dipcl 22164 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  x  e.  X  /\  A  e.  X )  ->  (
x P A )  e.  CC )
62, 5mp3an1 1266 . . . . 5  |-  ( ( x  e.  X  /\  A  e.  X )  ->  ( x P A )  e.  CC )
76ancoms 440 . . . 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 5852 . . 3  |-  ( A  e.  X  ->  F : X --> CC )
10 eqid 2404 . . . . . . . . . . 11  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
113, 10nvscl 22060 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  y  e.  CC  /\  z  e.  X )  ->  (
y ( .s OLD `  U ) z )  e.  X )
122, 11mp3an1 1266 . . . . . . . . 9  |-  ( ( y  e.  CC  /\  z  e.  X )  ->  ( y ( .s
OLD `  U )
z )  e.  X
)
1312ad2ant2lr 729 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( y ( .s
OLD `  U )
z )  e.  X
)
14 simprr 734 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  ->  w  e.  X )
15 simpll 731 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  ->  A  e.  X )
16 eqid 2404 . . . . . . . . . 10  |-  ( +v
`  U )  =  ( +v `  U
)
173, 16, 4dipdir 22296 . . . . . . . . 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 652 . . . . . . . 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 1184 . . . . . . 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 732 . . . . . . . . 9  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
y  e.  CC )
21 simprl 733 . . . . . . . . 9  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
z  e.  X )
223, 16, 10, 4, 1ipassi 22295 . . . . . . . . 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 1184 . . . . . . . 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 6055 . . . . . . 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 2436 . . . . . 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 695 . . . . . . . . 9  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  z  e.  X
)  ->  ( y
( .s OLD `  U
) z )  e.  X )
273, 16nvgcl 22052 . . . . . . . . . 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 1266 . . . . . . . . 9  |-  ( ( ( y ( .s
OLD `  U )
z )  e.  X  /\  w  e.  X
)  ->  ( (
y ( .s OLD `  U ) z ) ( +v `  U
) w )  e.  X )
2926, 28sylan 458 . . . . . . . 8  |-  ( ( ( ( A  e.  X  /\  y  e.  CC )  /\  z  e.  X )  /\  w  e.  X )  ->  (
( y ( .s
OLD `  U )
z ) ( +v
`  U ) w )  e.  X )
3029anasss 629 . . . . . . 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 6047 . . . . . . . 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 6065 . . . . . . . 8  |-  ( ( ( y ( .s
OLD `  U )
z ) ( +v
`  U ) w ) P A )  e.  _V
3331, 8, 32fvmpt 5765 . . . . . . 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 16 . . . . . 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 6047 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x P A )  =  ( z P A ) )
36 ovex 6065 . . . . . . . . . 10  |-  ( z P A )  e. 
_V
3735, 8, 36fvmpt 5765 . . . . . . . . 9  |-  ( z  e.  X  ->  ( F `  z )  =  ( z P A ) )
3837ad2antrl 709 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( F `  z
)  =  ( z P A ) )
3938oveq2d 6056 . . . . . . 7  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( y  x.  ( F `  z )
)  =  ( y  x.  ( z P A ) ) )
40 oveq1 6047 . . . . . . . . 9  |-  ( x  =  w  ->  (
x P A )  =  ( w P A ) )
41 ovex 6065 . . . . . . . . 9  |-  ( w P A )  e. 
_V
4240, 8, 41fvmpt 5765 . . . . . . . 8  |-  ( w  e.  X  ->  ( F `  w )  =  ( w P A ) )
4342ad2antll 710 . . . . . . 7  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( F `  w
)  =  ( w P A ) )
4439, 43oveq12d 6058 . . . . . 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 2446 . . . . 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 2758 . . . 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 2749 . . 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 22121 . . . 4  |-  C  e.  NrmCVec
5048cnnvba 22123 . . . . 5  |-  CC  =  ( BaseSet `  C )
5148cnnvg 22122 . . . . 5  |-  +  =  ( +v `  C )
5248cnnvs 22125 . . . . 5  |-  x.  =  ( .s OLD `  C
)
53 eqid 2404 . . . . 5  |-  ( U 
LnOp  C )  =  ( U  LnOp  C )
543, 50, 16, 51, 10, 52, 53islno 22207 . . . 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 654 . . 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 646 . 2  |-  ( A  e.  X  ->  F  e.  ( U  LnOp  C
) )
57 eqid 2404 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
583, 57nvcl 22101 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( normCV `  U ) `  A )  e.  RR )
592, 58mpan 652 . 2  |-  ( A  e.  X  ->  (
( normCV `  U ) `  A )  e.  RR )
603, 57, 4, 1sii 22308 . . . . 5  |-  ( ( z  e.  X  /\  A  e.  X )  ->  ( abs `  (
z P A ) )  <_  ( (
( normCV `  U ) `  z )  x.  (
( normCV `  U ) `  A ) ) )
6160ancoms 440 . . . 4  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( abs `  (
z P A ) )  <_  ( (
( normCV `  U ) `  z )  x.  (
( normCV `  U ) `  A ) ) )
6237adantl 453 . . . . 5  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( F `  z
)  =  ( z P A ) )
6362fveq2d 5691 . . . 4  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( abs `  ( F `  z )
)  =  ( abs `  ( z P A ) ) )
6459recnd 9070 . . . . 5  |-  ( A  e.  X  ->  (
( normCV `  U ) `  A )  e.  CC )
653, 57nvcl 22101 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  z  e.  X )  ->  (
( normCV `  U ) `  z )  e.  RR )
662, 65mpan 652 . . . . . 6  |-  ( z  e.  X  ->  (
( normCV `  U ) `  z )  e.  RR )
6766recnd 9070 . . . . 5  |-  ( z  e.  X  ->  (
( normCV `  U ) `  z )  e.  CC )
68 mulcom 9032 . . . . 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 464 . . . 4  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( ( ( normCV `  U ) `  A
)  x.  ( (
normCV
`  U ) `  z ) )  =  ( ( ( normCV `  U ) `  z
)  x.  ( (
normCV
`  U ) `  A ) ) )
7061, 63, 693brtr4d 4202 . . 3  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( abs `  ( F `  z )
)  <_  ( (
( normCV `  U ) `  A )  x.  (
( normCV `  U ) `  z ) ) )
7170ralrimiva 2749 . 2  |-  ( A  e.  X  ->  A. z  e.  X  ( abs `  ( F `  z
) )  <_  (
( ( normCV `  U
) `  A )  x.  ( ( normCV `  U
) `  z )
) )
7248cnnvnm 22126 . . 3  |-  abs  =  ( normCV `  C )
73 ipblnfi.l . . 3  |-  B  =  ( U  BLnOp  C )
743, 57, 72, 53, 73, 2, 49blo3i 22256 . 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 1184 1  |-  ( A  e.  X  ->  F  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   <.cop 3777   class class class wbr 4172    e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945    + caddc 8949    x. cmul 8951    <_ cle 9077   abscabs 11994   NrmCVeccnv 22016   +vcpv 22017   BaseSetcba 22018   .s
OLDcns 22019   normCVcnmcv 22022   .i OLDcdip 22149    LnOp clno 22194    BLnOp cblo 22196   CPreHil OLDccphlo 22266
This theorem is referenced by:  htthlem  22373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-cn 17245  df-cnp 17246  df-t1 17332  df-haus 17333  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305  df-grpo 21732  df-gid 21733  df-ginv 21734  df-gdiv 21735  df-ablo 21823  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-vs 22031  df-nmcv 22032  df-ims 22033  df-dip 22150  df-lno 22198  df-nmoo 22199  df-blo 22200  df-0o 22201  df-ph 22267
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