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Theorem ipblnfi 22359
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 22319 . . . . . 6  |-  U  e.  NrmCVec
3 ipblnfi.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
4 ipblnfi.7 . . . . . . 7  |-  P  =  ( .i OLD `  U
)
53, 4dipcl 22213 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  x  e.  X  /\  A  e.  X )  ->  (
x P A )  e.  CC )
62, 5mp3an1 1267 . . . . 5  |-  ( ( x  e.  X  /\  A  e.  X )  ->  ( x P A )  e.  CC )
76ancoms 441 . . . 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 5895 . . 3  |-  ( A  e.  X  ->  F : X --> CC )
10 eqid 2438 . . . . . . . . . . 11  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
113, 10nvscl 22109 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  y  e.  CC  /\  z  e.  X )  ->  (
y ( .s OLD `  U ) z )  e.  X )
122, 11mp3an1 1267 . . . . . . . . 9  |-  ( ( y  e.  CC  /\  z  e.  X )  ->  ( y ( .s
OLD `  U )
z )  e.  X
)
1312ad2ant2lr 730 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( y ( .s
OLD `  U )
z )  e.  X
)
14 simprr 735 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  ->  w  e.  X )
15 simpll 732 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  ->  A  e.  X )
16 eqid 2438 . . . . . . . . . 10  |-  ( +v
`  U )  =  ( +v `  U
)
173, 16, 4dipdir 22345 . . . . . . . . 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 653 . . . . . . . 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 1185 . . . . . . 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 733 . . . . . . . . 9  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
y  e.  CC )
21 simprl 734 . . . . . . . . 9  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
z  e.  X )
223, 16, 10, 4, 1ipassi 22344 . . . . . . . . 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 1185 . . . . . . . 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 6098 . . . . . . 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 2470 . . . . . 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 696 . . . . . . . . 9  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  z  e.  X
)  ->  ( y
( .s OLD `  U
) z )  e.  X )
273, 16nvgcl 22101 . . . . . . . . . 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 1267 . . . . . . . . 9  |-  ( ( ( y ( .s
OLD `  U )
z )  e.  X  /\  w  e.  X
)  ->  ( (
y ( .s OLD `  U ) z ) ( +v `  U
) w )  e.  X )
2926, 28sylan 459 . . . . . . . 8  |-  ( ( ( ( A  e.  X  /\  y  e.  CC )  /\  z  e.  X )  /\  w  e.  X )  ->  (
( y ( .s
OLD `  U )
z ) ( +v
`  U ) w )  e.  X )
3029anasss 630 . . . . . . 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 6090 . . . . . . . 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 6108 . . . . . . . 8  |-  ( ( ( y ( .s
OLD `  U )
z ) ( +v
`  U ) w ) P A )  e.  _V
3331, 8, 32fvmpt 5808 . . . . . . 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 6090 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x P A )  =  ( z P A ) )
36 ovex 6108 . . . . . . . . . 10  |-  ( z P A )  e. 
_V
3735, 8, 36fvmpt 5808 . . . . . . . . 9  |-  ( z  e.  X  ->  ( F `  z )  =  ( z P A ) )
3837ad2antrl 710 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( F `  z
)  =  ( z P A ) )
3938oveq2d 6099 . . . . . . 7  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( y  x.  ( F `  z )
)  =  ( y  x.  ( z P A ) ) )
40 oveq1 6090 . . . . . . . . 9  |-  ( x  =  w  ->  (
x P A )  =  ( w P A ) )
41 ovex 6108 . . . . . . . . 9  |-  ( w P A )  e. 
_V
4240, 8, 41fvmpt 5808 . . . . . . . 8  |-  ( w  e.  X  ->  ( F `  w )  =  ( w P A ) )
4342ad2antll 711 . . . . . . 7  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( F `  w
)  =  ( w P A ) )
4439, 43oveq12d 6101 . . . . . 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 2480 . . . . 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 2800 . . . 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 2791 . . 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 22170 . . . 4  |-  C  e.  NrmCVec
5048cnnvba 22172 . . . . 5  |-  CC  =  ( BaseSet `  C )
5148cnnvg 22171 . . . . 5  |-  +  =  ( +v `  C )
5248cnnvs 22174 . . . . 5  |-  x.  =  ( .s OLD `  C
)
53 eqid 2438 . . . . 5  |-  ( U 
LnOp  C )  =  ( U  LnOp  C )
543, 50, 16, 51, 10, 52, 53islno 22256 . . . 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 655 . . 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 647 . 2  |-  ( A  e.  X  ->  F  e.  ( U  LnOp  C
) )
57 eqid 2438 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
583, 57nvcl 22150 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( normCV `  U ) `  A )  e.  RR )
592, 58mpan 653 . 2  |-  ( A  e.  X  ->  (
( normCV `  U ) `  A )  e.  RR )
603, 57, 4, 1sii 22357 . . . . 5  |-  ( ( z  e.  X  /\  A  e.  X )  ->  ( abs `  (
z P A ) )  <_  ( (
( normCV `  U ) `  z )  x.  (
( normCV `  U ) `  A ) ) )
6160ancoms 441 . . . 4  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( abs `  (
z P A ) )  <_  ( (
( normCV `  U ) `  z )  x.  (
( normCV `  U ) `  A ) ) )
6237adantl 454 . . . . 5  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( F `  z
)  =  ( z P A ) )
6362fveq2d 5734 . . . 4  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( abs `  ( F `  z )
)  =  ( abs `  ( z P A ) ) )
6459recnd 9116 . . . . 5  |-  ( A  e.  X  ->  (
( normCV `  U ) `  A )  e.  CC )
653, 57nvcl 22150 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  z  e.  X )  ->  (
( normCV `  U ) `  z )  e.  RR )
662, 65mpan 653 . . . . . 6  |-  ( z  e.  X  ->  (
( normCV `  U ) `  z )  e.  RR )
6766recnd 9116 . . . . 5  |-  ( z  e.  X  ->  (
( normCV `  U ) `  z )  e.  CC )
68 mulcom 9078 . . . . 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 465 . . . 4  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( ( ( normCV `  U ) `  A
)  x.  ( (
normCV
`  U ) `  z ) )  =  ( ( ( normCV `  U ) `  z
)  x.  ( (
normCV
`  U ) `  A ) ) )
7061, 63, 693brtr4d 4244 . . 3  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( abs `  ( F `  z )
)  <_  ( (
( normCV `  U ) `  A )  x.  (
( normCV `  U ) `  z ) ) )
7170ralrimiva 2791 . 2  |-  ( A  e.  X  ->  A. z  e.  X  ( abs `  ( F `  z
) )  <_  (
( ( normCV `  U
) `  A )  x.  ( ( normCV `  U
) `  z )
) )
7248cnnvnm 22175 . . 3  |-  abs  =  ( normCV `  C )
73 ipblnfi.l . . 3  |-  B  =  ( U  BLnOp  C )
743, 57, 72, 53, 73, 2, 49blo3i 22305 . 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 1185 1  |-  ( A  e.  X  ->  F  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   <.cop 3819   class class class wbr 4214    e. cmpt 4268   -->wf 5452   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991    + caddc 8995    x. cmul 8997    <_ cle 9123   abscabs 12041   NrmCVeccnv 22065   +vcpv 22066   BaseSetcba 22067   .s
OLDcns 22068   normCVcnmcv 22071   .i OLDcdip 22198    LnOp clno 22243    BLnOp cblo 22245   CPreHil OLDccphlo 22315
This theorem is referenced by:  htthlem  22422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-icc 10925  df-fz 11046  df-fzo 11138  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-cn 17293  df-cnp 17294  df-t1 17380  df-haus 17381  df-tx 17596  df-hmeo 17789  df-xms 18352  df-ms 18353  df-tms 18354  df-grpo 21781  df-gid 21782  df-ginv 21783  df-gdiv 21784  df-ablo 21872  df-vc 22027  df-nv 22073  df-va 22076  df-ba 22077  df-sm 22078  df-0v 22079  df-vs 22080  df-nmcv 22081  df-ims 22082  df-dip 22199  df-lno 22247  df-nmoo 22248  df-blo 22249  df-0o 22250  df-ph 22316
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