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Theorem ipblnfi 24207
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  =  ( .iOLD `  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 24167 . . . . . 6  |-  U  e.  NrmCVec
3 ipblnfi.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
4 ipblnfi.7 . . . . . . 7  |-  P  =  ( .iOLD `  U )
53, 4dipcl 24061 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  x  e.  X  /\  A  e.  X )  ->  (
x P A )  e.  CC )
62, 5mp3an1 1301 . . . . 5  |-  ( ( x  e.  X  /\  A  e.  X )  ->  ( x P A )  e.  CC )
76ancoms 453 . . . 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 5862 . . 3  |-  ( A  e.  X  ->  F : X --> CC )
10 eqid 2438 . . . . . . . . . . 11  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
113, 10nvscl 23957 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  y  e.  CC  /\  z  e.  X )  ->  (
y ( .sOLD `  U ) z )  e.  X )
122, 11mp3an1 1301 . . . . . . . . 9  |-  ( ( y  e.  CC  /\  z  e.  X )  ->  ( y ( .sOLD `  U ) z )  e.  X
)
1312ad2ant2lr 747 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( y ( .sOLD `  U ) z )  e.  X
)
14 simprr 756 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  ->  w  e.  X )
15 simpll 753 . . . . . . . 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 24193 . . . . . . . . 9  |-  ( ( U  e.  CPreHil OLD  /\  ( ( y ( .sOLD `  U
) z )  e.  X  /\  w  e.  X  /\  A  e.  X ) )  -> 
( ( ( y ( .sOLD `  U ) z ) ( +v `  U
) w ) P A )  =  ( ( ( y ( .sOLD `  U
) z ) P A )  +  ( w P A ) ) )
181, 17mpan 670 . . . . . . . 8  |-  ( ( ( y ( .sOLD `  U ) z )  e.  X  /\  w  e.  X  /\  A  e.  X
)  ->  ( (
( y ( .sOLD `  U ) z ) ( +v
`  U ) w ) P A )  =  ( ( ( y ( .sOLD `  U ) z ) P A )  +  ( w P A ) ) )
1913, 14, 15, 18syl3anc 1218 . . . . . . 7  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( ( ( y ( .sOLD `  U ) z ) ( +v `  U
) w ) P A )  =  ( ( ( y ( .sOLD `  U
) z ) P A )  +  ( w P A ) ) )
20 simplr 754 . . . . . . . . 9  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
y  e.  CC )
21 simprl 755 . . . . . . . . 9  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
z  e.  X )
223, 16, 10, 4, 1ipassi 24192 . . . . . . . . 9  |-  ( ( y  e.  CC  /\  z  e.  X  /\  A  e.  X )  ->  ( ( y ( .sOLD `  U
) z ) P A )  =  ( y  x.  ( z P A ) ) )
2320, 21, 15, 22syl3anc 1218 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( ( y ( .sOLD `  U
) z ) P A )  =  ( y  x.  ( z P A ) ) )
2423oveq1d 6101 . . . . . . 7  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( ( ( y ( .sOLD `  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 ( .sOLD `  U ) z ) ( +v `  U
) w ) P A )  =  ( ( y  x.  (
z P A ) )  +  ( w P A ) ) )
2612adantll 713 . . . . . . . . 9  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  z  e.  X
)  ->  ( y
( .sOLD `  U ) z )  e.  X )
273, 16nvgcl 23949 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  (
y ( .sOLD `  U ) z )  e.  X  /\  w  e.  X )  ->  (
( y ( .sOLD `  U ) z ) ( +v
`  U ) w )  e.  X )
282, 27mp3an1 1301 . . . . . . . . 9  |-  ( ( ( y ( .sOLD `  U ) z )  e.  X  /\  w  e.  X
)  ->  ( (
y ( .sOLD `  U ) z ) ( +v `  U
) w )  e.  X )
2926, 28sylan 471 . . . . . . . 8  |-  ( ( ( ( A  e.  X  /\  y  e.  CC )  /\  z  e.  X )  /\  w  e.  X )  ->  (
( y ( .sOLD `  U ) z ) ( +v
`  U ) w )  e.  X )
3029anasss 647 . . . . . . 7  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( ( y ( .sOLD `  U
) z ) ( +v `  U ) w )  e.  X
)
31 oveq1 6093 . . . . . . . 8  |-  ( x  =  ( ( y ( .sOLD `  U ) z ) ( +v `  U
) w )  -> 
( x P A )  =  ( ( ( y ( .sOLD `  U ) z ) ( +v
`  U ) w ) P A ) )
32 ovex 6111 . . . . . . . 8  |-  ( ( ( y ( .sOLD `  U ) z ) ( +v
`  U ) w ) P A )  e.  _V
3331, 8, 32fvmpt 5769 . . . . . . 7  |-  ( ( ( y ( .sOLD `  U ) z ) ( +v
`  U ) w )  e.  X  -> 
( F `  (
( y ( .sOLD `  U ) z ) ( +v
`  U ) w ) )  =  ( ( ( y ( .sOLD `  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 ( .sOLD `  U ) z ) ( +v
`  U ) w ) )  =  ( ( ( y ( .sOLD `  U
) z ) ( +v `  U ) w ) P A ) )
35 oveq1 6093 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x P A )  =  ( z P A ) )
36 ovex 6111 . . . . . . . . . 10  |-  ( z P A )  e. 
_V
3735, 8, 36fvmpt 5769 . . . . . . . . 9  |-  ( z  e.  X  ->  ( F `  z )  =  ( z P A ) )
3837ad2antrl 727 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( F `  z
)  =  ( z P A ) )
3938oveq2d 6102 . . . . . . 7  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( y  x.  ( F `  z )
)  =  ( y  x.  ( z P A ) ) )
40 oveq1 6093 . . . . . . . . 9  |-  ( x  =  w  ->  (
x P A )  =  ( w P A ) )
41 ovex 6111 . . . . . . . . 9  |-  ( w P A )  e. 
_V
4240, 8, 41fvmpt 5769 . . . . . . . 8  |-  ( w  e.  X  ->  ( F `  w )  =  ( w P A ) )
4342ad2antll 728 . . . . . . 7  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( F `  w
)  =  ( w P A ) )
4439, 43oveq12d 6104 . . . . . 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 ( .sOLD `  U ) z ) ( +v
`  U ) w ) )  =  ( ( y  x.  ( F `  z )
)  +  ( F `
 w ) ) )
4645ralrimivva 2803 . . . 4  |-  ( ( A  e.  X  /\  y  e.  CC )  ->  A. z  e.  X  A. w  e.  X  ( F `  ( ( y ( .sOLD `  U ) z ) ( +v `  U
) w ) )  =  ( ( y  x.  ( F `  z ) )  +  ( F `  w
) ) )
4746ralrimiva 2794 . . 3  |-  ( A  e.  X  ->  A. y  e.  CC  A. z  e.  X  A. w  e.  X  ( F `  ( ( y ( .sOLD `  U
) z ) ( +v `  U ) w ) )  =  ( ( y  x.  ( F `  z
) )  +  ( F `  w ) ) )
48 ipblnfi.c . . . . 5  |-  C  = 
<. <.  +  ,  x.  >. ,  abs >.
4948cnnv 24018 . . . 4  |-  C  e.  NrmCVec
5048cnnvba 24020 . . . . 5  |-  CC  =  ( BaseSet `  C )
5148cnnvg 24019 . . . . 5  |-  +  =  ( +v `  C )
5248cnnvs 24022 . . . . 5  |-  x.  =  ( .sOLD `  C
)
53 eqid 2438 . . . . 5  |-  ( U 
LnOp  C )  =  ( U  LnOp  C )
543, 50, 16, 51, 10, 52, 53islno 24104 . . . 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 ( .sOLD `  U
) z ) ( +v `  U ) w ) )  =  ( ( y  x.  ( F `  z
) )  +  ( F `  w ) ) ) ) )
552, 49, 54mp2an 672 . . 3  |-  ( F  e.  ( U  LnOp  C )  <->  ( F : X
--> CC  /\  A. y  e.  CC  A. z  e.  X  A. w  e.  X  ( F `  ( ( y ( .sOLD `  U
) z ) ( +v `  U ) w ) )  =  ( ( y  x.  ( F `  z
) )  +  ( F `  w ) ) ) )
569, 47, 55sylanbrc 664 . 2  |-  ( A  e.  X  ->  F  e.  ( U  LnOp  C
) )
57 eqid 2438 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
583, 57nvcl 23998 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( normCV `  U ) `  A )  e.  RR )
592, 58mpan 670 . 2  |-  ( A  e.  X  ->  (
( normCV `  U ) `  A )  e.  RR )
603, 57, 4, 1sii 24205 . . . . 5  |-  ( ( z  e.  X  /\  A  e.  X )  ->  ( abs `  (
z P A ) )  <_  ( (
( normCV `  U ) `  z )  x.  (
( normCV `  U ) `  A ) ) )
6160ancoms 453 . . . 4  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( abs `  (
z P A ) )  <_  ( (
( normCV `  U ) `  z )  x.  (
( normCV `  U ) `  A ) ) )
6237adantl 466 . . . . 5  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( F `  z
)  =  ( z P A ) )
6362fveq2d 5690 . . . 4  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( abs `  ( F `  z )
)  =  ( abs `  ( z P A ) ) )
6459recnd 9404 . . . . 5  |-  ( A  e.  X  ->  (
( normCV `  U ) `  A )  e.  CC )
653, 57nvcl 23998 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  z  e.  X )  ->  (
( normCV `  U ) `  z )  e.  RR )
662, 65mpan 670 . . . . . 6  |-  ( z  e.  X  ->  (
( normCV `  U ) `  z )  e.  RR )
6766recnd 9404 . . . . 5  |-  ( z  e.  X  ->  (
( normCV `  U ) `  z )  e.  CC )
68 mulcom 9360 . . . . 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 477 . . . 4  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( ( ( normCV `  U ) `  A
)  x.  ( (
normCV
`  U ) `  z ) )  =  ( ( ( normCV `  U ) `  z
)  x.  ( (
normCV
`  U ) `  A ) ) )
7061, 63, 693brtr4d 4317 . . 3  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( abs `  ( F `  z )
)  <_  ( (
( normCV `  U ) `  A )  x.  (
( normCV `  U ) `  z ) ) )
7170ralrimiva 2794 . 2  |-  ( A  e.  X  ->  A. z  e.  X  ( abs `  ( F `  z
) )  <_  (
( ( normCV `  U
) `  A )  x.  ( ( normCV `  U
) `  z )
) )
7248cnnvnm 24023 . . 3  |-  abs  =  ( normCV `  C )
73 ipblnfi.l . . 3  |-  B  =  ( U  BLnOp  C )
743, 57, 72, 53, 73, 2, 49blo3i 24153 . 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 1218 1  |-  ( A  e.  X  ->  F  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   <.cop 3878   class class class wbr 4287    e. cmpt 4345   -->wf 5409   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273    + caddc 9277    x. cmul 9279    <_ cle 9411   abscabs 12715   NrmCVeccnv 23913   +vcpv 23914   BaseSetcba 23915   .sOLDcns 23916   normCVcnmcv 23919   .iOLDcdip 24046    LnOp clno 24091    BLnOp cblo 24093   CPreHil OLDccphlo 24163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-icc 11299  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-cn 18806  df-cnp 18807  df-t1 18893  df-haus 18894  df-tx 19110  df-hmeo 19303  df-xms 19870  df-ms 19871  df-tms 19872  df-grpo 23629  df-gid 23630  df-ginv 23631  df-gdiv 23632  df-ablo 23720  df-vc 23875  df-nv 23921  df-va 23924  df-ba 23925  df-sm 23926  df-0v 23927  df-vs 23928  df-nmcv 23929  df-ims 23930  df-dip 24047  df-lno 24095  df-nmoo 24096  df-blo 24097  df-0o 24098  df-ph 24164
This theorem is referenced by:  htthlem  24270
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