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Theorem ipblnfi 25463
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 25423 . . . . . 6  |-  U  e.  NrmCVec
3 ipblnfi.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
4 ipblnfi.7 . . . . . . 7  |-  P  =  ( .iOLD `  U )
53, 4dipcl 25317 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  x  e.  X  /\  A  e.  X )  ->  (
x P A )  e.  CC )
62, 5mp3an1 1311 . . . . 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 6044 . . 3  |-  ( A  e.  X  ->  F : X --> CC )
10 eqid 2467 . . . . . . . . . . 11  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
113, 10nvscl 25213 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  y  e.  CC  /\  z  e.  X )  ->  (
y ( .sOLD `  U ) z )  e.  X )
122, 11mp3an1 1311 . . . . . . . . 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 2467 . . . . . . . . . 10  |-  ( +v
`  U )  =  ( +v `  U
)
173, 16, 4dipdir 25449 . . . . . . . . 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 1228 . . . . . . 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 25448 . . . . . . . . 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 1228 . . . . . . . 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 6298 . . . . . . 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 2508 . . . . . 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 25205 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  (
y ( .sOLD `  U ) z )  e.  X  /\  w  e.  X )  ->  (
( y ( .sOLD `  U ) z ) ( +v
`  U ) w )  e.  X )
282, 27mp3an1 1311 . . . . . . . . 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 6290 . . . . . . . 8  |-  ( x  =  ( ( y ( .sOLD `  U ) z ) ( +v `  U
) w )  -> 
( x P A )  =  ( ( ( y ( .sOLD `  U ) z ) ( +v
`  U ) w ) P A ) )
32 ovex 6308 . . . . . . . 8  |-  ( ( ( y ( .sOLD `  U ) z ) ( +v
`  U ) w ) P A )  e.  _V
3331, 8, 32fvmpt 5949 . . . . . . 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 6290 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x P A )  =  ( z P A ) )
36 ovex 6308 . . . . . . . . . 10  |-  ( z P A )  e. 
_V
3735, 8, 36fvmpt 5949 . . . . . . . . 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 6299 . . . . . . 7  |-  ( ( ( A  e.  X  /\  y  e.  CC )  /\  ( z  e.  X  /\  w  e.  X ) )  -> 
( y  x.  ( F `  z )
)  =  ( y  x.  ( z P A ) ) )
40 oveq1 6290 . . . . . . . . 9  |-  ( x  =  w  ->  (
x P A )  =  ( w P A ) )
41 ovex 6308 . . . . . . . . 9  |-  ( w P A )  e. 
_V
4240, 8, 41fvmpt 5949 . . . . . . . 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 6301 . . . . . 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 2518 . . . . 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 2885 . . . 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 2878 . . 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 25274 . . . 4  |-  C  e.  NrmCVec
5048cnnvba 25276 . . . . 5  |-  CC  =  ( BaseSet `  C )
5148cnnvg 25275 . . . . 5  |-  +  =  ( +v `  C )
5248cnnvs 25278 . . . . 5  |-  x.  =  ( .sOLD `  C
)
53 eqid 2467 . . . . 5  |-  ( U 
LnOp  C )  =  ( U  LnOp  C )
543, 50, 16, 51, 10, 52, 53islno 25360 . . . 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 2467 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
583, 57nvcl 25254 . . 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 25461 . . . . 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 5869 . . . 4  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( abs `  ( F `  z )
)  =  ( abs `  ( z P A ) ) )
6459recnd 9621 . . . . 5  |-  ( A  e.  X  ->  (
( normCV `  U ) `  A )  e.  CC )
653, 57nvcl 25254 . . . . . . 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 9621 . . . . 5  |-  ( z  e.  X  ->  (
( normCV `  U ) `  z )  e.  CC )
68 mulcom 9577 . . . . 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 4477 . . 3  |-  ( ( A  e.  X  /\  z  e.  X )  ->  ( abs `  ( F `  z )
)  <_  ( (
( normCV `  U ) `  A )  x.  (
( normCV `  U ) `  z ) ) )
7170ralrimiva 2878 . 2  |-  ( A  e.  X  ->  A. z  e.  X  ( abs `  ( F `  z
) )  <_  (
( ( normCV `  U
) `  A )  x.  ( ( normCV `  U
) `  z )
) )
7248cnnvnm 25279 . . 3  |-  abs  =  ( normCV `  C )
73 ipblnfi.l . . 3  |-  B  =  ( U  BLnOp  C )
743, 57, 72, 53, 73, 2, 49blo3i 25409 . 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 1228 1  |-  ( A  e.  X  ->  F  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   <.cop 4033   class class class wbr 4447    |-> cmpt 4505   -->wf 5583   ` cfv 5587  (class class class)co 6283   CCcc 9489   RRcr 9490    + caddc 9494    x. cmul 9496    <_ cle 9628   abscabs 13029   NrmCVeccnv 25169   +vcpv 25170   BaseSetcba 25171   .sOLDcns 25172   normCVcnmcv 25175   .iOLDcdip 25302    LnOp clno 25347    BLnOp cblo 25349   CPreHil OLDccphlo 25419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-fi 7870  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-ioo 11532  df-icc 11535  df-fz 11672  df-fzo 11792  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-sum 13471  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-rest 14677  df-topn 14678  df-0g 14696  df-gsum 14697  df-topgen 14698  df-pt 14699  df-prds 14702  df-xrs 14756  df-qtop 14761  df-imas 14762  df-xps 14764  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-submnd 15784  df-mulg 15867  df-cntz 16157  df-cmn 16603  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-cnfld 18208  df-top 19182  df-bases 19184  df-topon 19185  df-topsp 19186  df-cld 19302  df-ntr 19303  df-cls 19304  df-cn 19510  df-cnp 19511  df-t1 19597  df-haus 19598  df-tx 19814  df-hmeo 20007  df-xms 20574  df-ms 20575  df-tms 20576  df-grpo 24885  df-gid 24886  df-ginv 24887  df-gdiv 24888  df-ablo 24976  df-vc 25131  df-nv 25177  df-va 25180  df-ba 25181  df-sm 25182  df-0v 25183  df-vs 25184  df-nmcv 25185  df-ims 25186  df-dip 25303  df-lno 25351  df-nmoo 25352  df-blo 25353  df-0o 25354  df-ph 25420
This theorem is referenced by:  htthlem  25526
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