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Theorem riesz4i 23523
Description: A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nlelch.1  |-  T  e. 
LinFn
nlelch.2  |-  T  e. 
ConFn
Assertion
Ref Expression
riesz4i  |-  E! w  e.  ~H  A. v  e. 
~H  ( T `  v )  =  ( v  .ih  w )
Distinct variable group:    w, v, T

Proof of Theorem riesz4i
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 nlelch.1 . . 3  |-  T  e. 
LinFn
2 nlelch.2 . . 3  |-  T  e. 
ConFn
31, 2riesz3i 23522 . 2  |-  E. w  e.  ~H  A. v  e. 
~H  ( T `  v )  =  ( v  .ih  w )
4 r19.26 2802 . . . . 5  |-  ( A. v  e.  ~H  (
( T `  v
)  =  ( v 
.ih  w )  /\  ( T `  v )  =  ( v  .ih  u ) )  <->  ( A. v  e.  ~H  ( T `  v )  =  ( v  .ih  w )  /\  A. v  e.  ~H  ( T `  v )  =  ( v  .ih  u ) ) )
5 oveq12 6053 . . . . . . . 8  |-  ( ( ( T `  v
)  =  ( v 
.ih  w )  /\  ( T `  v )  =  ( v  .ih  u ) )  -> 
( ( T `  v )  -  ( T `  v )
)  =  ( ( v  .ih  w )  -  ( v  .ih  u ) ) )
65adantl 453 . . . . . . 7  |-  ( ( v  e.  ~H  /\  ( ( T `  v )  =  ( v  .ih  w )  /\  ( T `  v )  =  ( v  .ih  u ) ) )  ->  (
( T `  v
)  -  ( T `
 v ) )  =  ( ( v 
.ih  w )  -  ( v  .ih  u
) ) )
71lnfnfi 23501 . . . . . . . . . 10  |-  T : ~H
--> CC
87ffvelrni 5832 . . . . . . . . 9  |-  ( v  e.  ~H  ->  ( T `  v )  e.  CC )
98subidd 9359 . . . . . . . 8  |-  ( v  e.  ~H  ->  (
( T `  v
)  -  ( T `
 v ) )  =  0 )
109adantr 452 . . . . . . 7  |-  ( ( v  e.  ~H  /\  ( ( T `  v )  =  ( v  .ih  w )  /\  ( T `  v )  =  ( v  .ih  u ) ) )  ->  (
( T `  v
)  -  ( T `
 v ) )  =  0 )
116, 10eqtr3d 2442 . . . . . 6  |-  ( ( v  e.  ~H  /\  ( ( T `  v )  =  ( v  .ih  w )  /\  ( T `  v )  =  ( v  .ih  u ) ) )  ->  (
( v  .ih  w
)  -  ( v 
.ih  u ) )  =  0 )
1211ralimiaa 2744 . . . . 5  |-  ( A. v  e.  ~H  (
( T `  v
)  =  ( v 
.ih  w )  /\  ( T `  v )  =  ( v  .ih  u ) )  ->  A. v  e.  ~H  ( ( v  .ih  w )  -  (
v  .ih  u )
)  =  0 )
134, 12sylbir 205 . . . 4  |-  ( ( A. v  e.  ~H  ( T `  v )  =  ( v  .ih  w )  /\  A. v  e.  ~H  ( T `  v )  =  ( v  .ih  u ) )  ->  A. v  e.  ~H  ( ( v  .ih  w )  -  (
v  .ih  u )
)  =  0 )
14 hvsubcl 22477 . . . . . 6  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( w  -h  u
)  e.  ~H )
15 oveq1 6051 . . . . . . . . 9  |-  ( v  =  ( w  -h  u )  ->  (
v  .ih  w )  =  ( ( w  -h  u )  .ih  w ) )
16 oveq1 6051 . . . . . . . . 9  |-  ( v  =  ( w  -h  u )  ->  (
v  .ih  u )  =  ( ( w  -h  u )  .ih  u ) )
1715, 16oveq12d 6062 . . . . . . . 8  |-  ( v  =  ( w  -h  u )  ->  (
( v  .ih  w
)  -  ( v 
.ih  u ) )  =  ( ( ( w  -h  u ) 
.ih  w )  -  ( ( w  -h  u )  .ih  u
) ) )
1817eqeq1d 2416 . . . . . . 7  |-  ( v  =  ( w  -h  u )  ->  (
( ( v  .ih  w )  -  (
v  .ih  u )
)  =  0  <->  (
( ( w  -h  u )  .ih  w
)  -  ( ( w  -h  u ) 
.ih  u ) )  =  0 ) )
1918rspcv 3012 . . . . . 6  |-  ( ( w  -h  u )  e.  ~H  ->  ( A. v  e.  ~H  ( ( v  .ih  w )  -  (
v  .ih  u )
)  =  0  -> 
( ( ( w  -h  u )  .ih  w )  -  (
( w  -h  u
)  .ih  u )
)  =  0 ) )
2014, 19syl 16 . . . . 5  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( A. v  e. 
~H  ( ( v 
.ih  w )  -  ( v  .ih  u
) )  =  0  ->  ( ( ( w  -h  u ) 
.ih  w )  -  ( ( w  -h  u )  .ih  u
) )  =  0 ) )
21 normcl 22584 . . . . . . . . . 10  |-  ( ( w  -h  u )  e.  ~H  ->  ( normh `  ( w  -h  u ) )  e.  RR )
2221recnd 9074 . . . . . . . . 9  |-  ( ( w  -h  u )  e.  ~H  ->  ( normh `  ( w  -h  u ) )  e.  CC )
23 sqeq0 11405 . . . . . . . . 9  |-  ( (
normh `  ( w  -h  u ) )  e.  CC  ->  ( (
( normh `  ( w  -h  u ) ) ^
2 )  =  0  <-> 
( normh `  ( w  -h  u ) )  =  0 ) )
2422, 23syl 16 . . . . . . . 8  |-  ( ( w  -h  u )  e.  ~H  ->  (
( ( normh `  (
w  -h  u ) ) ^ 2 )  =  0  <->  ( normh `  ( w  -h  u
) )  =  0 ) )
25 norm-i 22588 . . . . . . . 8  |-  ( ( w  -h  u )  e.  ~H  ->  (
( normh `  ( w  -h  u ) )  =  0  <->  ( w  -h  u )  =  0h ) )
2624, 25bitrd 245 . . . . . . 7  |-  ( ( w  -h  u )  e.  ~H  ->  (
( ( normh `  (
w  -h  u ) ) ^ 2 )  =  0  <->  ( w  -h  u )  =  0h ) )
2714, 26syl 16 . . . . . 6  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( ( normh `  ( w  -h  u
) ) ^ 2 )  =  0  <->  (
w  -h  u )  =  0h ) )
28 normsq 22593 . . . . . . . . 9  |-  ( ( w  -h  u )  e.  ~H  ->  (
( normh `  ( w  -h  u ) ) ^
2 )  =  ( ( w  -h  u
)  .ih  ( w  -h  u ) ) )
2914, 28syl 16 . . . . . . . 8  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( normh `  (
w  -h  u ) ) ^ 2 )  =  ( ( w  -h  u )  .ih  ( w  -h  u
) ) )
30 simpl 444 . . . . . . . . 9  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  w  e.  ~H )
31 simpr 448 . . . . . . . . 9  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  u  e.  ~H )
32 his2sub2 22552 . . . . . . . . 9  |-  ( ( ( w  -h  u
)  e.  ~H  /\  w  e.  ~H  /\  u  e.  ~H )  ->  (
( w  -h  u
)  .ih  ( w  -h  u ) )  =  ( ( ( w  -h  u )  .ih  w )  -  (
( w  -h  u
)  .ih  u )
) )
3314, 30, 31, 32syl3anc 1184 . . . . . . . 8  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( w  -h  u )  .ih  (
w  -h  u ) )  =  ( ( ( w  -h  u
)  .ih  w )  -  ( ( w  -h  u )  .ih  u ) ) )
3429, 33eqtrd 2440 . . . . . . 7  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( normh `  (
w  -h  u ) ) ^ 2 )  =  ( ( ( w  -h  u ) 
.ih  w )  -  ( ( w  -h  u )  .ih  u
) ) )
3534eqeq1d 2416 . . . . . 6  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( ( normh `  ( w  -h  u
) ) ^ 2 )  =  0  <->  (
( ( w  -h  u )  .ih  w
)  -  ( ( w  -h  u ) 
.ih  u ) )  =  0 ) )
36 hvsubeq0 22527 . . . . . 6  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( w  -h  u )  =  0h  <->  w  =  u ) )
3727, 35, 363bitr3d 275 . . . . 5  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( ( ( w  -h  u ) 
.ih  w )  -  ( ( w  -h  u )  .ih  u
) )  =  0  <-> 
w  =  u ) )
3820, 37sylibd 206 . . . 4  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( A. v  e. 
~H  ( ( v 
.ih  w )  -  ( v  .ih  u
) )  =  0  ->  w  =  u ) )
3913, 38syl5 30 . . 3  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( A. v  e.  ~H  ( T `  v )  =  ( v  .ih  w )  /\  A. v  e. 
~H  ( T `  v )  =  ( v  .ih  u ) )  ->  w  =  u ) )
4039rgen2a 2736 . 2  |-  A. w  e.  ~H  A. u  e. 
~H  ( ( A. v  e.  ~H  ( T `  v )  =  ( v  .ih  w )  /\  A. v  e.  ~H  ( T `  v )  =  ( v  .ih  u ) )  ->  w  =  u )
41 oveq2 6052 . . . . 5  |-  ( w  =  u  ->  (
v  .ih  w )  =  ( v  .ih  u ) )
4241eqeq2d 2419 . . . 4  |-  ( w  =  u  ->  (
( T `  v
)  =  ( v 
.ih  w )  <->  ( T `  v )  =  ( v  .ih  u ) ) )
4342ralbidv 2690 . . 3  |-  ( w  =  u  ->  ( A. v  e.  ~H  ( T `  v )  =  ( v  .ih  w )  <->  A. v  e.  ~H  ( T `  v )  =  ( v  .ih  u ) ) )
4443reu4 3092 . 2  |-  ( E! w  e.  ~H  A. v  e.  ~H  ( T `  v )  =  ( v  .ih  w )  <->  ( E. w  e.  ~H  A. v  e.  ~H  ( T `  v )  =  ( v  .ih  w )  /\  A. w  e. 
~H  A. u  e.  ~H  ( ( A. v  e.  ~H  ( T `  v )  =  ( v  .ih  w )  /\  A. v  e. 
~H  ( T `  v )  =  ( v  .ih  u ) )  ->  w  =  u ) ) )
453, 40, 44mpbir2an 887 1  |-  E! w  e.  ~H  A. v  e. 
~H  ( T `  v )  =  ( v  .ih  w )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670   E.wrex 2671   E!wreu 2672   ` cfv 5417  (class class class)co 6044   CCcc 8948   0cc0 8950    - cmin 9251   2c2 10009   ^cexp 11341   ~Hchil 22379    .ih csp 22382   normhcno 22383   0hc0v 22384    -h cmv 22385   ConFnccnfn 22413   LinFnclf 22414
This theorem is referenced by:  riesz4  23524
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cc 8275  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028  ax-addf 9029  ax-mulf 9030  ax-hilex 22459  ax-hfvadd 22460  ax-hvcom 22461  ax-hvass 22462  ax-hv0cl 22463  ax-hvaddid 22464  ax-hfvmul 22465  ax-hvmulid 22466  ax-hvmulass 22467  ax-hvdistr1 22468  ax-hvdistr2 22469  ax-hvmul0 22470  ax-hfi 22538  ax-his1 22541  ax-his2 22542  ax-his3 22543  ax-his4 22544  ax-hcompl 22661
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6868  df-map 6983  df-pm 6984  df-ixp 7027  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-fi 7378  df-sup 7408  df-oi 7439  df-card 7786  df-acn 7789  df-cda 8008  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-7 10023  df-8 10024  df-9 10025  df-10 10026  df-n0 10182  df-z 10243  df-dec 10343  df-uz 10449  df-q 10535  df-rp 10573  df-xneg 10670  df-xadd 10671  df-xmul 10672  df-ioo 10880  df-ico 10882  df-icc 10883  df-fz 11004  df-fzo 11095  df-fl 11161  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-rlim 12242  df-sum 12439  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-starv 13503  df-sca 13504  df-vsca 13505  df-tset 13507  df-ple 13508  df-ds 13510  df-unif 13511  df-hom 13512  df-cco 13513  df-rest 13609  df-topn 13610  df-topgen 13626  df-pt 13627  df-prds 13630  df-xrs 13685  df-0g 13686  df-gsum 13687  df-qtop 13692  df-imas 13693  df-xps 13695  df-mre 13770  df-mrc 13771  df-acs 13773  df-mnd 14649  df-submnd 14698  df-mulg 14774  df-cntz 15075  df-cmn 15373  df-psmet 16653  df-xmet 16654  df-met 16655  df-bl 16656  df-mopn 16657  df-fbas 16658  df-fg 16659  df-cnfld 16663  df-top 16922  df-bases 16924  df-topon 16925  df-topsp 16926  df-cld 17042  df-ntr 17043  df-cls 17044  df-nei 17121  df-cn 17249  df-cnp 17250  df-lm 17251  df-haus 17337  df-tx 17551  df-hmeo 17744  df-fil 17835  df-fm 17927  df-flim 17928  df-flf 17929  df-xms 18307  df-ms 18308  df-tms 18309  df-cfil 19165  df-cau 19166  df-cmet 19167  df-grpo 21736  df-gid 21737  df-ginv 21738  df-gdiv 21739  df-ablo 21827  df-subgo 21847  df-vc 21982  df-nv 22028  df-va 22031  df-ba 22032  df-sm 22033  df-0v 22034  df-vs 22035  df-nmcv 22036  df-ims 22037  df-dip 22154  df-ssp 22178  df-ph 22271  df-cbn 22322  df-hnorm 22428  df-hba 22429  df-hvsub 22431  df-hlim 22432  df-hcau 22433  df-sh 22666  df-ch 22681  df-oc 22711  df-ch0 22712  df-nlfn 23306  df-cnfn 23307  df-lnfn 23308
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