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Theorem riesz4i 23415
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 23414 . 2  |-  E. w  e.  ~H  A. v  e. 
~H  ( T `  v )  =  ( v  .ih  w )
4 r19.26 2782 . . . . 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 6030 . . . . . . . 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 23393 . . . . . . . . . 10  |-  T : ~H
--> CC
87ffvelrni 5809 . . . . . . . . 9  |-  ( v  e.  ~H  ->  ( T `  v )  e.  CC )
98subidd 9332 . . . . . . . 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 2422 . . . . . 6  |-  ( ( v  e.  ~H  /\  ( ( T `  v )  =  ( v  .ih  w )  /\  ( T `  v )  =  ( v  .ih  u ) ) )  ->  (
( v  .ih  w
)  -  ( v 
.ih  u ) )  =  0 )
1211ralimiaa 2724 . . . . 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 22369 . . . . . 6  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( w  -h  u
)  e.  ~H )
15 oveq1 6028 . . . . . . . . 9  |-  ( v  =  ( w  -h  u )  ->  (
v  .ih  w )  =  ( ( w  -h  u )  .ih  w ) )
16 oveq1 6028 . . . . . . . . 9  |-  ( v  =  ( w  -h  u )  ->  (
v  .ih  u )  =  ( ( w  -h  u )  .ih  u ) )
1715, 16oveq12d 6039 . . . . . . . 8  |-  ( v  =  ( w  -h  u )  ->  (
( v  .ih  w
)  -  ( v 
.ih  u ) )  =  ( ( ( w  -h  u ) 
.ih  w )  -  ( ( w  -h  u )  .ih  u
) ) )
1817eqeq1d 2396 . . . . . . 7  |-  ( v  =  ( w  -h  u )  ->  (
( ( v  .ih  w )  -  (
v  .ih  u )
)  =  0  <->  (
( ( w  -h  u )  .ih  w
)  -  ( ( w  -h  u ) 
.ih  u ) )  =  0 ) )
1918rspcv 2992 . . . . . 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 22476 . . . . . . . . . 10  |-  ( ( w  -h  u )  e.  ~H  ->  ( normh `  ( w  -h  u ) )  e.  RR )
2221recnd 9048 . . . . . . . . 9  |-  ( ( w  -h  u )  e.  ~H  ->  ( normh `  ( w  -h  u ) )  e.  CC )
23 sqeq0 11374 . . . . . . . . 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 22480 . . . . . . . 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 22485 . . . . . . . . 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 22444 . . . . . . . . 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 2420 . . . . . . 7  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( normh `  (
w  -h  u ) ) ^ 2 )  =  ( ( ( w  -h  u ) 
.ih  w )  -  ( ( w  -h  u )  .ih  u
) ) )
3534eqeq1d 2396 . . . . . 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 22419 . . . . . 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 2716 . 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 6029 . . . . 5  |-  ( w  =  u  ->  (
v  .ih  w )  =  ( v  .ih  u ) )
4241eqeq2d 2399 . . . 4  |-  ( w  =  u  ->  (
( T `  v
)  =  ( v 
.ih  w )  <->  ( T `  v )  =  ( v  .ih  u ) ) )
4342ralbidv 2670 . . 3  |-  ( w  =  u  ->  ( A. v  e.  ~H  ( T `  v )  =  ( v  .ih  w )  <->  A. v  e.  ~H  ( T `  v )  =  ( v  .ih  u ) ) )
4443reu4 3072 . 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 1717   A.wral 2650   E.wrex 2651   E!wreu 2652   ` cfv 5395  (class class class)co 6021   CCcc 8922   0cc0 8924    - cmin 9224   2c2 9982   ^cexp 11310   ~Hchil 22271    .ih csp 22274   normhcno 22275   0hc0v 22276    -h cmv 22277   ConFnccnfn 22305   LinFnclf 22306
This theorem is referenced by:  riesz4  23416
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cc 8249  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004  ax-hilex 22351  ax-hfvadd 22352  ax-hvcom 22353  ax-hvass 22354  ax-hv0cl 22355  ax-hvaddid 22356  ax-hfvmul 22357  ax-hvmulid 22358  ax-hvmulass 22359  ax-hvdistr1 22360  ax-hvdistr2 22361  ax-hvmul0 22362  ax-hfi 22430  ax-his1 22433  ax-his2 22434  ax-his3 22435  ax-his4 22436  ax-hcompl 22553
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-omul 6666  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-acn 7763  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-rlim 12211  df-sum 12408  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-submnd 14667  df-mulg 14743  df-cntz 15044  df-cmn 15342  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-cn 17214  df-cnp 17215  df-lm 17216  df-haus 17302  df-tx 17516  df-hmeo 17709  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-tms 18262  df-cfil 19080  df-cau 19081  df-cmet 19082  df-grpo 21628  df-gid 21629  df-ginv 21630  df-gdiv 21631  df-ablo 21719  df-subgo 21739  df-vc 21874  df-nv 21920  df-va 21923  df-ba 21924  df-sm 21925  df-0v 21926  df-vs 21927  df-nmcv 21928  df-ims 21929  df-dip 22046  df-ssp 22070  df-ph 22163  df-cbn 22214  df-hnorm 22320  df-hba 22321  df-hvsub 22323  df-hlim 22324  df-hcau 22325  df-sh 22558  df-ch 22573  df-oc 22603  df-ch0 22604  df-nlfn 23198  df-cnfn 23199  df-lnfn 23200
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