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Theorem riesz4i 23571
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 23570 . 2  |-  E. w  e.  ~H  A. v  e. 
~H  ( T `  v )  =  ( v  .ih  w )
4 r19.26 2840 . . . . 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 6093 . . . . . . . 8  |-  ( ( ( T `  v
)  =  ( v 
.ih  w )  /\  ( T `  v )  =  ( v  .ih  u ) )  -> 
( ( T `  v )  -  ( T `  v )
)  =  ( ( v  .ih  w )  -  ( v  .ih  u ) ) )
65adantl 454 . . . . . . 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 23549 . . . . . . . . . 10  |-  T : ~H
--> CC
87ffvelrni 5872 . . . . . . . . 9  |-  ( v  e.  ~H  ->  ( T `  v )  e.  CC )
98subidd 9404 . . . . . . . 8  |-  ( v  e.  ~H  ->  (
( T `  v
)  -  ( T `
 v ) )  =  0 )
109adantr 453 . . . . . . 7  |-  ( ( v  e.  ~H  /\  ( ( T `  v )  =  ( v  .ih  w )  /\  ( T `  v )  =  ( v  .ih  u ) ) )  ->  (
( T `  v
)  -  ( T `
 v ) )  =  0 )
116, 10eqtr3d 2472 . . . . . 6  |-  ( ( v  e.  ~H  /\  ( ( T `  v )  =  ( v  .ih  w )  /\  ( T `  v )  =  ( v  .ih  u ) ) )  ->  (
( v  .ih  w
)  -  ( v 
.ih  u ) )  =  0 )
1211ralimiaa 2782 . . . . 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 206 . . . 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 22525 . . . . . 6  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( w  -h  u
)  e.  ~H )
15 oveq1 6091 . . . . . . . . 9  |-  ( v  =  ( w  -h  u )  ->  (
v  .ih  w )  =  ( ( w  -h  u )  .ih  w ) )
16 oveq1 6091 . . . . . . . . 9  |-  ( v  =  ( w  -h  u )  ->  (
v  .ih  u )  =  ( ( w  -h  u )  .ih  u ) )
1715, 16oveq12d 6102 . . . . . . . 8  |-  ( v  =  ( w  -h  u )  ->  (
( v  .ih  w
)  -  ( v 
.ih  u ) )  =  ( ( ( w  -h  u ) 
.ih  w )  -  ( ( w  -h  u )  .ih  u
) ) )
1817eqeq1d 2446 . . . . . . 7  |-  ( v  =  ( w  -h  u )  ->  (
( ( v  .ih  w )  -  (
v  .ih  u )
)  =  0  <->  (
( ( w  -h  u )  .ih  w
)  -  ( ( w  -h  u ) 
.ih  u ) )  =  0 ) )
1918rspcv 3050 . . . . . 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 22632 . . . . . . . . . 10  |-  ( ( w  -h  u )  e.  ~H  ->  ( normh `  ( w  -h  u ) )  e.  RR )
2221recnd 9119 . . . . . . . . 9  |-  ( ( w  -h  u )  e.  ~H  ->  ( normh `  ( w  -h  u ) )  e.  CC )
23 sqeq0 11451 . . . . . . . . 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 22636 . . . . . . . 8  |-  ( ( w  -h  u )  e.  ~H  ->  (
( normh `  ( w  -h  u ) )  =  0  <->  ( w  -h  u )  =  0h ) )
2624, 25bitrd 246 . . . . . . 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 22641 . . . . . . . . 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 445 . . . . . . . . 9  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  w  e.  ~H )
31 simpr 449 . . . . . . . . 9  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  u  e.  ~H )
32 his2sub2 22600 . . . . . . . . 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 1185 . . . . . . . 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 2470 . . . . . . 7  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( normh `  (
w  -h  u ) ) ^ 2 )  =  ( ( ( w  -h  u ) 
.ih  w )  -  ( ( w  -h  u )  .ih  u
) ) )
3534eqeq1d 2446 . . . . . 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 22575 . . . . . 6  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( w  -h  u )  =  0h  <->  w  =  u ) )
3727, 35, 363bitr3d 276 . . . . 5  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( ( ( w  -h  u ) 
.ih  w )  -  ( ( w  -h  u )  .ih  u
) )  =  0  <-> 
w  =  u ) )
3820, 37sylibd 207 . . . 4  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( A. v  e. 
~H  ( ( v 
.ih  w )  -  ( v  .ih  u
) )  =  0  ->  w  =  u ) )
3913, 38syl5 31 . . 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 2774 . 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 6092 . . . . 5  |-  ( w  =  u  ->  (
v  .ih  w )  =  ( v  .ih  u ) )
4241eqeq2d 2449 . . . 4  |-  ( w  =  u  ->  (
( T `  v
)  =  ( v 
.ih  w )  <->  ( T `  v )  =  ( v  .ih  u ) ) )
4342ralbidv 2727 . . 3  |-  ( w  =  u  ->  ( A. v  e.  ~H  ( T `  v )  =  ( v  .ih  w )  <->  A. v  e.  ~H  ( T `  v )  =  ( v  .ih  u ) ) )
4443reu4 3130 . 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 888 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   E!wreu 2709   ` cfv 5457  (class class class)co 6084   CCcc 8993   0cc0 8995    - cmin 9296   2c2 10054   ^cexp 11387   ~Hchil 22427    .ih csp 22430   normhcno 22431   0hc0v 22432    -h cmv 22433   ConFnccnfn 22461   LinFnclf 22462
This theorem is referenced by:  riesz4  23572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cc 8320  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075  ax-hilex 22507  ax-hfvadd 22508  ax-hvcom 22509  ax-hvass 22510  ax-hv0cl 22511  ax-hvaddid 22512  ax-hfvmul 22513  ax-hvmulid 22514  ax-hvmulass 22515  ax-hvdistr1 22516  ax-hvdistr2 22517  ax-hvmul0 22518  ax-hfi 22586  ax-his1 22589  ax-his2 22590  ax-his3 22591  ax-his4 22592  ax-hcompl 22709
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 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-omul 6732  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-acn 7834  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-cn 17296  df-cnp 17297  df-lm 17298  df-haus 17384  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cfil 19213  df-cau 19214  df-cmet 19215  df-grpo 21784  df-gid 21785  df-ginv 21786  df-gdiv 21787  df-ablo 21875  df-subgo 21895  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-vs 22083  df-nmcv 22084  df-ims 22085  df-dip 22202  df-ssp 22226  df-ph 22319  df-cbn 22370  df-hnorm 22476  df-hba 22477  df-hvsub 22479  df-hlim 22480  df-hcau 22481  df-sh 22714  df-ch 22729  df-oc 22759  df-ch0 22760  df-nlfn 23354  df-cnfn 23355  df-lnfn 23356
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