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Theorem riesz4i 25292
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 25291 . 2  |-  E. w  e.  ~H  A. v  e. 
~H  ( T `  v )  =  ( v  .ih  w )
4 r19.26 2841 . . . . 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 6091 . . . . . . . 8  |-  ( ( ( T `  v
)  =  ( v 
.ih  w )  /\  ( T `  v )  =  ( v  .ih  u ) )  -> 
( ( T `  v )  -  ( T `  v )
)  =  ( ( v  .ih  w )  -  ( v  .ih  u ) ) )
65adantl 463 . . . . . . 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 25270 . . . . . . . . . 10  |-  T : ~H
--> CC
87ffvelrni 5832 . . . . . . . . 9  |-  ( v  e.  ~H  ->  ( T `  v )  e.  CC )
98subidd 9697 . . . . . . . 8  |-  ( v  e.  ~H  ->  (
( T `  v
)  -  ( T `
 v ) )  =  0 )
109adantr 462 . . . . . . 7  |-  ( ( v  e.  ~H  /\  ( ( T `  v )  =  ( v  .ih  w )  /\  ( T `  v )  =  ( v  .ih  u ) ) )  ->  (
( T `  v
)  -  ( T `
 v ) )  =  0 )
116, 10eqtr3d 2469 . . . . . 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 213 . . . 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 24244 . . . . . 6  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( w  -h  u
)  e.  ~H )
15 oveq1 6089 . . . . . . . . 9  |-  ( v  =  ( w  -h  u )  ->  (
v  .ih  w )  =  ( ( w  -h  u )  .ih  w ) )
16 oveq1 6089 . . . . . . . . 9  |-  ( v  =  ( w  -h  u )  ->  (
v  .ih  u )  =  ( ( w  -h  u )  .ih  u ) )
1715, 16oveq12d 6100 . . . . . . . 8  |-  ( v  =  ( w  -h  u )  ->  (
( v  .ih  w
)  -  ( v 
.ih  u ) )  =  ( ( ( w  -h  u ) 
.ih  w )  -  ( ( w  -h  u )  .ih  u
) ) )
1817eqeq1d 2443 . . . . . . 7  |-  ( v  =  ( w  -h  u )  ->  (
( ( v  .ih  w )  -  (
v  .ih  u )
)  =  0  <->  (
( ( w  -h  u )  .ih  w
)  -  ( ( w  -h  u ) 
.ih  u ) )  =  0 ) )
1918rspcv 3060 . . . . . 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 24352 . . . . . . . . . 10  |-  ( ( w  -h  u )  e.  ~H  ->  ( normh `  ( w  -h  u ) )  e.  RR )
2221recnd 9402 . . . . . . . . 9  |-  ( ( w  -h  u )  e.  ~H  ->  ( normh `  ( w  -h  u ) )  e.  CC )
23 sqeq0 11916 . . . . . . . . 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 24356 . . . . . . . 8  |-  ( ( w  -h  u )  e.  ~H  ->  (
( normh `  ( w  -h  u ) )  =  0  <->  ( w  -h  u )  =  0h ) )
2624, 25bitrd 253 . . . . . . 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 24361 . . . . . . . . 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 454 . . . . . . . . 9  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  w  e.  ~H )
31 simpr 458 . . . . . . . . 9  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  u  e.  ~H )
32 his2sub2 24320 . . . . . . . . 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 1213 . . . . . . . 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 2467 . . . . . . 7  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( normh `  (
w  -h  u ) ) ^ 2 )  =  ( ( ( w  -h  u ) 
.ih  w )  -  ( ( w  -h  u )  .ih  u
) ) )
3534eqeq1d 2443 . . . . . 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 24295 . . . . . 6  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( w  -h  u )  =  0h  <->  w  =  u ) )
3727, 35, 363bitr3d 283 . . . . 5  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( ( ( w  -h  u ) 
.ih  w )  -  ( ( w  -h  u )  .ih  u
) )  =  0  <-> 
w  =  u ) )
3820, 37sylibd 214 . . . 4  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( A. v  e. 
~H  ( ( v 
.ih  w )  -  ( v  .ih  u
) )  =  0  ->  w  =  u ) )
3913, 38syl5 32 . . 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 6090 . . . . 5  |-  ( w  =  u  ->  (
v  .ih  w )  =  ( v  .ih  u ) )
4241eqeq2d 2446 . . . 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 3144 . 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 906 1  |-  E! w  e.  ~H  A. v  e. 
~H  ( T `  v )  =  ( v  .ih  w )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1757   A.wral 2707   E.wrex 2708   E!wreu 2709   ` cfv 5408  (class class class)co 6082   CCcc 9270   0cc0 9272    - cmin 9585   2c2 10361   ^cexp 11851   ~Hchil 24146    .ih csp 24149   normhcno 24150   0hc0v 24151    -h cmv 24152   ConFnccnfn 24180   LinFnclf 24181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-rep 4393  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-inf2 7837  ax-cc 8594  ax-cnex 9328  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348  ax-pre-mulgt0 9349  ax-pre-sup 9350  ax-addf 9351  ax-mulf 9352  ax-hilex 24226  ax-hfvadd 24227  ax-hvcom 24228  ax-hvass 24229  ax-hv0cl 24230  ax-hvaddid 24231  ax-hfvmul 24232  ax-hvmulid 24233  ax-hvmulass 24234  ax-hvdistr1 24235  ax-hvdistr2 24236  ax-hvmul0 24237  ax-hfi 24306  ax-his1 24309  ax-his2 24310  ax-his3 24311  ax-his4 24312  ax-hcompl 24429
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-pss 3334  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-tp 3872  df-op 3874  df-uni 4082  df-int 4119  df-iun 4163  df-iin 4164  df-br 4283  df-opab 4341  df-mpt 4342  df-tr 4376  df-eprel 4621  df-id 4625  df-po 4630  df-so 4631  df-fr 4668  df-se 4669  df-we 4670  df-ord 4711  df-on 4712  df-lim 4713  df-suc 4714  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-isom 5417  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6311  df-om 6468  df-1st 6568  df-2nd 6569  df-supp 6682  df-recs 6820  df-rdg 6854  df-1o 6910  df-2o 6911  df-oadd 6914  df-omul 6915  df-er 7091  df-map 7206  df-pm 7207  df-ixp 7254  df-en 7301  df-dom 7302  df-sdom 7303  df-fin 7304  df-fsupp 7611  df-fi 7651  df-sup 7681  df-oi 7714  df-card 8099  df-acn 8102  df-cda 8327  df-pnf 9410  df-mnf 9411  df-xr 9412  df-ltxr 9413  df-le 9414  df-sub 9587  df-neg 9588  df-div 9984  df-nn 10313  df-2 10370  df-3 10371  df-4 10372  df-5 10373  df-6 10374  df-7 10375  df-8 10376  df-9 10377  df-10 10378  df-n0 10570  df-z 10637  df-dec 10746  df-uz 10852  df-q 10944  df-rp 10982  df-xneg 11079  df-xadd 11080  df-xmul 11081  df-ioo 11294  df-ico 11296  df-icc 11297  df-fz 11427  df-fzo 11535  df-fl 11628  df-seq 11793  df-exp 11852  df-hash 12090  df-cj 12574  df-re 12575  df-im 12576  df-sqr 12710  df-abs 12711  df-clim 12952  df-rlim 12953  df-sum 13150  df-struct 14161  df-ndx 14162  df-slot 14163  df-base 14164  df-sets 14165  df-ress 14166  df-plusg 14236  df-mulr 14237  df-starv 14238  df-sca 14239  df-vsca 14240  df-ip 14241  df-tset 14242  df-ple 14243  df-ds 14245  df-unif 14246  df-hom 14247  df-cco 14248  df-rest 14346  df-topn 14347  df-0g 14365  df-gsum 14366  df-topgen 14367  df-pt 14368  df-prds 14371  df-xrs 14425  df-qtop 14430  df-imas 14431  df-xps 14433  df-mre 14509  df-mrc 14510  df-acs 14512  df-mnd 15400  df-submnd 15450  df-mulg 15530  df-cntz 15817  df-cmn 16261  df-psmet 17655  df-xmet 17656  df-met 17657  df-bl 17658  df-mopn 17659  df-fbas 17660  df-fg 17661  df-cnfld 17665  df-top 18347  df-bases 18349  df-topon 18350  df-topsp 18351  df-cld 18467  df-ntr 18468  df-cls 18469  df-nei 18546  df-cn 18675  df-cnp 18676  df-lm 18677  df-haus 18763  df-tx 18979  df-hmeo 19172  df-fil 19263  df-fm 19355  df-flim 19356  df-flf 19357  df-xms 19739  df-ms 19740  df-tms 19741  df-cfil 20610  df-cau 20611  df-cmet 20612  df-grpo 23503  df-gid 23504  df-ginv 23505  df-gdiv 23506  df-ablo 23594  df-subgo 23614  df-vc 23749  df-nv 23795  df-va 23798  df-ba 23799  df-sm 23800  df-0v 23801  df-vs 23802  df-nmcv 23803  df-ims 23804  df-dip 23921  df-ssp 23945  df-ph 24038  df-cbn 24089  df-hnorm 24195  df-hba 24196  df-hvsub 24198  df-hlim 24199  df-hcau 24200  df-sh 24434  df-ch 24449  df-oc 24480  df-ch0 24481  df-nlfn 25075  df-cnfn 25076  df-lnfn 25077
This theorem is referenced by:  riesz4  25293
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