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Theorem riesz4i 22645
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 22644 . 2  |-  E. w  e.  ~H  A. v  e. 
~H  ( T `  v )  =  ( v  .ih  w )
4 r19.26 2677 . . . . 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 5869 . . . . . . . 8  |-  ( ( ( T `  v
)  =  ( v 
.ih  w )  /\  ( T `  v )  =  ( v  .ih  u ) )  -> 
( ( T `  v )  -  ( T `  v )
)  =  ( ( v  .ih  w )  -  ( v  .ih  u ) ) )
65adantl 452 . . . . . . 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 22623 . . . . . . . . . 10  |-  T : ~H
--> CC
87ffvelrni 5666 . . . . . . . . 9  |-  ( v  e.  ~H  ->  ( T `  v )  e.  CC )
98subidd 9147 . . . . . . . 8  |-  ( v  e.  ~H  ->  (
( T `  v
)  -  ( T `
 v ) )  =  0 )
109adantr 451 . . . . . . 7  |-  ( ( v  e.  ~H  /\  ( ( T `  v )  =  ( v  .ih  w )  /\  ( T `  v )  =  ( v  .ih  u ) ) )  ->  (
( T `  v
)  -  ( T `
 v ) )  =  0 )
116, 10eqtr3d 2319 . . . . . 6  |-  ( ( v  e.  ~H  /\  ( ( T `  v )  =  ( v  .ih  w )  /\  ( T `  v )  =  ( v  .ih  u ) ) )  ->  (
( v  .ih  w
)  -  ( v 
.ih  u ) )  =  0 )
1211ralimiaa 2619 . . . . 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 204 . . . 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 21599 . . . . . 6  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( w  -h  u
)  e.  ~H )
15 oveq1 5867 . . . . . . . . 9  |-  ( v  =  ( w  -h  u )  ->  (
v  .ih  w )  =  ( ( w  -h  u )  .ih  w ) )
16 oveq1 5867 . . . . . . . . 9  |-  ( v  =  ( w  -h  u )  ->  (
v  .ih  u )  =  ( ( w  -h  u )  .ih  u ) )
1715, 16oveq12d 5878 . . . . . . . 8  |-  ( v  =  ( w  -h  u )  ->  (
( v  .ih  w
)  -  ( v 
.ih  u ) )  =  ( ( ( w  -h  u ) 
.ih  w )  -  ( ( w  -h  u )  .ih  u
) ) )
1817eqeq1d 2293 . . . . . . 7  |-  ( v  =  ( w  -h  u )  ->  (
( ( v  .ih  w )  -  (
v  .ih  u )
)  =  0  <->  (
( ( w  -h  u )  .ih  w
)  -  ( ( w  -h  u ) 
.ih  u ) )  =  0 ) )
1918rspcv 2882 . . . . . 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 15 . . . . 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 21706 . . . . . . . . . 10  |-  ( ( w  -h  u )  e.  ~H  ->  ( normh `  ( w  -h  u ) )  e.  RR )
2221recnd 8863 . . . . . . . . 9  |-  ( ( w  -h  u )  e.  ~H  ->  ( normh `  ( w  -h  u ) )  e.  CC )
23 sqeq0 11170 . . . . . . . . 9  |-  ( (
normh `  ( w  -h  u ) )  e.  CC  ->  ( (
( normh `  ( w  -h  u ) ) ^
2 )  =  0  <-> 
( normh `  ( w  -h  u ) )  =  0 ) )
2422, 23syl 15 . . . . . . . 8  |-  ( ( w  -h  u )  e.  ~H  ->  (
( ( normh `  (
w  -h  u ) ) ^ 2 )  =  0  <->  ( normh `  ( w  -h  u
) )  =  0 ) )
25 norm-i 21710 . . . . . . . 8  |-  ( ( w  -h  u )  e.  ~H  ->  (
( normh `  ( w  -h  u ) )  =  0  <->  ( w  -h  u )  =  0h ) )
2624, 25bitrd 244 . . . . . . 7  |-  ( ( w  -h  u )  e.  ~H  ->  (
( ( normh `  (
w  -h  u ) ) ^ 2 )  =  0  <->  ( w  -h  u )  =  0h ) )
2714, 26syl 15 . . . . . 6  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( ( normh `  ( w  -h  u
) ) ^ 2 )  =  0  <->  (
w  -h  u )  =  0h ) )
28 normsq 21715 . . . . . . . . 9  |-  ( ( w  -h  u )  e.  ~H  ->  (
( normh `  ( w  -h  u ) ) ^
2 )  =  ( ( w  -h  u
)  .ih  ( w  -h  u ) ) )
2914, 28syl 15 . . . . . . . 8  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( normh `  (
w  -h  u ) ) ^ 2 )  =  ( ( w  -h  u )  .ih  ( w  -h  u
) ) )
30 simpl 443 . . . . . . . . 9  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  w  e.  ~H )
31 simpr 447 . . . . . . . . 9  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  u  e.  ~H )
32 his2sub2 21674 . . . . . . . . 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 1182 . . . . . . . 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 2317 . . . . . . 7  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( normh `  (
w  -h  u ) ) ^ 2 )  =  ( ( ( w  -h  u ) 
.ih  w )  -  ( ( w  -h  u )  .ih  u
) ) )
3534eqeq1d 2293 . . . . . 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 21649 . . . . . 6  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( w  -h  u )  =  0h  <->  w  =  u ) )
3727, 35, 363bitr3d 274 . . . . 5  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( ( ( ( w  -h  u ) 
.ih  w )  -  ( ( w  -h  u )  .ih  u
) )  =  0  <-> 
w  =  u ) )
3820, 37sylibd 205 . . . 4  |-  ( ( w  e.  ~H  /\  u  e.  ~H )  ->  ( A. v  e. 
~H  ( ( v 
.ih  w )  -  ( v  .ih  u
) )  =  0  ->  w  =  u ) )
3913, 38syl5 28 . . 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 2611 . 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 5868 . . . . 5  |-  ( w  =  u  ->  (
v  .ih  w )  =  ( v  .ih  u ) )
4241eqeq2d 2296 . . . 4  |-  ( w  =  u  ->  (
( T `  v
)  =  ( v 
.ih  w )  <->  ( T `  v )  =  ( v  .ih  u ) ) )
4342ralbidv 2565 . . 3  |-  ( w  =  u  ->  ( A. v  e.  ~H  ( T `  v )  =  ( v  .ih  w )  <->  A. v  e.  ~H  ( T `  v )  =  ( v  .ih  u ) ) )
4443reu4 2961 . 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 886 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 176    /\ wa 358    = wceq 1625    e. wcel 1686   A.wral 2545   E.wrex 2546   E!wreu 2547   ` cfv 5257  (class class class)co 5860   CCcc 8737   0cc0 8739    - cmin 9039   2c2 9797   ^cexp 11106   ~Hchil 21501    .ih csp 21504   normhcno 21505   0hc0v 21506    -h cmv 21507   ConFnccnfn 21535   LinFnclf 21536
This theorem is referenced by:  riesz4  22646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cc 8063  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819  ax-hilex 21581  ax-hfvadd 21582  ax-hvcom 21583  ax-hvass 21584  ax-hv0cl 21585  ax-hvaddid 21586  ax-hfvmul 21587  ax-hvmulid 21588  ax-hvmulass 21589  ax-hvdistr1 21590  ax-hvdistr2 21591  ax-hvmul0 21592  ax-hfi 21660  ax-his1 21663  ax-his2 21664  ax-his3 21665  ax-his4 21666  ax-hcompl 21783
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-omul 6486  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-acn 7577  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-seq 11049  df-exp 11107  df-hash 11340  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-clim 11964  df-rlim 11965  df-sum 12161  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-cn 16959  df-cnp 16960  df-lm 16961  df-haus 17045  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cfil 18683  df-cau 18684  df-cmet 18685  df-grpo 20860  df-gid 20861  df-ginv 20862  df-gdiv 20863  df-ablo 20951  df-subgo 20971  df-vc 21104  df-nv 21150  df-va 21153  df-ba 21154  df-sm 21155  df-0v 21156  df-vs 21157  df-nmcv 21158  df-ims 21159  df-dip 21276  df-ssp 21300  df-ph 21393  df-cbn 21444  df-hnorm 21550  df-hba 21551  df-hvsub 21553  df-hlim 21554  df-hcau 21555  df-sh 21788  df-ch 21803  df-oc 21833  df-ch0 21834  df-nlfn 22428  df-cnfn 22429  df-lnfn 22430
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