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Theorem hhph 22637
Description: The Hilbert space of the Hilbert Space Explorer is an inner product space. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
hhnv.1  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
Assertion
Ref Expression
hhph  |-  U  e.  CPreHil
OLD

Proof of Theorem hhph
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2408 . . 3  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
21hhnv 22624 . 2  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
3 normpar 22614 . . . 4  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( ( normh `  ( x  -h  y
) ) ^ 2 )  +  ( (
normh `  ( x  +h  y ) ) ^
2 ) )  =  ( ( 2  x.  ( ( normh `  x
) ^ 2 ) )  +  ( 2  x.  ( ( normh `  y ) ^ 2 ) ) ) )
4 hvsubval 22476 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  -h  y
)  =  ( x  +h  ( -u 1  .h  y ) ) )
54fveq2d 5695 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  -h  y ) )  =  ( normh `  ( x  +h  ( -u 1  .h  y ) ) ) )
65oveq1d 6059 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( normh `  (
x  -h  y ) ) ^ 2 )  =  ( ( normh `  ( x  +h  ( -u 1  .h  y ) ) ) ^ 2 ) )
76oveq2d 6060 . . . . 5  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( ( normh `  ( x  +h  y
) ) ^ 2 )  +  ( (
normh `  ( x  -h  y ) ) ^
2 ) )  =  ( ( ( normh `  ( x  +h  y
) ) ^ 2 )  +  ( (
normh `  ( x  +h  ( -u 1  .h  y
) ) ) ^
2 ) ) )
8 hvaddcl 22472 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  e.  ~H )
9 normcl 22584 . . . . . . . . 9  |-  ( ( x  +h  y )  e.  ~H  ->  ( normh `  ( x  +h  y ) )  e.  RR )
108, 9syl 16 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  +h  y ) )  e.  RR )
1110recnd 9074 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  +h  y ) )  e.  CC )
1211sqcld 11480 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( normh `  (
x  +h  y ) ) ^ 2 )  e.  CC )
13 hvsubcl 22477 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  -h  y
)  e.  ~H )
14 normcl 22584 . . . . . . . . 9  |-  ( ( x  -h  y )  e.  ~H  ->  ( normh `  ( x  -h  y ) )  e.  RR )
1514recnd 9074 . . . . . . . 8  |-  ( ( x  -h  y )  e.  ~H  ->  ( normh `  ( x  -h  y ) )  e.  CC )
1613, 15syl 16 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  -h  y ) )  e.  CC )
1716sqcld 11480 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( normh `  (
x  -h  y ) ) ^ 2 )  e.  CC )
1812, 17addcomd 9228 . . . . 5  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( ( normh `  ( x  +h  y
) ) ^ 2 )  +  ( (
normh `  ( x  -h  y ) ) ^
2 ) )  =  ( ( ( normh `  ( x  -h  y
) ) ^ 2 )  +  ( (
normh `  ( x  +h  y ) ) ^
2 ) ) )
197, 18eqtr3d 2442 . . . 4  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( ( normh `  ( x  +h  y
) ) ^ 2 )  +  ( (
normh `  ( x  +h  ( -u 1  .h  y
) ) ) ^
2 ) )  =  ( ( ( normh `  ( x  -h  y
) ) ^ 2 )  +  ( (
normh `  ( x  +h  y ) ) ^
2 ) ) )
20 normcl 22584 . . . . . . 7  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  RR )
2120recnd 9074 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  CC )
2221sqcld 11480 . . . . 5  |-  ( x  e.  ~H  ->  (
( normh `  x ) ^ 2 )  e.  CC )
23 normcl 22584 . . . . . . 7  |-  ( y  e.  ~H  ->  ( normh `  y )  e.  RR )
2423recnd 9074 . . . . . 6  |-  ( y  e.  ~H  ->  ( normh `  y )  e.  CC )
2524sqcld 11480 . . . . 5  |-  ( y  e.  ~H  ->  (
( normh `  y ) ^ 2 )  e.  CC )
26 2cn 10030 . . . . . 6  |-  2  e.  CC
27 adddi 9039 . . . . . 6  |-  ( ( 2  e.  CC  /\  ( ( normh `  x
) ^ 2 )  e.  CC  /\  (
( normh `  y ) ^ 2 )  e.  CC )  ->  (
2  x.  ( ( ( normh `  x ) ^ 2 )  +  ( ( normh `  y
) ^ 2 ) ) )  =  ( ( 2  x.  (
( normh `  x ) ^ 2 ) )  +  ( 2  x.  ( ( normh `  y
) ^ 2 ) ) ) )
2826, 27mp3an1 1266 . . . . 5  |-  ( ( ( ( normh `  x
) ^ 2 )  e.  CC  /\  (
( normh `  y ) ^ 2 )  e.  CC )  ->  (
2  x.  ( ( ( normh `  x ) ^ 2 )  +  ( ( normh `  y
) ^ 2 ) ) )  =  ( ( 2  x.  (
( normh `  x ) ^ 2 ) )  +  ( 2  x.  ( ( normh `  y
) ^ 2 ) ) ) )
2922, 25, 28syl2an 464 . . . 4  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( 2  x.  (
( ( normh `  x
) ^ 2 )  +  ( ( normh `  y ) ^ 2 ) ) )  =  ( ( 2  x.  ( ( normh `  x
) ^ 2 ) )  +  ( 2  x.  ( ( normh `  y ) ^ 2 ) ) ) )
303, 19, 293eqtr4d 2450 . . 3  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( ( normh `  ( x  +h  y
) ) ^ 2 )  +  ( (
normh `  ( x  +h  ( -u 1  .h  y
) ) ) ^
2 ) )  =  ( 2  x.  (
( ( normh `  x
) ^ 2 )  +  ( ( normh `  y ) ^ 2 ) ) ) )
3130rgen2a 2736 . 2  |-  A. x  e.  ~H  A. y  e. 
~H  ( ( (
normh `  ( x  +h  y ) ) ^
2 )  +  ( ( normh `  ( x  +h  ( -u 1  .h  y ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( normh `  x ) ^ 2 )  +  ( (
normh `  y ) ^
2 ) ) )
32 hilablo 22619 . . . 4  |-  +h  e.  AbelOp
3332elexi 2929 . . 3  |-  +h  e.  _V
34 hvmulex 22471 . . 3  |-  .h  e.  _V
35 normf 22582 . . . 4  |-  normh : ~H --> RR
36 ax-hilex 22459 . . . 4  |-  ~H  e.  _V
37 fex 5932 . . . 4  |-  ( (
normh : ~H --> RR  /\  ~H  e.  _V )  ->  normh  e.  _V )
3835, 36, 37mp2an 654 . . 3  |-  normh  e.  _V
39 hhnv.1 . . . . 5  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
4039eleq1i 2471 . . . 4  |-  ( U  e.  CPreHil OLD  <->  <. <.  +h  ,  .h  >. ,  normh >.  e.  CPreHil OLD )
41 ablogrpo 21829 . . . . . . 7  |-  (  +h  e.  AbelOp  ->  +h  e.  GrpOp )
4232, 41ax-mp 8 . . . . . 6  |-  +h  e.  GrpOp
43 ax-hfvadd 22460 . . . . . . 7  |-  +h  :
( ~H  X.  ~H )
--> ~H
4443fdmi 5559 . . . . . 6  |-  dom  +h  =  ( ~H  X.  ~H )
4542, 44grporn 21757 . . . . 5  |-  ~H  =  ran  +h
4645isphg 22275 . . . 4  |-  ( (  +h  e.  _V  /\  .h  e.  _V  /\  normh  e. 
_V )  ->  ( <. <.  +h  ,  .h  >. ,  normh >.  e.  CPreHil OLD  <->  ( <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec  /\  A. x  e.  ~H  A. y  e. 
~H  ( ( (
normh `  ( x  +h  y ) ) ^
2 )  +  ( ( normh `  ( x  +h  ( -u 1  .h  y ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( normh `  x ) ^ 2 )  +  ( (
normh `  y ) ^
2 ) ) ) ) ) )
4740, 46syl5bb 249 . . 3  |-  ( (  +h  e.  _V  /\  .h  e.  _V  /\  normh  e. 
_V )  ->  ( U  e.  CPreHil OLD  <->  ( <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec  /\  A. x  e.  ~H  A. y  e. 
~H  ( ( (
normh `  ( x  +h  y ) ) ^
2 )  +  ( ( normh `  ( x  +h  ( -u 1  .h  y ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( normh `  x ) ^ 2 )  +  ( (
normh `  y ) ^
2 ) ) ) ) ) )
4833, 34, 38, 47mp3an 1279 . 2  |-  ( U  e.  CPreHil OLD  <->  ( <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec  /\  A. x  e.  ~H  A. y  e.  ~H  (
( ( normh `  (
x  +h  y ) ) ^ 2 )  +  ( ( normh `  ( x  +h  ( -u 1  .h  y ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( normh `  x ) ^ 2 )  +  ( ( normh `  y
) ^ 2 ) ) ) ) )
492, 31, 48mpbir2an 887 1  |-  U  e.  CPreHil
OLD
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   _Vcvv 2920   <.cop 3781    X. cxp 4839   -->wf 5413   ` cfv 5417  (class class class)co 6044   CCcc 8948   RRcr 8949   1c1 8951    + caddc 8953    x. cmul 8955   -ucneg 9252   2c2 10009   ^cexp 11341   GrpOpcgr 21731   AbelOpcablo 21826   NrmCVeccnv 22020   CPreHil OLDccphlo 22270   ~Hchil 22379    +h cva 22380    .h csm 22381   normhcno 22383    -h cmv 22385
This theorem is referenced by:  bcsiHIL  22639  hhhl  22663  hhssph  22731  pjhthlem2  22851
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-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-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
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-iun 4059  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-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-ov 6047  df-oprab 6048  df-mpt2 6049  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-sup 7408  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-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-seq 11283  df-exp 11342  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-grpo 21736  df-gid 21737  df-ablo 21827  df-vc 21982  df-nv 22028  df-ph 22271  df-hnorm 22428  df-hvsub 22431
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