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Theorem hhph 26509
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 2402 . . 3  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
21hhnv 26496 . 2  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
3 normpar 26486 . . . 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 26347 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  -h  y
)  =  ( x  +h  ( -u 1  .h  y ) ) )
54fveq2d 5853 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  -h  y ) )  =  ( normh `  ( x  +h  ( -u 1  .h  y ) ) ) )
65oveq1d 6293 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( normh `  (
x  -h  y ) ) ^ 2 )  =  ( ( normh `  ( x  +h  ( -u 1  .h  y ) ) ) ^ 2 ) )
76oveq2d 6294 . . . . 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 26343 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  e.  ~H )
9 normcl 26456 . . . . . . . . 9  |-  ( ( x  +h  y )  e.  ~H  ->  ( normh `  ( x  +h  y ) )  e.  RR )
108, 9syl 17 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  +h  y ) )  e.  RR )
1110recnd 9652 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  +h  y ) )  e.  CC )
1211sqcld 12352 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( normh `  (
x  +h  y ) ) ^ 2 )  e.  CC )
13 hvsubcl 26348 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  -h  y
)  e.  ~H )
14 normcl 26456 . . . . . . . . 9  |-  ( ( x  -h  y )  e.  ~H  ->  ( normh `  ( x  -h  y ) )  e.  RR )
1514recnd 9652 . . . . . . . 8  |-  ( ( x  -h  y )  e.  ~H  ->  ( normh `  ( x  -h  y ) )  e.  CC )
1613, 15syl 17 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  -h  y ) )  e.  CC )
1716sqcld 12352 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( normh `  (
x  -h  y ) ) ^ 2 )  e.  CC )
1812, 17addcomd 9816 . . . . 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 2445 . . . 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 26456 . . . . . . 7  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  RR )
2120recnd 9652 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  CC )
2221sqcld 12352 . . . . 5  |-  ( x  e.  ~H  ->  (
( normh `  x ) ^ 2 )  e.  CC )
23 normcl 26456 . . . . . . 7  |-  ( y  e.  ~H  ->  ( normh `  y )  e.  RR )
2423recnd 9652 . . . . . 6  |-  ( y  e.  ~H  ->  ( normh `  y )  e.  CC )
2524sqcld 12352 . . . . 5  |-  ( y  e.  ~H  ->  (
( normh `  y ) ^ 2 )  e.  CC )
26 2cn 10647 . . . . . 6  |-  2  e.  CC
27 adddi 9611 . . . . . 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 1313 . . . . 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 475 . . . 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 2453 . . 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 2831 . 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 26491 . . . 4  |-  +h  e.  AbelOp
3332elexi 3069 . . 3  |-  +h  e.  _V
34 hvmulex 26342 . . 3  |-  .h  e.  _V
35 normf 26454 . . . 4  |-  normh : ~H --> RR
36 ax-hilex 26330 . . . 4  |-  ~H  e.  _V
37 fex 6126 . . . 4  |-  ( (
normh : ~H --> RR  /\  ~H  e.  _V )  ->  normh  e.  _V )
3835, 36, 37mp2an 670 . . 3  |-  normh  e.  _V
39 hhnv.1 . . . . 5  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
4039eleq1i 2479 . . . 4  |-  ( U  e.  CPreHil OLD  <->  <. <.  +h  ,  .h  >. ,  normh >.  e.  CPreHil OLD )
41 ablogrpo 25700 . . . . . . 7  |-  (  +h  e.  AbelOp  ->  +h  e.  GrpOp )
4232, 41ax-mp 5 . . . . . 6  |-  +h  e.  GrpOp
43 ax-hfvadd 26331 . . . . . . 7  |-  +h  :
( ~H  X.  ~H )
--> ~H
4443fdmi 5719 . . . . . 6  |-  dom  +h  =  ( ~H  X.  ~H )
4542, 44grporn 25628 . . . . 5  |-  ~H  =  ran  +h
4645isphg 26146 . . . 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 257 . . 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 1326 . 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 921 1  |-  U  e.  CPreHil
OLD
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   _Vcvv 3059   <.cop 3978    X. cxp 4821   -->wf 5565   ` cfv 5569  (class class class)co 6278   CCcc 9520   RRcr 9521   1c1 9523    + caddc 9525    x. cmul 9527   -ucneg 9842   2c2 10626   ^cexp 12210   GrpOpcgr 25602   AbelOpcablo 25697   NrmCVeccnv 25891   CPreHil OLDccphlo 26141   ~Hchil 26250    +h cva 26251    .h csm 26252   normhcno 26254    -h cmv 26256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-hilex 26330  ax-hfvadd 26331  ax-hvcom 26332  ax-hvass 26333  ax-hv0cl 26334  ax-hvaddid 26335  ax-hfvmul 26336  ax-hvmulid 26337  ax-hvmulass 26338  ax-hvdistr1 26339  ax-hvdistr2 26340  ax-hvmul0 26341  ax-hfi 26410  ax-his1 26413  ax-his2 26414  ax-his3 26415  ax-his4 26416
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-seq 12152  df-exp 12211  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-grpo 25607  df-gid 25608  df-ablo 25698  df-vc 25853  df-nv 25899  df-ph 26142  df-hnorm 26299  df-hvsub 26302
This theorem is referenced by:  bcsiHIL  26511  hhhl  26535  hhssph  26604  pjhthlem2  26724
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