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Theorem hhph 26817
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 2422 . . 3  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
21hhnv 26804 . 2  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
3 normpar 26794 . . . 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 26655 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  -h  y
)  =  ( x  +h  ( -u 1  .h  y ) ) )
54fveq2d 5882 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  -h  y ) )  =  ( normh `  ( x  +h  ( -u 1  .h  y ) ) ) )
65oveq1d 6317 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( normh `  (
x  -h  y ) ) ^ 2 )  =  ( ( normh `  ( x  +h  ( -u 1  .h  y ) ) ) ^ 2 ) )
76oveq2d 6318 . . . . 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 26651 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  e.  ~H )
9 normcl 26764 . . . . . . . . 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 9670 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  +h  y ) )  e.  CC )
1211sqcld 12414 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( normh `  (
x  +h  y ) ) ^ 2 )  e.  CC )
13 hvsubcl 26656 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  -h  y
)  e.  ~H )
14 normcl 26764 . . . . . . . . 9  |-  ( ( x  -h  y )  e.  ~H  ->  ( normh `  ( x  -h  y ) )  e.  RR )
1514recnd 9670 . . . . . . . 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 12414 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( normh `  (
x  -h  y ) ) ^ 2 )  e.  CC )
1812, 17addcomd 9836 . . . . 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 2465 . . . 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 26764 . . . . . . 7  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  RR )
2120recnd 9670 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  CC )
2221sqcld 12414 . . . . 5  |-  ( x  e.  ~H  ->  (
( normh `  x ) ^ 2 )  e.  CC )
23 normcl 26764 . . . . . . 7  |-  ( y  e.  ~H  ->  ( normh `  y )  e.  RR )
2423recnd 9670 . . . . . 6  |-  ( y  e.  ~H  ->  ( normh `  y )  e.  CC )
2524sqcld 12414 . . . . 5  |-  ( y  e.  ~H  ->  (
( normh `  y ) ^ 2 )  e.  CC )
26 2cn 10681 . . . . . 6  |-  2  e.  CC
27 adddi 9629 . . . . . 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 1347 . . . . 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 479 . . . 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 2473 . . 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 2852 . 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 26799 . . . 4  |-  +h  e.  AbelOp
3332elexi 3091 . . 3  |-  +h  e.  _V
34 hvmulex 26650 . . 3  |-  .h  e.  _V
35 normf 26762 . . . 4  |-  normh : ~H --> RR
36 ax-hilex 26638 . . . 4  |-  ~H  e.  _V
37 fex 6150 . . . 4  |-  ( (
normh : ~H --> RR  /\  ~H  e.  _V )  ->  normh  e.  _V )
3835, 36, 37mp2an 676 . . 3  |-  normh  e.  _V
39 hhnv.1 . . . . 5  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
4039eleq1i 2499 . . . 4  |-  ( U  e.  CPreHil OLD  <->  <. <.  +h  ,  .h  >. ,  normh >.  e.  CPreHil OLD )
41 ablogrpo 25998 . . . . . . 7  |-  (  +h  e.  AbelOp  ->  +h  e.  GrpOp )
4232, 41ax-mp 5 . . . . . 6  |-  +h  e.  GrpOp
43 ax-hfvadd 26639 . . . . . . 7  |-  +h  :
( ~H  X.  ~H )
--> ~H
4443fdmi 5748 . . . . . 6  |-  dom  +h  =  ( ~H  X.  ~H )
4542, 44grporn 25926 . . . . 5  |-  ~H  =  ran  +h
4645isphg 26444 . . . 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 260 . . 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 1360 . 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 928 1  |-  U  e.  CPreHil
OLD
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775   _Vcvv 3081   <.cop 4002    X. cxp 4848   -->wf 5594   ` cfv 5598  (class class class)co 6302   CCcc 9538   RRcr 9539   1c1 9541    + caddc 9543    x. cmul 9545   -ucneg 9862   2c2 10660   ^cexp 12272   GrpOpcgr 25900   AbelOpcablo 25995   NrmCVeccnv 26189   CPreHil OLDccphlo 26439   ~Hchil 26558    +h cva 26559    .h csm 26560   normhcno 26562    -h cmv 26564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-hilex 26638  ax-hfvadd 26639  ax-hvcom 26640  ax-hvass 26641  ax-hv0cl 26642  ax-hvaddid 26643  ax-hfvmul 26644  ax-hvmulid 26645  ax-hvmulass 26646  ax-hvdistr1 26647  ax-hvdistr2 26648  ax-hvmul0 26649  ax-hfi 26718  ax-his1 26721  ax-his2 26722  ax-his3 26723  ax-his4 26724
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-sup 7959  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-n0 10871  df-z 10939  df-uz 11161  df-rp 11304  df-seq 12214  df-exp 12273  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-grpo 25905  df-gid 25906  df-ablo 25996  df-vc 26151  df-nv 26197  df-ph 26440  df-hnorm 26607  df-hvsub 26610
This theorem is referenced by:  bcsiHIL  26819  hhhl  26843  hhssph  26911  pjhthlem2  27031
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