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Theorem hhph 21757
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 2283 . . 3  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
21hhnv 21744 . 2  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
3 normpar 21734 . . . 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 21596 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  -h  y
)  =  ( x  +h  ( -u 1  .h  y ) ) )
54fveq2d 5529 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  -h  y ) )  =  ( normh `  ( x  +h  ( -u 1  .h  y ) ) ) )
65oveq1d 5873 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( normh `  (
x  -h  y ) ) ^ 2 )  =  ( ( normh `  ( x  +h  ( -u 1  .h  y ) ) ) ^ 2 ) )
76oveq2d 5874 . . . . 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 21592 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  e.  ~H )
9 normcl 21704 . . . . . . . . 9  |-  ( ( x  +h  y )  e.  ~H  ->  ( normh `  ( x  +h  y ) )  e.  RR )
108, 9syl 15 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  +h  y ) )  e.  RR )
1110recnd 8861 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  +h  y ) )  e.  CC )
1211sqcld 11243 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( normh `  (
x  +h  y ) ) ^ 2 )  e.  CC )
13 hvsubcl 21597 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  -h  y
)  e.  ~H )
14 normcl 21704 . . . . . . . . 9  |-  ( ( x  -h  y )  e.  ~H  ->  ( normh `  ( x  -h  y ) )  e.  RR )
1514recnd 8861 . . . . . . . 8  |-  ( ( x  -h  y )  e.  ~H  ->  ( normh `  ( x  -h  y ) )  e.  CC )
1613, 15syl 15 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  -h  y ) )  e.  CC )
1716sqcld 11243 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( normh `  (
x  -h  y ) ) ^ 2 )  e.  CC )
1812, 17addcomd 9014 . . . . 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 2317 . . . 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 21704 . . . . . . 7  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  RR )
2120recnd 8861 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  CC )
2221sqcld 11243 . . . . 5  |-  ( x  e.  ~H  ->  (
( normh `  x ) ^ 2 )  e.  CC )
23 normcl 21704 . . . . . . 7  |-  ( y  e.  ~H  ->  ( normh `  y )  e.  RR )
2423recnd 8861 . . . . . 6  |-  ( y  e.  ~H  ->  ( normh `  y )  e.  CC )
2524sqcld 11243 . . . . 5  |-  ( y  e.  ~H  ->  (
( normh `  y ) ^ 2 )  e.  CC )
26 2cn 9816 . . . . . 6  |-  2  e.  CC
27 adddi 8826 . . . . . 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 1264 . . . . 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 463 . . . 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 2325 . . 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 2609 . 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 21739 . . . 4  |-  +h  e.  AbelOp
3332elexi 2797 . . 3  |-  +h  e.  _V
34 hvmulex 21591 . . 3  |-  .h  e.  _V
35 normf 21702 . . . 4  |-  normh : ~H --> RR
36 ax-hilex 21579 . . . 4  |-  ~H  e.  _V
37 fex 5749 . . . 4  |-  ( (
normh : ~H --> RR  /\  ~H  e.  _V )  ->  normh  e.  _V )
3835, 36, 37mp2an 653 . . 3  |-  normh  e.  _V
39 hhnv.1 . . . . 5  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
4039eleq1i 2346 . . . 4  |-  ( U  e.  CPreHil OLD  <->  <. <.  +h  ,  .h  >. ,  normh >.  e.  CPreHil OLD )
41 ablogrpo 20951 . . . . . . 7  |-  (  +h  e.  AbelOp  ->  +h  e.  GrpOp )
4232, 41ax-mp 8 . . . . . 6  |-  +h  e.  GrpOp
43 ax-hfvadd 21580 . . . . . . 7  |-  +h  :
( ~H  X.  ~H )
--> ~H
4443fdmi 5394 . . . . . 6  |-  dom  +h  =  ( ~H  X.  ~H )
4542, 44grporn 20879 . . . . 5  |-  ~H  =  ran  +h
4645isphg 21395 . . . 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 248 . . 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 1277 . 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 886 1  |-  U  e.  CPreHil
OLD
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   <.cop 3643    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   1c1 8738    + caddc 8740    x. cmul 8742   -ucneg 9038   2c2 9795   ^cexp 11104   GrpOpcgr 20853   AbelOpcablo 20948   NrmCVeccnv 21140   CPreHil OLDccphlo 21390   ~Hchil 21499    +h cva 21500    .h csm 21501   normhcno 21503    -h cmv 21505
This theorem is referenced by:  bcsiHIL  21759  hhhl  21783  hhssph  21851  pjhthlem2  21971
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-grpo 20858  df-gid 20859  df-ablo 20949  df-vc 21102  df-nv 21148  df-ph 21391  df-hnorm 21548  df-hvsub 21551
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