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Theorem hhph 24403
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 2433 . . 3  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
21hhnv 24390 . 2  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
3 normpar 24380 . . . 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 24241 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  -h  y
)  =  ( x  +h  ( -u 1  .h  y ) ) )
54fveq2d 5683 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  -h  y ) )  =  ( normh `  ( x  +h  ( -u 1  .h  y ) ) ) )
65oveq1d 6095 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( normh `  (
x  -h  y ) ) ^ 2 )  =  ( ( normh `  ( x  +h  ( -u 1  .h  y ) ) ) ^ 2 ) )
76oveq2d 6096 . . . . 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 24237 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  e.  ~H )
9 normcl 24350 . . . . . . . . 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 9400 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( normh `  ( x  +h  y ) )  e.  CC )
1211sqcld 11990 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( normh `  (
x  +h  y ) ) ^ 2 )  e.  CC )
13 hvsubcl 24242 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  -h  y
)  e.  ~H )
14 normcl 24350 . . . . . . . . 9  |-  ( ( x  -h  y )  e.  ~H  ->  ( normh `  ( x  -h  y ) )  e.  RR )
1514recnd 9400 . . . . . . . 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 11990 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( normh `  (
x  -h  y ) ) ^ 2 )  e.  CC )
1812, 17addcomd 9559 . . . . 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 2467 . . . 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 24350 . . . . . . 7  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  RR )
2120recnd 9400 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  CC )
2221sqcld 11990 . . . . 5  |-  ( x  e.  ~H  ->  (
( normh `  x ) ^ 2 )  e.  CC )
23 normcl 24350 . . . . . . 7  |-  ( y  e.  ~H  ->  ( normh `  y )  e.  RR )
2423recnd 9400 . . . . . 6  |-  ( y  e.  ~H  ->  ( normh `  y )  e.  CC )
2524sqcld 11990 . . . . 5  |-  ( y  e.  ~H  ->  (
( normh `  y ) ^ 2 )  e.  CC )
26 2cn 10380 . . . . . 6  |-  2  e.  CC
27 adddi 9359 . . . . . 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 1294 . . . . 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 474 . . . 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 2475 . . 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 2772 . 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 24385 . . . 4  |-  +h  e.  AbelOp
3332elexi 2972 . . 3  |-  +h  e.  _V
34 hvmulex 24236 . . 3  |-  .h  e.  _V
35 normf 24348 . . . 4  |-  normh : ~H --> RR
36 ax-hilex 24224 . . . 4  |-  ~H  e.  _V
37 fex 5937 . . . 4  |-  ( (
normh : ~H --> RR  /\  ~H  e.  _V )  ->  normh  e.  _V )
3835, 36, 37mp2an 665 . . 3  |-  normh  e.  _V
39 hhnv.1 . . . . 5  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
4039eleq1i 2496 . . . 4  |-  ( U  e.  CPreHil OLD  <->  <. <.  +h  ,  .h  >. ,  normh >.  e.  CPreHil OLD )
41 ablogrpo 23594 . . . . . . 7  |-  (  +h  e.  AbelOp  ->  +h  e.  GrpOp )
4232, 41ax-mp 5 . . . . . 6  |-  +h  e.  GrpOp
43 ax-hfvadd 24225 . . . . . . 7  |-  +h  :
( ~H  X.  ~H )
--> ~H
4443fdmi 5552 . . . . . 6  |-  dom  +h  =  ( ~H  X.  ~H )
4542, 44grporn 23522 . . . . 5  |-  ~H  =  ran  +h
4645isphg 24040 . . . 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 1307 . 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 904 1  |-  U  e.  CPreHil
OLD
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755   A.wral 2705   _Vcvv 2962   <.cop 3871    X. cxp 4825   -->wf 5402   ` cfv 5406  (class class class)co 6080   CCcc 9268   RRcr 9269   1c1 9271    + caddc 9273    x. cmul 9275   -ucneg 9584   2c2 10359   ^cexp 11849   GrpOpcgr 23496   AbelOpcablo 23591   NrmCVeccnv 23785   CPreHil OLDccphlo 24035   ~Hchil 24144    +h cva 24145    .h csm 24146   normhcno 24148    -h cmv 24150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-hilex 24224  ax-hfvadd 24225  ax-hvcom 24226  ax-hvass 24227  ax-hv0cl 24228  ax-hvaddid 24229  ax-hfvmul 24230  ax-hvmulid 24231  ax-hvmulass 24232  ax-hvdistr1 24233  ax-hvdistr2 24234  ax-hvmul0 24235  ax-hfi 24304  ax-his1 24307  ax-his2 24308  ax-his3 24309  ax-his4 24310
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-sup 7679  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-n0 10568  df-z 10635  df-uz 10850  df-rp 10980  df-seq 11791  df-exp 11850  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-grpo 23501  df-gid 23502  df-ablo 23592  df-vc 23747  df-nv 23793  df-ph 24036  df-hnorm 24193  df-hvsub 24196
This theorem is referenced by:  bcsiHIL  24405  hhhl  24429  hhssph  24498  pjhthlem2  24618
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