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Theorem phpar 21402
Description: The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
phpar.1  |-  X  =  ( BaseSet `  U )
phpar.2  |-  G  =  ( +v `  U
)
phpar.4  |-  S  =  ( .s OLD `  U
)
phpar.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
phpar  |-  ( ( U  e.  CPreHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( N `
 ( A G B ) ) ^
2 )  +  ( ( N `  ( A G ( -u 1 S B ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )

Proof of Theorem phpar
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 phpar.2 . . . . . . 7  |-  G  =  ( +v `  U
)
21vafval 21159 . . . . . 6  |-  G  =  ( 1st `  ( 1st `  U ) )
3 fvex 5539 . . . . . 6  |-  ( 1st `  ( 1st `  U
) )  e.  _V
42, 3eqeltri 2353 . . . . 5  |-  G  e. 
_V
5 phpar.4 . . . . . . 7  |-  S  =  ( .s OLD `  U
)
65smfval 21161 . . . . . 6  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 fvex 5539 . . . . . 6  |-  ( 2nd `  ( 1st `  U
) )  e.  _V
86, 7eqeltri 2353 . . . . 5  |-  S  e. 
_V
9 phpar.6 . . . . . . 7  |-  N  =  ( normCV `  U )
109nmcvfval 21163 . . . . . 6  |-  N  =  ( 2nd `  U
)
11 fvex 5539 . . . . . 6  |-  ( 2nd `  U )  e.  _V
1210, 11eqeltri 2353 . . . . 5  |-  N  e. 
_V
134, 8, 123pm3.2i 1130 . . . 4  |-  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V )
141, 5, 9phop 21396 . . . . . 6  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. <. G ,  S >. ,  N >. )
1514eleq1d 2349 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  ( U  e.  CPreHil OLD  <->  <. <. G ,  S >. ,  N >.  e.  CPreHil OLD ) )
1615ibi 232 . . . 4  |-  ( U  e.  CPreHil OLD  ->  <. <. G ,  S >. ,  N >.  e.  CPreHil
OLD )
17 phpar.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
1817, 1bafval 21160 . . . . . 6  |-  X  =  ran  G
1918isphg 21395 . . . . 5  |-  ( ( G  e.  _V  /\  S  e.  _V  /\  N  e.  _V )  ->  ( <. <. G ,  S >. ,  N >.  e.  CPreHil OLD  <->  (
<. <. G ,  S >. ,  N >.  e.  NrmCVec  /\  A. x  e.  X  A. y  e.  X  (
( ( N `  ( x G y ) ) ^ 2 )  +  ( ( N `  ( x G ( -u 1 S y ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) ) ) ) )
2019simplbda 607 . . . 4  |-  ( ( ( G  e.  _V  /\  S  e.  _V  /\  N  e.  _V )  /\  <. <. G ,  S >. ,  N >.  e.  CPreHil OLD )  ->  A. x  e.  X  A. y  e.  X  ( (
( N `  (
x G y ) ) ^ 2 )  +  ( ( N `
 ( x G ( -u 1 S y ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
 x ) ^
2 )  +  ( ( N `  y
) ^ 2 ) ) ) )
2113, 16, 20sylancr 644 . . 3  |-  ( U  e.  CPreHil OLD  ->  A. x  e.  X  A. y  e.  X  ( (
( N `  (
x G y ) ) ^ 2 )  +  ( ( N `
 ( x G ( -u 1 S y ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
 x ) ^
2 )  +  ( ( N `  y
) ^ 2 ) ) ) )
22213ad2ant1 976 . 2  |-  ( ( U  e.  CPreHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  A. x  e.  X  A. y  e.  X  ( ( ( N `
 ( x G y ) ) ^
2 )  +  ( ( N `  (
x G ( -u
1 S y ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  x
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) ) )
23 oveq1 5865 . . . . . . . 8  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
2423fveq2d 5529 . . . . . . 7  |-  ( x  =  A  ->  ( N `  ( x G y ) )  =  ( N `  ( A G y ) ) )
2524oveq1d 5873 . . . . . 6  |-  ( x  =  A  ->  (
( N `  (
x G y ) ) ^ 2 )  =  ( ( N `
 ( A G y ) ) ^
2 ) )
26 oveq1 5865 . . . . . . . 8  |-  ( x  =  A  ->  (
x G ( -u
1 S y ) )  =  ( A G ( -u 1 S y ) ) )
2726fveq2d 5529 . . . . . . 7  |-  ( x  =  A  ->  ( N `  ( x G ( -u 1 S y ) ) )  =  ( N `
 ( A G ( -u 1 S y ) ) ) )
2827oveq1d 5873 . . . . . 6  |-  ( x  =  A  ->  (
( N `  (
x G ( -u
1 S y ) ) ) ^ 2 )  =  ( ( N `  ( A G ( -u 1 S y ) ) ) ^ 2 ) )
2925, 28oveq12d 5876 . . . . 5  |-  ( x  =  A  ->  (
( ( N `  ( x G y ) ) ^ 2 )  +  ( ( N `  ( x G ( -u 1 S y ) ) ) ^ 2 ) )  =  ( ( ( N `  ( A G y ) ) ^ 2 )  +  ( ( N `  ( A G ( -u
1 S y ) ) ) ^ 2 ) ) )
30 fveq2 5525 . . . . . . . 8  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
3130oveq1d 5873 . . . . . . 7  |-  ( x  =  A  ->  (
( N `  x
) ^ 2 )  =  ( ( N `
 A ) ^
2 ) )
3231oveq1d 5873 . . . . . 6  |-  ( x  =  A  ->  (
( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) )  =  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) )
3332oveq2d 5874 . . . . 5  |-  ( x  =  A  ->  (
2  x.  ( ( ( N `  x
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  y
) ^ 2 ) ) ) )
3429, 33eqeq12d 2297 . . . 4  |-  ( x  =  A  ->  (
( ( ( N `
 ( x G y ) ) ^
2 )  +  ( ( N `  (
x G ( -u
1 S y ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  x
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  <-> 
( ( ( N `
 ( A G y ) ) ^
2 )  +  ( ( N `  ( A G ( -u 1 S y ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) ) ) )
35 oveq2 5866 . . . . . . . 8  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
3635fveq2d 5529 . . . . . . 7  |-  ( y  =  B  ->  ( N `  ( A G y ) )  =  ( N `  ( A G B ) ) )
3736oveq1d 5873 . . . . . 6  |-  ( y  =  B  ->  (
( N `  ( A G y ) ) ^ 2 )  =  ( ( N `  ( A G B ) ) ^ 2 ) )
38 oveq2 5866 . . . . . . . . 9  |-  ( y  =  B  ->  ( -u 1 S y )  =  ( -u 1 S B ) )
3938oveq2d 5874 . . . . . . . 8  |-  ( y  =  B  ->  ( A G ( -u 1 S y ) )  =  ( A G ( -u 1 S B ) ) )
4039fveq2d 5529 . . . . . . 7  |-  ( y  =  B  ->  ( N `  ( A G ( -u 1 S y ) ) )  =  ( N `
 ( A G ( -u 1 S B ) ) ) )
4140oveq1d 5873 . . . . . 6  |-  ( y  =  B  ->  (
( N `  ( A G ( -u 1 S y ) ) ) ^ 2 )  =  ( ( N `
 ( A G ( -u 1 S B ) ) ) ^ 2 ) )
4237, 41oveq12d 5876 . . . . 5  |-  ( y  =  B  ->  (
( ( N `  ( A G y ) ) ^ 2 )  +  ( ( N `
 ( A G ( -u 1 S y ) ) ) ^ 2 ) )  =  ( ( ( N `  ( A G B ) ) ^ 2 )  +  ( ( N `  ( A G ( -u
1 S B ) ) ) ^ 2 ) ) )
43 fveq2 5525 . . . . . . . 8  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
4443oveq1d 5873 . . . . . . 7  |-  ( y  =  B  ->  (
( N `  y
) ^ 2 )  =  ( ( N `
 B ) ^
2 ) )
4544oveq2d 5874 . . . . . 6  |-  ( y  =  B  ->  (
( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) )  =  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) )
4645oveq2d 5874 . . . . 5  |-  ( y  =  B  ->  (
2  x.  ( ( ( N `  A
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )
4742, 46eqeq12d 2297 . . . 4  |-  ( y  =  B  ->  (
( ( ( N `
 ( A G y ) ) ^
2 )  +  ( ( N `  ( A G ( -u 1 S y ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) )  <->  ( (
( N `  ( A G B ) ) ^ 2 )  +  ( ( N `  ( A G ( -u
1 S B ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  A
) ^ 2 )  +  ( ( N `
 B ) ^
2 ) ) ) ) )
4834, 47rspc2v 2890 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( ( ( N `  ( x G y ) ) ^ 2 )  +  ( ( N `  ( x G (
-u 1 S y ) ) ) ^
2 ) )  =  ( 2  x.  (
( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) )  ->  ( (
( N `  ( A G B ) ) ^ 2 )  +  ( ( N `  ( A G ( -u
1 S B ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  A
) ^ 2 )  +  ( ( N `
 B ) ^
2 ) ) ) ) )
49483adant1 973 . 2  |-  ( ( U  e.  CPreHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( ( ( N `  ( x G y ) ) ^ 2 )  +  ( ( N `  ( x G (
-u 1 S y ) ) ) ^
2 ) )  =  ( 2  x.  (
( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) )  ->  ( (
( N `  ( A G B ) ) ^ 2 )  +  ( ( N `  ( A G ( -u
1 S B ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  A
) ^ 2 )  +  ( ( N `
 B ) ^
2 ) ) ) ) )
5022, 49mpd 14 1  |-  ( ( U  e.  CPreHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( N `
 ( A G B ) ) ^
2 )  +  ( ( N `  ( A G ( -u 1 S B ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   <.cop 3643   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   1c1 8738    + caddc 8740    x. cmul 8742   -ucneg 9038   2c2 9795   ^cexp 11104   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   normCVcnmcv 21146   CPreHil OLDccphlo 21390
This theorem is referenced by:  ip0i  21403  hlpar  21476
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-reu 2550  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-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-1st 6122  df-2nd 6123  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156  df-ph 21391
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