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Theorem phpar 26310
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  =  ( .sOLD `  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 26067 . . . . . 6  |-  G  =  ( 1st `  ( 1st `  U ) )
3 fvex 5891 . . . . . 6  |-  ( 1st `  ( 1st `  U
) )  e.  _V
42, 3eqeltri 2513 . . . . 5  |-  G  e. 
_V
5 phpar.4 . . . . . . 7  |-  S  =  ( .sOLD `  U )
65smfval 26069 . . . . . 6  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 fvex 5891 . . . . . 6  |-  ( 2nd `  ( 1st `  U
) )  e.  _V
86, 7eqeltri 2513 . . . . 5  |-  S  e. 
_V
9 phpar.6 . . . . . . 7  |-  N  =  ( normCV `  U )
109nmcvfval 26071 . . . . . 6  |-  N  =  ( 2nd `  U
)
11 fvex 5891 . . . . . 6  |-  ( 2nd `  U )  e.  _V
1210, 11eqeltri 2513 . . . . 5  |-  N  e. 
_V
134, 8, 123pm3.2i 1183 . . . 4  |-  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V )
141, 5, 9phop 26304 . . . . . 6  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. <. G ,  S >. ,  N >. )
1514eleq1d 2498 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  ( U  e.  CPreHil OLD  <->  <. <. G ,  S >. ,  N >.  e.  CPreHil OLD ) )
1615ibi 244 . . . 4  |-  ( U  e.  CPreHil OLD  ->  <. <. G ,  S >. ,  N >.  e.  CPreHil
OLD )
17 phpar.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
1817, 1bafval 26068 . . . . . 6  |-  X  =  ran  G
1918isphg 26303 . . . . 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 628 . . . 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 667 . . 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 1026 . 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 6312 . . . . . . . 8  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
2423fveq2d 5885 . . . . . . 7  |-  ( x  =  A  ->  ( N `  ( x G y ) )  =  ( N `  ( A G y ) ) )
2524oveq1d 6320 . . . . . 6  |-  ( x  =  A  ->  (
( N `  (
x G y ) ) ^ 2 )  =  ( ( N `
 ( A G y ) ) ^
2 ) )
26 oveq1 6312 . . . . . . . 8  |-  ( x  =  A  ->  (
x G ( -u
1 S y ) )  =  ( A G ( -u 1 S y ) ) )
2726fveq2d 5885 . . . . . . 7  |-  ( x  =  A  ->  ( N `  ( x G ( -u 1 S y ) ) )  =  ( N `
 ( A G ( -u 1 S y ) ) ) )
2827oveq1d 6320 . . . . . 6  |-  ( x  =  A  ->  (
( N `  (
x G ( -u
1 S y ) ) ) ^ 2 )  =  ( ( N `  ( A G ( -u 1 S y ) ) ) ^ 2 ) )
2925, 28oveq12d 6323 . . . . 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 5881 . . . . . . . 8  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
3130oveq1d 6320 . . . . . . 7  |-  ( x  =  A  ->  (
( N `  x
) ^ 2 )  =  ( ( N `
 A ) ^
2 ) )
3231oveq1d 6320 . . . . . 6  |-  ( x  =  A  ->  (
( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) )  =  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) )
3332oveq2d 6321 . . . . 5  |-  ( x  =  A  ->  (
2  x.  ( ( ( N `  x
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  y
) ^ 2 ) ) ) )
3429, 33eqeq12d 2451 . . . 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 6313 . . . . . . . 8  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
3635fveq2d 5885 . . . . . . 7  |-  ( y  =  B  ->  ( N `  ( A G y ) )  =  ( N `  ( A G B ) ) )
3736oveq1d 6320 . . . . . 6  |-  ( y  =  B  ->  (
( N `  ( A G y ) ) ^ 2 )  =  ( ( N `  ( A G B ) ) ^ 2 ) )
38 oveq2 6313 . . . . . . . . 9  |-  ( y  =  B  ->  ( -u 1 S y )  =  ( -u 1 S B ) )
3938oveq2d 6321 . . . . . . . 8  |-  ( y  =  B  ->  ( A G ( -u 1 S y ) )  =  ( A G ( -u 1 S B ) ) )
4039fveq2d 5885 . . . . . . 7  |-  ( y  =  B  ->  ( N `  ( A G ( -u 1 S y ) ) )  =  ( N `
 ( A G ( -u 1 S B ) ) ) )
4140oveq1d 6320 . . . . . 6  |-  ( y  =  B  ->  (
( N `  ( A G ( -u 1 S y ) ) ) ^ 2 )  =  ( ( N `
 ( A G ( -u 1 S B ) ) ) ^ 2 ) )
4237, 41oveq12d 6323 . . . . 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 5881 . . . . . . . 8  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
4443oveq1d 6320 . . . . . . 7  |-  ( y  =  B  ->  (
( N `  y
) ^ 2 )  =  ( ( N `
 B ) ^
2 ) )
4544oveq2d 6321 . . . . . 6  |-  ( y  =  B  ->  (
( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) )  =  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) )
4645oveq2d 6321 . . . . 5  |-  ( y  =  B  ->  (
2  x.  ( ( ( N `  A
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )
4742, 46eqeq12d 2451 . . . 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 3197 . . 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 1023 . 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 15 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 setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087   <.cop 4008   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   1c1 9539    + caddc 9541    x. cmul 9543   -ucneg 9860   2c2 10659   ^cexp 12269   NrmCVeccnv 26048   +vcpv 26049   BaseSetcba 26050   .sOLDcns 26051   normCVcnmcv 26054   CPreHil OLDccphlo 26298
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-1st 6807  df-2nd 6808  df-vc 26010  df-nv 26056  df-va 26059  df-ba 26060  df-sm 26061  df-0v 26062  df-nmcv 26064  df-ph 26299
This theorem is referenced by:  ip0i  26311  hlpar  26384
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