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Theorem phpar 25443
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 25200 . . . . . 6  |-  G  =  ( 1st `  ( 1st `  U ) )
3 fvex 5876 . . . . . 6  |-  ( 1st `  ( 1st `  U
) )  e.  _V
42, 3eqeltri 2551 . . . . 5  |-  G  e. 
_V
5 phpar.4 . . . . . . 7  |-  S  =  ( .sOLD `  U )
65smfval 25202 . . . . . 6  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 fvex 5876 . . . . . 6  |-  ( 2nd `  ( 1st `  U
) )  e.  _V
86, 7eqeltri 2551 . . . . 5  |-  S  e. 
_V
9 phpar.6 . . . . . . 7  |-  N  =  ( normCV `  U )
109nmcvfval 25204 . . . . . 6  |-  N  =  ( 2nd `  U
)
11 fvex 5876 . . . . . 6  |-  ( 2nd `  U )  e.  _V
1210, 11eqeltri 2551 . . . . 5  |-  N  e. 
_V
134, 8, 123pm3.2i 1174 . . . 4  |-  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V )
141, 5, 9phop 25437 . . . . . 6  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. <. G ,  S >. ,  N >. )
1514eleq1d 2536 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  ( U  e.  CPreHil OLD  <->  <. <. G ,  S >. ,  N >.  e.  CPreHil OLD ) )
1615ibi 241 . . . 4  |-  ( U  e.  CPreHil OLD  ->  <. <. G ,  S >. ,  N >.  e.  CPreHil
OLD )
17 phpar.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
1817, 1bafval 25201 . . . . . 6  |-  X  =  ran  G
1918isphg 25436 . . . . 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 624 . . . 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 663 . . 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 1017 . 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 6291 . . . . . . . 8  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
2423fveq2d 5870 . . . . . . 7  |-  ( x  =  A  ->  ( N `  ( x G y ) )  =  ( N `  ( A G y ) ) )
2524oveq1d 6299 . . . . . 6  |-  ( x  =  A  ->  (
( N `  (
x G y ) ) ^ 2 )  =  ( ( N `
 ( A G y ) ) ^
2 ) )
26 oveq1 6291 . . . . . . . 8  |-  ( x  =  A  ->  (
x G ( -u
1 S y ) )  =  ( A G ( -u 1 S y ) ) )
2726fveq2d 5870 . . . . . . 7  |-  ( x  =  A  ->  ( N `  ( x G ( -u 1 S y ) ) )  =  ( N `
 ( A G ( -u 1 S y ) ) ) )
2827oveq1d 6299 . . . . . 6  |-  ( x  =  A  ->  (
( N `  (
x G ( -u
1 S y ) ) ) ^ 2 )  =  ( ( N `  ( A G ( -u 1 S y ) ) ) ^ 2 ) )
2925, 28oveq12d 6302 . . . . 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 5866 . . . . . . . 8  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
3130oveq1d 6299 . . . . . . 7  |-  ( x  =  A  ->  (
( N `  x
) ^ 2 )  =  ( ( N `
 A ) ^
2 ) )
3231oveq1d 6299 . . . . . 6  |-  ( x  =  A  ->  (
( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) )  =  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) )
3332oveq2d 6300 . . . . 5  |-  ( x  =  A  ->  (
2  x.  ( ( ( N `  x
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  y
) ^ 2 ) ) ) )
3429, 33eqeq12d 2489 . . . 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 6292 . . . . . . . 8  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
3635fveq2d 5870 . . . . . . 7  |-  ( y  =  B  ->  ( N `  ( A G y ) )  =  ( N `  ( A G B ) ) )
3736oveq1d 6299 . . . . . 6  |-  ( y  =  B  ->  (
( N `  ( A G y ) ) ^ 2 )  =  ( ( N `  ( A G B ) ) ^ 2 ) )
38 oveq2 6292 . . . . . . . . 9  |-  ( y  =  B  ->  ( -u 1 S y )  =  ( -u 1 S B ) )
3938oveq2d 6300 . . . . . . . 8  |-  ( y  =  B  ->  ( A G ( -u 1 S y ) )  =  ( A G ( -u 1 S B ) ) )
4039fveq2d 5870 . . . . . . 7  |-  ( y  =  B  ->  ( N `  ( A G ( -u 1 S y ) ) )  =  ( N `
 ( A G ( -u 1 S B ) ) ) )
4140oveq1d 6299 . . . . . 6  |-  ( y  =  B  ->  (
( N `  ( A G ( -u 1 S y ) ) ) ^ 2 )  =  ( ( N `
 ( A G ( -u 1 S B ) ) ) ^ 2 ) )
4237, 41oveq12d 6302 . . . . 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 5866 . . . . . . . 8  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
4443oveq1d 6299 . . . . . . 7  |-  ( y  =  B  ->  (
( N `  y
) ^ 2 )  =  ( ( N `
 B ) ^
2 ) )
4544oveq2d 6300 . . . . . 6  |-  ( y  =  B  ->  (
( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) )  =  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) )
4645oveq2d 6300 . . . . 5  |-  ( y  =  B  ->  (
2  x.  ( ( ( N `  A
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )
4742, 46eqeq12d 2489 . . . 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 3223 . . 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 1014 . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   <.cop 4033   ` cfv 5588  (class class class)co 6284   1stc1st 6782   2ndc2nd 6783   1c1 9493    + caddc 9495    x. cmul 9497   -ucneg 9806   2c2 10585   ^cexp 12134   NrmCVeccnv 25181   +vcpv 25182   BaseSetcba 25183   .sOLDcns 25184   normCVcnmcv 25187   CPreHil OLDccphlo 25431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-1st 6784  df-2nd 6785  df-vc 25143  df-nv 25189  df-va 25192  df-ba 25193  df-sm 25194  df-0v 25195  df-nmcv 25197  df-ph 25432
This theorem is referenced by:  ip0i  25444  hlpar  25517
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