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Theorem phpar 22317
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 22074 . . . . . 6  |-  G  =  ( 1st `  ( 1st `  U ) )
3 fvex 5734 . . . . . 6  |-  ( 1st `  ( 1st `  U
) )  e.  _V
42, 3eqeltri 2505 . . . . 5  |-  G  e. 
_V
5 phpar.4 . . . . . . 7  |-  S  =  ( .s OLD `  U
)
65smfval 22076 . . . . . 6  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 fvex 5734 . . . . . 6  |-  ( 2nd `  ( 1st `  U
) )  e.  _V
86, 7eqeltri 2505 . . . . 5  |-  S  e. 
_V
9 phpar.6 . . . . . . 7  |-  N  =  ( normCV `  U )
109nmcvfval 22078 . . . . . 6  |-  N  =  ( 2nd `  U
)
11 fvex 5734 . . . . . 6  |-  ( 2nd `  U )  e.  _V
1210, 11eqeltri 2505 . . . . 5  |-  N  e. 
_V
134, 8, 123pm3.2i 1132 . . . 4  |-  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V )
141, 5, 9phop 22311 . . . . . 6  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. <. G ,  S >. ,  N >. )
1514eleq1d 2501 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  ( U  e.  CPreHil OLD  <->  <. <. G ,  S >. ,  N >.  e.  CPreHil OLD ) )
1615ibi 233 . . . 4  |-  ( U  e.  CPreHil OLD  ->  <. <. G ,  S >. ,  N >.  e.  CPreHil
OLD )
17 phpar.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
1817, 1bafval 22075 . . . . . 6  |-  X  =  ran  G
1918isphg 22310 . . . . 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 608 . . . 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 645 . . 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 978 . 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 6080 . . . . . . . 8  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
2423fveq2d 5724 . . . . . . 7  |-  ( x  =  A  ->  ( N `  ( x G y ) )  =  ( N `  ( A G y ) ) )
2524oveq1d 6088 . . . . . 6  |-  ( x  =  A  ->  (
( N `  (
x G y ) ) ^ 2 )  =  ( ( N `
 ( A G y ) ) ^
2 ) )
26 oveq1 6080 . . . . . . . 8  |-  ( x  =  A  ->  (
x G ( -u
1 S y ) )  =  ( A G ( -u 1 S y ) ) )
2726fveq2d 5724 . . . . . . 7  |-  ( x  =  A  ->  ( N `  ( x G ( -u 1 S y ) ) )  =  ( N `
 ( A G ( -u 1 S y ) ) ) )
2827oveq1d 6088 . . . . . 6  |-  ( x  =  A  ->  (
( N `  (
x G ( -u
1 S y ) ) ) ^ 2 )  =  ( ( N `  ( A G ( -u 1 S y ) ) ) ^ 2 ) )
2925, 28oveq12d 6091 . . . . 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 5720 . . . . . . . 8  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
3130oveq1d 6088 . . . . . . 7  |-  ( x  =  A  ->  (
( N `  x
) ^ 2 )  =  ( ( N `
 A ) ^
2 ) )
3231oveq1d 6088 . . . . . 6  |-  ( x  =  A  ->  (
( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) )  =  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) )
3332oveq2d 6089 . . . . 5  |-  ( x  =  A  ->  (
2  x.  ( ( ( N `  x
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  y
) ^ 2 ) ) ) )
3429, 33eqeq12d 2449 . . . 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 6081 . . . . . . . 8  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
3635fveq2d 5724 . . . . . . 7  |-  ( y  =  B  ->  ( N `  ( A G y ) )  =  ( N `  ( A G B ) ) )
3736oveq1d 6088 . . . . . 6  |-  ( y  =  B  ->  (
( N `  ( A G y ) ) ^ 2 )  =  ( ( N `  ( A G B ) ) ^ 2 ) )
38 oveq2 6081 . . . . . . . . 9  |-  ( y  =  B  ->  ( -u 1 S y )  =  ( -u 1 S B ) )
3938oveq2d 6089 . . . . . . . 8  |-  ( y  =  B  ->  ( A G ( -u 1 S y ) )  =  ( A G ( -u 1 S B ) ) )
4039fveq2d 5724 . . . . . . 7  |-  ( y  =  B  ->  ( N `  ( A G ( -u 1 S y ) ) )  =  ( N `
 ( A G ( -u 1 S B ) ) ) )
4140oveq1d 6088 . . . . . 6  |-  ( y  =  B  ->  (
( N `  ( A G ( -u 1 S y ) ) ) ^ 2 )  =  ( ( N `
 ( A G ( -u 1 S B ) ) ) ^ 2 ) )
4237, 41oveq12d 6091 . . . . 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 5720 . . . . . . . 8  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
4443oveq1d 6088 . . . . . . 7  |-  ( y  =  B  ->  (
( N `  y
) ^ 2 )  =  ( ( N `
 B ) ^
2 ) )
4544oveq2d 6089 . . . . . 6  |-  ( y  =  B  ->  (
( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) )  =  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) )
4645oveq2d 6089 . . . . 5  |-  ( y  =  B  ->  (
2  x.  ( ( ( N `  A
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )
4742, 46eqeq12d 2449 . . . 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 3050 . . 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 975 . 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 set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   <.cop 3809   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   1c1 8983    + caddc 8985    x. cmul 8987   -ucneg 9284   2c2 10041   ^cexp 11374   NrmCVeccnv 22055   +vcpv 22056   BaseSetcba 22057   .s
OLDcns 22058   normCVcnmcv 22061   CPreHil OLDccphlo 22305
This theorem is referenced by:  ip0i  22318  hlpar  22391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-1st 6341  df-2nd 6342  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071  df-ph 22306
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