MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  phpar2 Unicode version

Theorem phpar2 22172
Description: The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
isph.1  |-  X  =  ( BaseSet `  U )
isph.2  |-  G  =  ( +v `  U
)
isph.3  |-  M  =  ( -v `  U
)
isph.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
phpar2  |-  ( ( U  e.  CPreHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( N `
 ( A G B ) ) ^
2 )  +  ( ( N `  ( A M B ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )

Proof of Theorem phpar2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isph.1 . . . . 5  |-  X  =  ( BaseSet `  U )
2 isph.2 . . . . 5  |-  G  =  ( +v `  U
)
3 isph.3 . . . . 5  |-  M  =  ( -v `  U
)
4 isph.6 . . . . 5  |-  N  =  ( normCV `  U )
51, 2, 3, 4isph 22171 . . . 4  |-  ( U  e.  CPreHil OLD  <->  ( U  e.  NrmCVec 
/\  A. x  e.  X  A. y  e.  X  ( ( ( N `
 ( x G y ) ) ^
2 )  +  ( ( N `  (
x M y ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) ) ) )
65simprbi 451 . . 3  |-  ( U  e.  CPreHil OLD  ->  A. x  e.  X  A. y  e.  X  ( (
( N `  (
x G y ) ) ^ 2 )  +  ( ( N `
 ( x M y ) ) ^
2 ) )  =  ( 2  x.  (
( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) ) )
763ad2ant1 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 M y ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) ) )
8 oveq1 6027 . . . . . . . 8  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
98fveq2d 5672 . . . . . . 7  |-  ( x  =  A  ->  ( N `  ( x G y ) )  =  ( N `  ( A G y ) ) )
109oveq1d 6035 . . . . . 6  |-  ( x  =  A  ->  (
( N `  (
x G y ) ) ^ 2 )  =  ( ( N `
 ( A G y ) ) ^
2 ) )
11 oveq1 6027 . . . . . . . 8  |-  ( x  =  A  ->  (
x M y )  =  ( A M y ) )
1211fveq2d 5672 . . . . . . 7  |-  ( x  =  A  ->  ( N `  ( x M y ) )  =  ( N `  ( A M y ) ) )
1312oveq1d 6035 . . . . . 6  |-  ( x  =  A  ->  (
( N `  (
x M y ) ) ^ 2 )  =  ( ( N `
 ( A M y ) ) ^
2 ) )
1410, 13oveq12d 6038 . . . . 5  |-  ( x  =  A  ->  (
( ( N `  ( x G y ) ) ^ 2 )  +  ( ( N `  ( x M y ) ) ^ 2 ) )  =  ( ( ( N `  ( A G y ) ) ^ 2 )  +  ( ( N `  ( A M y ) ) ^ 2 ) ) )
15 fveq2 5668 . . . . . . . 8  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
1615oveq1d 6035 . . . . . . 7  |-  ( x  =  A  ->  (
( N `  x
) ^ 2 )  =  ( ( N `
 A ) ^
2 ) )
1716oveq1d 6035 . . . . . 6  |-  ( x  =  A  ->  (
( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) )  =  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) )
1817oveq2d 6036 . . . . 5  |-  ( x  =  A  ->  (
2  x.  ( ( ( N `  x
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  y
) ^ 2 ) ) ) )
1914, 18eqeq12d 2401 . . . 4  |-  ( x  =  A  ->  (
( ( ( N `
 ( x G y ) ) ^
2 )  +  ( ( N `  (
x M y ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) )  <->  ( (
( N `  ( A G y ) ) ^ 2 )  +  ( ( N `  ( A M y ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) ) ) )
20 oveq2 6028 . . . . . . . 8  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
2120fveq2d 5672 . . . . . . 7  |-  ( y  =  B  ->  ( N `  ( A G y ) )  =  ( N `  ( A G B ) ) )
2221oveq1d 6035 . . . . . 6  |-  ( y  =  B  ->  (
( N `  ( A G y ) ) ^ 2 )  =  ( ( N `  ( A G B ) ) ^ 2 ) )
23 oveq2 6028 . . . . . . . 8  |-  ( y  =  B  ->  ( A M y )  =  ( A M B ) )
2423fveq2d 5672 . . . . . . 7  |-  ( y  =  B  ->  ( N `  ( A M y ) )  =  ( N `  ( A M B ) ) )
2524oveq1d 6035 . . . . . 6  |-  ( y  =  B  ->  (
( N `  ( A M y ) ) ^ 2 )  =  ( ( N `  ( A M B ) ) ^ 2 ) )
2622, 25oveq12d 6038 . . . . 5  |-  ( y  =  B  ->  (
( ( N `  ( A G y ) ) ^ 2 )  +  ( ( N `
 ( A M y ) ) ^
2 ) )  =  ( ( ( N `
 ( A G B ) ) ^
2 )  +  ( ( N `  ( A M B ) ) ^ 2 ) ) )
27 fveq2 5668 . . . . . . . 8  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
2827oveq1d 6035 . . . . . . 7  |-  ( y  =  B  ->  (
( N `  y
) ^ 2 )  =  ( ( N `
 B ) ^
2 ) )
2928oveq2d 6036 . . . . . 6  |-  ( y  =  B  ->  (
( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) )  =  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) )
3029oveq2d 6036 . . . . 5  |-  ( y  =  B  ->  (
2  x.  ( ( ( N `  A
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )
3126, 30eqeq12d 2401 . . . 4  |-  ( y  =  B  ->  (
( ( ( N `
 ( A G y ) ) ^
2 )  +  ( ( N `  ( A M y ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  y
) ^ 2 ) ) )  <->  ( (
( N `  ( A G B ) ) ^ 2 )  +  ( ( N `  ( A M B ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) ) ) )
3219, 31rspc2v 3001 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( ( ( N `  ( x G y ) ) ^ 2 )  +  ( ( N `  ( x M y ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  x
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  ->  ( ( ( N `  ( A G B ) ) ^ 2 )  +  ( ( N `  ( A M B ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) ) ) )
33323adant1 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 M y ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  x
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  ->  ( ( ( N `  ( A G B ) ) ^ 2 )  +  ( ( N `  ( A M B ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) ) ) )
347, 33mpd 15 1  |-  ( ( U  e.  CPreHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( N `
 ( A G B ) ) ^
2 )  +  ( ( N `  ( A M B ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649   ` cfv 5394  (class class class)co 6020    + caddc 8926    x. cmul 8928   2c2 9981   ^cexp 11309   NrmCVeccnv 21911   +vcpv 21912   BaseSetcba 21913   -vcnsb 21916   normCVcnmcv 21917   CPreHil OLDccphlo 22161
This theorem is referenced by:  sspph  22204  minvecolem2  22225  hlpar2  22246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-ltxr 9058  df-sub 9225  df-neg 9226  df-grpo 21627  df-gid 21628  df-ginv 21629  df-gdiv 21630  df-ablo 21718  df-vc 21873  df-nv 21919  df-va 21922  df-ba 21923  df-sm 21924  df-0v 21925  df-vs 21926  df-nmcv 21927  df-ph 22162
  Copyright terms: Public domain W3C validator