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Theorem phpar 21418
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 21175 . . . . . 6  |-  G  =  ( 1st `  ( 1st `  U ) )
3 fvex 5555 . . . . . 6  |-  ( 1st `  ( 1st `  U
) )  e.  _V
42, 3eqeltri 2366 . . . . 5  |-  G  e. 
_V
5 phpar.4 . . . . . . 7  |-  S  =  ( .s OLD `  U
)
65smfval 21177 . . . . . 6  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 fvex 5555 . . . . . 6  |-  ( 2nd `  ( 1st `  U
) )  e.  _V
86, 7eqeltri 2366 . . . . 5  |-  S  e. 
_V
9 phpar.6 . . . . . . 7  |-  N  =  ( normCV `  U )
109nmcvfval 21179 . . . . . 6  |-  N  =  ( 2nd `  U
)
11 fvex 5555 . . . . . 6  |-  ( 2nd `  U )  e.  _V
1210, 11eqeltri 2366 . . . . 5  |-  N  e. 
_V
134, 8, 123pm3.2i 1130 . . . 4  |-  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V )
141, 5, 9phop 21412 . . . . . 6  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. <. G ,  S >. ,  N >. )
1514eleq1d 2362 . . . . 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 21176 . . . . . 6  |-  X  =  ran  G
1918isphg 21411 . . . . 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 5881 . . . . . . . 8  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
2423fveq2d 5545 . . . . . . 7  |-  ( x  =  A  ->  ( N `  ( x G y ) )  =  ( N `  ( A G y ) ) )
2524oveq1d 5889 . . . . . 6  |-  ( x  =  A  ->  (
( N `  (
x G y ) ) ^ 2 )  =  ( ( N `
 ( A G y ) ) ^
2 ) )
26 oveq1 5881 . . . . . . . 8  |-  ( x  =  A  ->  (
x G ( -u
1 S y ) )  =  ( A G ( -u 1 S y ) ) )
2726fveq2d 5545 . . . . . . 7  |-  ( x  =  A  ->  ( N `  ( x G ( -u 1 S y ) ) )  =  ( N `
 ( A G ( -u 1 S y ) ) ) )
2827oveq1d 5889 . . . . . 6  |-  ( x  =  A  ->  (
( N `  (
x G ( -u
1 S y ) ) ) ^ 2 )  =  ( ( N `  ( A G ( -u 1 S y ) ) ) ^ 2 ) )
2925, 28oveq12d 5892 . . . . 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 5541 . . . . . . . 8  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
3130oveq1d 5889 . . . . . . 7  |-  ( x  =  A  ->  (
( N `  x
) ^ 2 )  =  ( ( N `
 A ) ^
2 ) )
3231oveq1d 5889 . . . . . 6  |-  ( x  =  A  ->  (
( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) )  =  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) )
3332oveq2d 5890 . . . . 5  |-  ( x  =  A  ->  (
2  x.  ( ( ( N `  x
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  y
) ^ 2 ) ) ) )
3429, 33eqeq12d 2310 . . . 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 5882 . . . . . . . 8  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
3635fveq2d 5545 . . . . . . 7  |-  ( y  =  B  ->  ( N `  ( A G y ) )  =  ( N `  ( A G B ) ) )
3736oveq1d 5889 . . . . . 6  |-  ( y  =  B  ->  (
( N `  ( A G y ) ) ^ 2 )  =  ( ( N `  ( A G B ) ) ^ 2 ) )
38 oveq2 5882 . . . . . . . . 9  |-  ( y  =  B  ->  ( -u 1 S y )  =  ( -u 1 S B ) )
3938oveq2d 5890 . . . . . . . 8  |-  ( y  =  B  ->  ( A G ( -u 1 S y ) )  =  ( A G ( -u 1 S B ) ) )
4039fveq2d 5545 . . . . . . 7  |-  ( y  =  B  ->  ( N `  ( A G ( -u 1 S y ) ) )  =  ( N `
 ( A G ( -u 1 S B ) ) ) )
4140oveq1d 5889 . . . . . 6  |-  ( y  =  B  ->  (
( N `  ( A G ( -u 1 S y ) ) ) ^ 2 )  =  ( ( N `
 ( A G ( -u 1 S B ) ) ) ^ 2 ) )
4237, 41oveq12d 5892 . . . . 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 5541 . . . . . . . 8  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
4443oveq1d 5889 . . . . . . 7  |-  ( y  =  B  ->  (
( N `  y
) ^ 2 )  =  ( ( N `
 B ) ^
2 ) )
4544oveq2d 5890 . . . . . 6  |-  ( y  =  B  ->  (
( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) )  =  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) )
4645oveq2d 5890 . . . . 5  |-  ( y  =  B  ->  (
2  x.  ( ( ( N `  A
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )
4742, 46eqeq12d 2310 . . . 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 2903 . . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   <.cop 3656   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   1c1 8754    + caddc 8756    x. cmul 8758   -ucneg 9054   2c2 9811   ^cexp 11120   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s
OLDcns 21159   normCVcnmcv 21162   CPreHil OLDccphlo 21406
This theorem is referenced by:  ip0i  21419  hlpar  21492
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-1st 6138  df-2nd 6139  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172  df-ph 21407
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