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Theorem phpar2 21401
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 21400 . . . 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 450 . . 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 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 M y ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) ) )
8 oveq1 5865 . . . . . . . 8  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
98fveq2d 5529 . . . . . . 7  |-  ( x  =  A  ->  ( N `  ( x G y ) )  =  ( N `  ( A G y ) ) )
109oveq1d 5873 . . . . . 6  |-  ( x  =  A  ->  (
( N `  (
x G y ) ) ^ 2 )  =  ( ( N `
 ( A G y ) ) ^
2 ) )
11 oveq1 5865 . . . . . . . 8  |-  ( x  =  A  ->  (
x M y )  =  ( A M y ) )
1211fveq2d 5529 . . . . . . 7  |-  ( x  =  A  ->  ( N `  ( x M y ) )  =  ( N `  ( A M y ) ) )
1312oveq1d 5873 . . . . . 6  |-  ( x  =  A  ->  (
( N `  (
x M y ) ) ^ 2 )  =  ( ( N `
 ( A M y ) ) ^
2 ) )
1410, 13oveq12d 5876 . . . . 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 5525 . . . . . . . 8  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
1615oveq1d 5873 . . . . . . 7  |-  ( x  =  A  ->  (
( N `  x
) ^ 2 )  =  ( ( N `
 A ) ^
2 ) )
1716oveq1d 5873 . . . . . 6  |-  ( x  =  A  ->  (
( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) )  =  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) ) )
1817oveq2d 5874 . . . . 5  |-  ( x  =  A  ->  (
2  x.  ( ( ( N `  x
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  y
) ^ 2 ) ) ) )
1914, 18eqeq12d 2297 . . . 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 5866 . . . . . . . 8  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
2120fveq2d 5529 . . . . . . 7  |-  ( y  =  B  ->  ( N `  ( A G y ) )  =  ( N `  ( A G B ) ) )
2221oveq1d 5873 . . . . . 6  |-  ( y  =  B  ->  (
( N `  ( A G y ) ) ^ 2 )  =  ( ( N `  ( A G B ) ) ^ 2 ) )
23 oveq2 5866 . . . . . . . 8  |-  ( y  =  B  ->  ( A M y )  =  ( A M B ) )
2423fveq2d 5529 . . . . . . 7  |-  ( y  =  B  ->  ( N `  ( A M y ) )  =  ( N `  ( A M B ) ) )
2524oveq1d 5873 . . . . . 6  |-  ( y  =  B  ->  (
( N `  ( A M y ) ) ^ 2 )  =  ( ( N `  ( A M B ) ) ^ 2 ) )
2622, 25oveq12d 5876 . . . . 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 5525 . . . . . . . 8  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
2827oveq1d 5873 . . . . . . 7  |-  ( y  =  B  ->  (
( N `  y
) ^ 2 )  =  ( ( N `
 B ) ^
2 ) )
2928oveq2d 5874 . . . . . 6  |-  ( y  =  B  ->  (
( ( N `  A ) ^ 2 )  +  ( ( N `  y ) ^ 2 ) )  =  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) )
3029oveq2d 5874 . . . . 5  |-  ( y  =  B  ->  (
2  x.  ( ( ( N `  A
) ^ 2 )  +  ( ( N `
 y ) ^
2 ) ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )
3126, 30eqeq12d 2297 . . . 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 2890 . . 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 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 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 14 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 934    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858    + caddc 8740    x. cmul 8742   2c2 9795   ^cexp 11104   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   -vcnsb 21145   normCVcnmcv 21146   CPreHil OLDccphlo 21390
This theorem is referenced by:  sspph  21433  minvecolem2  21454  hlpar2  21475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ph 21391
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