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Theorem phpar 27679
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 27458 . . . . . 6 |- G = ( 1st ` ( 1st ` U ) )
3 fvex 6201 . . . . . 6 |- ( 1st ` ( 1st ` U ) ) e. _V
42, 3eqeltri 2697 . . . . 5 |- G e. _V
5 phpar.4 . . . . . . 7 |- S = ( .sOLD ` U )
65smfval 27460 . . . . . 6 |- S = ( 2nd ` ( 1st ` U ) )
7 fvex 6201 . . . . . 6 |- ( 2nd ` ( 1st ` U ) ) e. _V
86, 7eqeltri 2697 . . . . 5 |- S e. _V
9 phpar.6 . . . . . . 7 |- N = ( normCV ` U )
109nmcvfval 27462 . . . . . 6 |- N = ( 2nd ` U )
11 fvex 6201 . . . . . 6 |- ( 2nd ` U ) e. _V
1210, 11eqeltri 2697 . . . . 5 |- N e. _V
134, 8, 123pm3.2i 1239 . . . 4 |- ( G e. _V /\ S e. _V /\ N e. _V )
141, 5, 9phop 27673 . . . . . 6 |- ( U e. CPreHil OLD -> U = <. <. G , S >. , N >. )
1514eleq1d 2686 . . . . 5 |- ( U e. CPreHil OLD -> ( U e. CPreHil OLD <-> <. <. G , S >. , N >. e. CPreHil OLD ) )
1615ibi 256 . . . 4 |- ( U e. CPreHil OLD -> <. <. G , S >. , N >. e. CPreHil OLD )
17 phpar.1 . . . . . . 7 |- X = ( BaseSet ` U )
1817, 1bafval 27459 . . . . . 6 |- X = ran G
1918isphg 27672 . . . . 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 654 . . . 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 695 . . 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 1082 . 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 6657 . . . . . . . 8 |- ( x = A -> ( x G y ) = ( A G y ) )
2423fveq2d 6195 . . . . . . 7 |- ( x = A -> ( N ` ( x G y ) ) = ( N ` ( A G y ) ) )
2524oveq1d 6665 . . . . . 6 |- ( x = A -> ( ( N ` ( x G y ) ) ^ 2 ) = ( ( N ` ( A G y ) ) ^ 2 ) )
26 oveq1 6657 . . . . . . . 8 |- ( x = A -> ( x G ( -u 1 S y ) ) = ( A G ( -u 1 S y ) ) )
2726fveq2d 6195 . . . . . . 7 |- ( x = A -> ( N ` ( x G ( -u 1 S y ) ) ) = ( N ` ( A G ( -u 1 S y ) ) ) )
2827oveq1d 6665 . . . . . 6 |- ( x = A -> ( ( N ` ( x G ( -u 1 S y ) ) ) ^ 2 ) = ( ( N ` ( A G ( -u 1 S y ) ) ) ^ 2 ) )
2925, 28oveq12d 6668 . . . . 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 6191 . . . . . . . 8 |- ( x = A -> ( N ` x ) = ( N ` A ) )
3130oveq1d 6665 . . . . . . 7 |- ( x = A -> ( ( N ` x ) ^ 2 ) = ( ( N ` A ) ^ 2 ) )
3231oveq1d 6665 . . . . . 6 |- ( x = A -> ( ( ( N ` x ) ^ 2 ) + ( ( N ` y ) ^ 2 ) ) = ( ( ( N ` A ) ^ 2 ) + ( ( N ` y ) ^ 2 ) ) )
3332oveq2d 6666 . . . . 5 |- ( x = A -> ( 2 x. ( ( ( N ` x ) ^ 2 ) + ( ( N ` y ) ^ 2 ) ) ) = ( 2 x. ( ( ( N ` A ) ^ 2 ) + ( ( N ` y ) ^ 2 ) ) ) )
3429, 33eqeq12d 2637 . . . 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 6658 . . . . . . . 8 |- ( y = B -> ( A G y ) = ( A G B ) )
3635fveq2d 6195 . . . . . . 7 |- ( y = B -> ( N ` ( A G y ) ) = ( N ` ( A G B ) ) )
3736oveq1d 6665 . . . . . 6 |- ( y = B -> ( ( N ` ( A G y ) ) ^ 2 ) = ( ( N ` ( A G B ) ) ^ 2 ) )
38 oveq2 6658 . . . . . . . . 9 |- ( y = B -> ( -u 1 S y ) = ( -u 1 S B ) )
3938oveq2d 6666 . . . . . . . 8 |- ( y = B -> ( A G ( -u 1 S y ) ) = ( A G ( -u 1 S B ) ) )
4039fveq2d 6195 . . . . . . 7 |- ( y = B -> ( N ` ( A G ( -u 1 S y ) ) ) = ( N ` ( A G ( -u 1 S B ) ) ) )
4140oveq1d 6665 . . . . . 6 |- ( y = B -> ( ( N ` ( A G ( -u 1 S y ) ) ) ^ 2 ) = ( ( N ` ( A G ( -u 1 S B ) ) ) ^ 2 ) )
4237, 41oveq12d 6668 . . . . 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 6191 . . . . . . . 8 |- ( y = B -> ( N ` y ) = ( N ` B ) )
4443oveq1d 6665 . . . . . . 7 |- ( y = B -> ( ( N ` y ) ^ 2 ) = ( ( N ` B ) ^ 2 ) )
4544oveq2d 6666 . . . . . 6 |- ( y = B -> ( ( ( N ` A ) ^ 2 ) + ( ( N ` y ) ^ 2 ) ) = ( ( ( N ` A ) ^ 2 ) + ( ( N ` B ) ^ 2 ) ) )
4645oveq2d 6666 . . . . 5 |- ( y = B -> ( 2 x. ( ( ( N ` A ) ^ 2 ) + ( ( N ` y ) ^ 2 ) ) ) = ( 2 x. ( ( ( N ` A ) ^ 2 ) + ( ( N ` B ) ^ 2 ) ) ) )
4742, 46eqeq12d 2637 . . . 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 3322 . . 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 1079 . 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 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   <.cop 4183   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   1c1 9937   + caddc 9939   x. cmul 9941   -ucneg 10267   2c2 11070   ^cexp 12860   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442   normCVcnmcv 27445   CPreHil OLDccphlo 27667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-1st 7168  df-2nd 7169  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455  df-ph 27668
This theorem is referenced by:  ip0i  27680  hlpar  27753
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