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Theorem phpar2 27678
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 27677 . . . 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 480 . . 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 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 M y ) ) ^ 2 ) ) = ( 2 x. ( ( ( N ` x ) ^ 2 ) + ( ( N ` y ) ^ 2 ) ) ) )
8 oveq1 6657 . . . . . . . 8 |- ( x = A -> ( x G y ) = ( A G y ) )
98fveq2d 6195 . . . . . . 7 |- ( x = A -> ( N ` ( x G y ) ) = ( N ` ( A G y ) ) )
109oveq1d 6665 . . . . . 6 |- ( x = A -> ( ( N ` ( x G y ) ) ^ 2 ) = ( ( N ` ( A G y ) ) ^ 2 ) )
11 oveq1 6657 . . . . . . . 8 |- ( x = A -> ( x M y ) = ( A M y ) )
1211fveq2d 6195 . . . . . . 7 |- ( x = A -> ( N ` ( x M y ) ) = ( N ` ( A M y ) ) )
1312oveq1d 6665 . . . . . 6 |- ( x = A -> ( ( N ` ( x M y ) ) ^ 2 ) = ( ( N ` ( A M y ) ) ^ 2 ) )
1410, 13oveq12d 6668 . . . . 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 6191 . . . . . . . 8 |- ( x = A -> ( N ` x ) = ( N ` A ) )
1615oveq1d 6665 . . . . . . 7 |- ( x = A -> ( ( N ` x ) ^ 2 ) = ( ( N ` A ) ^ 2 ) )
1716oveq1d 6665 . . . . . 6 |- ( x = A -> ( ( ( N ` x ) ^ 2 ) + ( ( N ` y ) ^ 2 ) ) = ( ( ( N ` A ) ^ 2 ) + ( ( N ` y ) ^ 2 ) ) )
1817oveq2d 6666 . . . . 5 |- ( x = A -> ( 2 x. ( ( ( N ` x ) ^ 2 ) + ( ( N ` y ) ^ 2 ) ) ) = ( 2 x. ( ( ( N ` A ) ^ 2 ) + ( ( N ` y ) ^ 2 ) ) ) )
1914, 18eqeq12d 2637 . . . 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 6658 . . . . . . . 8 |- ( y = B -> ( A G y ) = ( A G B ) )
2120fveq2d 6195 . . . . . . 7 |- ( y = B -> ( N ` ( A G y ) ) = ( N ` ( A G B ) ) )
2221oveq1d 6665 . . . . . 6 |- ( y = B -> ( ( N ` ( A G y ) ) ^ 2 ) = ( ( N ` ( A G B ) ) ^ 2 ) )
23 oveq2 6658 . . . . . . . 8 |- ( y = B -> ( A M y ) = ( A M B ) )
2423fveq2d 6195 . . . . . . 7 |- ( y = B -> ( N ` ( A M y ) ) = ( N ` ( A M B ) ) )
2524oveq1d 6665 . . . . . 6 |- ( y = B -> ( ( N ` ( A M y ) ) ^ 2 ) = ( ( N ` ( A M B ) ) ^ 2 ) )
2622, 25oveq12d 6668 . . . . 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 6191 . . . . . . . 8 |- ( y = B -> ( N ` y ) = ( N ` B ) )
2827oveq1d 6665 . . . . . . 7 |- ( y = B -> ( ( N ` y ) ^ 2 ) = ( ( N ` B ) ^ 2 ) )
2928oveq2d 6666 . . . . . 6 |- ( y = B -> ( ( ( N ` A ) ^ 2 ) + ( ( N ` y ) ^ 2 ) ) = ( ( ( N ` A ) ^ 2 ) + ( ( N ` B ) ^ 2 ) ) )
3029oveq2d 6666 . . . . 5 |- ( y = B -> ( 2 x. ( ( ( N ` A ) ^ 2 ) + ( ( N ` y ) ^ 2 ) ) ) = ( 2 x. ( ( ( N ` A ) ^ 2 ) + ( ( N ` B ) ^ 2 ) ) ) )
3126, 30eqeq12d 2637 . . . 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 3322 . . 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 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 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 setvar class
Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   + caddc 9939   x. cmul 9941   2c2 11070   ^cexp 12860   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   -vcnsb 27444   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  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-pw 4160  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-po 5035  df-so 5036  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-vs 27454  df-nmcv 27455  df-ph 27668
This theorem is referenced by:  sspph  27710  minvecolem2  27731  hlpar2  27752
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