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Theorem ipval3 24241
Description: Expansion of the inner product value ipval 24235. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
dipfval.1  |-  X  =  ( BaseSet `  U )
dipfval.2  |-  G  =  ( +v `  U
)
dipfval.4  |-  S  =  ( .sOLD `  U )
dipfval.6  |-  N  =  ( normCV `  U )
dipfval.7  |-  P  =  ( .iOLD `  U )
ipval3.3  |-  M  =  ( -v `  U
)
Assertion
Ref Expression
ipval3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  =  ( ( ( ( ( N `  ( A G B ) ) ^ 2 )  -  ( ( N `  ( A M B ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `  ( A G ( _i S B ) ) ) ^ 2 )  -  ( ( N `  ( A M ( _i S B ) ) ) ^ 2 ) ) ) )  / 
4 ) )

Proof of Theorem ipval3
StepHypRef Expression
1 dipfval.1 . . 3  |-  X  =  ( BaseSet `  U )
2 dipfval.2 . . 3  |-  G  =  ( +v `  U
)
3 dipfval.4 . . 3  |-  S  =  ( .sOLD `  U )
4 dipfval.6 . . 3  |-  N  =  ( normCV `  U )
5 dipfval.7 . . 3  |-  P  =  ( .iOLD `  U )
61, 2, 3, 4, 5ipval2 24239 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  =  ( ( ( ( ( N `  ( A G B ) ) ^ 2 )  -  ( ( N `  ( A G ( -u
1 S B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `  ( A G ( _i S B ) ) ) ^ 2 )  -  ( ( N `  ( A G ( -u _i S B ) ) ) ^ 2 ) ) ) )  / 
4 ) )
7 ipval3.3 . . . . . . . 8  |-  M  =  ( -v `  U
)
81, 2, 3, 7nvmval 24159 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A G (
-u 1 S B ) ) )
98fveq2d 5795 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M B ) )  =  ( N `  ( A G ( -u 1 S B ) ) ) )
109oveq1d 6207 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  ( A M B ) ) ^ 2 )  =  ( ( N `  ( A G ( -u
1 S B ) ) ) ^ 2 ) )
1110oveq2d 6208 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( N `  ( A G B ) ) ^ 2 )  -  ( ( N `
 ( A M B ) ) ^
2 ) )  =  ( ( ( N `
 ( A G B ) ) ^
2 )  -  (
( N `  ( A G ( -u 1 S B ) ) ) ^ 2 ) ) )
12 ax-icn 9444 . . . . . . . . . . . 12  |-  _i  e.  CC
131, 3nvscl 24143 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  _i  e.  CC  /\  B  e.  X )  ->  (
_i S B )  e.  X )
1412, 13mp3an2 1303 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
_i S B )  e.  X )
15143adant2 1007 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
_i S B )  e.  X )
161, 2, 3, 7nvmval 24159 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  (
_i S B )  e.  X )  -> 
( A M ( _i S B ) )  =  ( A G ( -u 1 S ( _i S B ) ) ) )
1715, 16syld3an3 1264 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( _i S B ) )  =  ( A G (
-u 1 S ( _i S B ) ) ) )
1812mulm1i 9892 . . . . . . . . . . . . 13  |-  ( -u
1  x.  _i )  =  -u _i
1918oveq1i 6202 . . . . . . . . . . . 12  |-  ( (
-u 1  x.  _i ) S B )  =  ( -u _i S B )
20 neg1cn 10528 . . . . . . . . . . . . . 14  |-  -u 1  e.  CC
211, 3nvsass 24145 . . . . . . . . . . . . . 14  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  _i  e.  CC  /\  B  e.  X ) )  -> 
( ( -u 1  x.  _i ) S B )  =  ( -u
1 S ( _i S B ) ) )
2220, 21mp3anr1 1312 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  (
_i  e.  CC  /\  B  e.  X )
)  ->  ( ( -u 1  x.  _i ) S B )  =  ( -u 1 S ( _i S B ) ) )
2312, 22mpanr1 683 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( -u 1  x.  _i ) S B )  =  ( -u 1 S ( _i S B ) ) )
2419, 23syl5reqr 2507 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 S ( _i S B ) )  =  ( -u _i S B ) )
25243adant2 1007 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 S ( _i S B ) )  =  ( -u _i S B ) )
2625oveq2d 6208 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( -u 1 S ( _i S B ) ) )  =  ( A G ( -u _i S B ) ) )
2717, 26eqtrd 2492 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( _i S B ) )  =  ( A G (
-u _i S B ) ) )
2827fveq2d 5795 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M ( _i S B ) ) )  =  ( N `  ( A G ( -u _i S B ) ) ) )
2928oveq1d 6207 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  ( A M ( _i S B ) ) ) ^ 2 )  =  ( ( N `  ( A G ( -u _i S B ) ) ) ^ 2 ) )
3029oveq2d 6208 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( N `  ( A G ( _i S B ) ) ) ^ 2 )  -  ( ( N `
 ( A M ( _i S B ) ) ) ^
2 ) )  =  ( ( ( N `
 ( A G ( _i S B ) ) ) ^
2 )  -  (
( N `  ( A G ( -u _i S B ) ) ) ^ 2 ) ) )
3130oveq2d 6208 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
_i  x.  ( (
( N `  ( A G ( _i S B ) ) ) ^ 2 )  -  ( ( N `  ( A M ( _i S B ) ) ) ^ 2 ) ) )  =  ( _i  x.  ( ( ( N `  ( A G ( _i S B ) ) ) ^ 2 )  -  ( ( N `  ( A G ( -u _i S B ) ) ) ^ 2 ) ) ) )
3211, 31oveq12d 6210 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( N `
 ( A G B ) ) ^
2 )  -  (
( N `  ( A M B ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `
 ( A G ( _i S B ) ) ) ^
2 )  -  (
( N `  ( A M ( _i S B ) ) ) ^ 2 ) ) ) )  =  ( ( ( ( N `
 ( A G B ) ) ^
2 )  -  (
( N `  ( A G ( -u 1 S B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `
 ( A G ( _i S B ) ) ) ^
2 )  -  (
( N `  ( A G ( -u _i S B ) ) ) ^ 2 ) ) ) ) )
3332oveq1d 6207 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( ( N `  ( A G B ) ) ^ 2 )  -  ( ( N `  ( A M B ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `  ( A G ( _i S B ) ) ) ^ 2 )  -  ( ( N `  ( A M ( _i S B ) ) ) ^ 2 ) ) ) )  / 
4 )  =  ( ( ( ( ( N `  ( A G B ) ) ^ 2 )  -  ( ( N `  ( A G ( -u
1 S B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `  ( A G ( _i S B ) ) ) ^ 2 )  -  ( ( N `  ( A G ( -u _i S B ) ) ) ^ 2 ) ) ) )  / 
4 ) )
346, 33eqtr4d 2495 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  =  ( ( ( ( ( N `  ( A G B ) ) ^ 2 )  -  ( ( N `  ( A M B ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `  ( A G ( _i S B ) ) ) ^ 2 )  -  ( ( N `  ( A M ( _i S B ) ) ) ^ 2 ) ) ) )  / 
4 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5518  (class class class)co 6192   CCcc 9383   1c1 9386   _ici 9387    + caddc 9388    x. cmul 9390    - cmin 9698   -ucneg 9699    / cdiv 10096   2c2 10474   4c4 10476   ^cexp 11968   NrmCVeccnv 24099   +vcpv 24100   BaseSetcba 24101   .sOLDcns 24102   -vcnsb 24104   normCVcnmcv 24105   .iOLDcdip 24232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-sup 7794  df-oi 7827  df-card 8212  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-n0 10683  df-z 10750  df-uz 10965  df-rp 11095  df-fz 11541  df-fzo 11652  df-seq 11910  df-exp 11969  df-hash 12207  df-cj 12692  df-re 12693  df-im 12694  df-sqr 12828  df-abs 12829  df-clim 13070  df-sum 13268  df-grpo 23815  df-gid 23816  df-ginv 23817  df-gdiv 23818  df-ablo 23906  df-vc 24061  df-nv 24107  df-va 24110  df-ba 24111  df-sm 24112  df-0v 24113  df-vs 24114  df-nmcv 24115  df-dip 24233
This theorem is referenced by:  4ipval3  24244  hhip  24716
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