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Theorem 4ipval3 26193
Description: Four times the inner product value ipval3 26190, useful for simplifying certain proofs. (Contributed by NM, 1-Feb-2008.) (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
4ipval3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
4  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 ) ) ) ) )

Proof of Theorem 4ipval3
StepHypRef Expression
1 dipfval.1 . . . 4  |-  X  =  ( BaseSet `  U )
2 dipfval.2 . . . 4  |-  G  =  ( +v `  U
)
3 dipfval.4 . . . 4  |-  S  =  ( .sOLD `  U )
4 dipfval.6 . . . 4  |-  N  =  ( normCV `  U )
5 dipfval.7 . . . 4  |-  P  =  ( .iOLD `  U )
6 ipval3.3 . . . 4  |-  M  =  ( -v `  U
)
71, 2, 3, 4, 5, 6ipval3 26190 . . 3  |-  ( ( 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 ) )
87oveq2d 6321 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
4  x.  ( A P B ) )  =  ( 4  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 ) ) )
91, 2, 3, 4, 5ipval2lem3 26186 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  ( A G B ) ) ^ 2 )  e.  RR )
109recnd 9668 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  ( A G B ) ) ^ 2 )  e.  CC )
111, 2, 3, 4, 5, 6ipval2lem6 26192 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  ( A M B ) ) ^ 2 )  e.  RR )
1211recnd 9668 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  ( A M B ) ) ^ 2 )  e.  CC )
1310, 12subcld 9985 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( N `  ( A G B ) ) ^ 2 )  -  ( ( N `
 ( A M B ) ) ^
2 ) )  e.  CC )
14 ax-icn 9597 . . . . 5  |-  _i  e.  CC
151, 2, 3, 4, 5ipval2lem2 26185 . . . . . . . 8  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  _i  e.  CC )  ->  ( ( N `
 ( A G ( _i S B ) ) ) ^
2 )  e.  RR )
1614, 15mpan2 675 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  ( A G ( _i S B ) ) ) ^ 2 )  e.  RR )
1716recnd 9668 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  ( A G ( _i S B ) ) ) ^ 2 )  e.  CC )
181, 2, 3, 4, 5, 6ipval2lem5 26191 . . . . . . . 8  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  _i  e.  CC )  ->  ( ( N `
 ( A M ( _i S B ) ) ) ^
2 )  e.  RR )
1914, 18mpan2 675 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  ( A M ( _i S B ) ) ) ^ 2 )  e.  RR )
2019recnd 9668 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  ( A M ( _i S B ) ) ) ^ 2 )  e.  CC )
2117, 20subcld 9985 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( N `  ( A G ( _i S B ) ) ) ^ 2 )  -  ( ( N `
 ( A M ( _i S B ) ) ) ^
2 ) )  e.  CC )
22 mulcl 9622 . . . . 5  |-  ( ( _i  e.  CC  /\  ( ( ( N `
 ( A G ( _i S B ) ) ) ^
2 )  -  (
( N `  ( A M ( _i S B ) ) ) ^ 2 ) )  e.  CC )  -> 
( _i  x.  (
( ( N `  ( A G ( _i S B ) ) ) ^ 2 )  -  ( ( N `
 ( A M ( _i S B ) ) ) ^
2 ) ) )  e.  CC )
2314, 21, 22sylancr 667 . . . 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 ) ) )  e.  CC )
2413, 23addcld 9661 . . 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 ) ) ) )  e.  CC )
25 4cn 10687 . . . 4  |-  4  e.  CC
26 4ne0 10706 . . . 4  |-  4  =/=  0
27 divcan2 10277 . . . 4  |-  ( ( ( ( ( ( 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 ) ) ) )  e.  CC  /\  4  e.  CC  /\  4  =/=  0 )  ->  (
4  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 M B ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `
 ( A G ( _i S B ) ) ) ^
2 )  -  (
( N `  ( A M ( _i S B ) ) ) ^ 2 ) ) ) ) )
2825, 26, 27mp3an23 1352 . . 3  |-  ( ( ( ( ( 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 ) ) ) )  e.  CC  ->  ( 4  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 M B ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `  ( A G ( _i S B ) ) ) ^ 2 )  -  ( ( N `  ( A M ( _i S B ) ) ) ^ 2 ) ) ) ) )
2924, 28syl 17 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
4  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 M B ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `
 ( A G ( _i S B ) ) ) ^
2 )  -  (
( N `  ( A M ( _i S B ) ) ) ^ 2 ) ) ) ) )
308, 29eqtrd 2470 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
4  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 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   _ici 9540    + caddc 9541    x. cmul 9543    - cmin 9859    / cdiv 10268   2c2 10659   4c4 10661   ^cexp 12269   NrmCVeccnv 26048   +vcpv 26049   BaseSetcba 26050   .sOLDcns 26051   -vcnsb 26053   normCVcnmcv 26054   .iOLDcdip 26181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-grpo 25764  df-gid 25765  df-ginv 25766  df-gdiv 25767  df-ablo 25855  df-vc 26010  df-nv 26056  df-va 26059  df-ba 26060  df-sm 26061  df-0v 26062  df-vs 26063  df-nmcv 26064  df-dip 26182
This theorem is referenced by: (None)
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