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Theorem ipval3 25145
Description: Expansion of the inner product value ipval 25139. (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 25143 . 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 25063 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A G (
-u 1 S B ) ) )
98fveq2d 5861 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M B ) )  =  ( N `  ( A G ( -u 1 S B ) ) ) )
109oveq1d 6290 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  ( A M B ) ) ^ 2 )  =  ( ( N `  ( A G ( -u
1 S B ) ) ) ^ 2 ) )
1110oveq2d 6291 . . . 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 9540 . . . . . . . . . . . 12  |-  _i  e.  CC
131, 3nvscl 25047 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  _i  e.  CC  /\  B  e.  X )  ->  (
_i S B )  e.  X )
1412, 13mp3an2 1307 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
_i S B )  e.  X )
15143adant2 1010 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
_i S B )  e.  X )
161, 2, 3, 7nvmval 25063 . . . . . . . . . 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 1268 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( _i S B ) )  =  ( A G (
-u 1 S ( _i S B ) ) ) )
1812mulm1i 9990 . . . . . . . . . . . . 13  |-  ( -u
1  x.  _i )  =  -u _i
1918oveq1i 6285 . . . . . . . . . . . 12  |-  ( (
-u 1  x.  _i ) S B )  =  ( -u _i S B )
20 neg1cn 10628 . . . . . . . . . . . . . 14  |-  -u 1  e.  CC
211, 3nvsass 25049 . . . . . . . . . . . . . 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 1316 . . . . . . . . . . . . 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 2516 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 S ( _i S B ) )  =  ( -u _i S B ) )
25243adant2 1010 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 S ( _i S B ) )  =  ( -u _i S B ) )
2625oveq2d 6291 . . . . . . . . 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 2501 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( _i S B ) )  =  ( A G (
-u _i S B ) ) )
2827fveq2d 5861 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M ( _i S B ) ) )  =  ( N `  ( A G ( -u _i S B ) ) ) )
2928oveq1d 6290 . . . . . 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 6291 . . . . 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 6291 . . . 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 6293 . . 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 6290 . 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 2504 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 968    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   CCcc 9479   1c1 9482   _ici 9483    + caddc 9484    x. cmul 9486    - cmin 9794   -ucneg 9795    / cdiv 10195   2c2 10574   4c4 10576   ^cexp 12122   NrmCVeccnv 25003   +vcpv 25004   BaseSetcba 25005   .sOLDcns 25006   -vcnsb 25008   normCVcnmcv 25009   .iOLDcdip 25136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-grpo 24719  df-gid 24720  df-ginv 24721  df-gdiv 24722  df-ablo 24810  df-vc 24965  df-nv 25011  df-va 25014  df-ba 25015  df-sm 25016  df-0v 25017  df-vs 25018  df-nmcv 25019  df-dip 25137
This theorem is referenced by:  4ipval3  25148  hhip  25620
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