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Theorem ipval3 25484
Description: Expansion of the inner product value ipval 25478. (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 25482 . 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 25402 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A G (
-u 1 S B ) ) )
98fveq2d 5856 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M B ) )  =  ( N `  ( A G ( -u 1 S B ) ) ) )
109oveq1d 6292 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  ( A M B ) ) ^ 2 )  =  ( ( N `  ( A G ( -u
1 S B ) ) ) ^ 2 ) )
1110oveq2d 6293 . . . 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 9549 . . . . . . . . . . . 12  |-  _i  e.  CC
131, 3nvscl 25386 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  _i  e.  CC  /\  B  e.  X )  ->  (
_i S B )  e.  X )
1412, 13mp3an2 1311 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
_i S B )  e.  X )
15143adant2 1014 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
_i S B )  e.  X )
161, 2, 3, 7nvmval 25402 . . . . . . . . . 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 1272 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( _i S B ) )  =  ( A G (
-u 1 S ( _i S B ) ) ) )
1812mulm1i 10002 . . . . . . . . . . . . 13  |-  ( -u
1  x.  _i )  =  -u _i
1918oveq1i 6287 . . . . . . . . . . . 12  |-  ( (
-u 1  x.  _i ) S B )  =  ( -u _i S B )
20 neg1cn 10640 . . . . . . . . . . . . . 14  |-  -u 1  e.  CC
211, 3nvsass 25388 . . . . . . . . . . . . . 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 1320 . . . . . . . . . . . . 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 2497 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 S ( _i S B ) )  =  ( -u _i S B ) )
25243adant2 1014 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 S ( _i S B ) )  =  ( -u _i S B ) )
2625oveq2d 6293 . . . . . . . . 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 2482 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( _i S B ) )  =  ( A G (
-u _i S B ) ) )
2827fveq2d 5856 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M ( _i S B ) ) )  =  ( N `  ( A G ( -u _i S B ) ) ) )
2928oveq1d 6292 . . . . . 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 6293 . . . . 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 6293 . . . 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 6295 . . 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 6292 . 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 2485 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 972    = wceq 1381    e. wcel 1802   ` cfv 5574  (class class class)co 6277   CCcc 9488   1c1 9491   _ici 9492    + caddc 9493    x. cmul 9495    - cmin 9805   -ucneg 9806    / cdiv 10207   2c2 10586   4c4 10588   ^cexp 12140   NrmCVeccnv 25342   +vcpv 25343   BaseSetcba 25344   .sOLDcns 25345   -vcnsb 25347   normCVcnmcv 25348   .iOLDcdip 25475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-fz 11677  df-fzo 11799  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-sum 13483  df-grpo 25058  df-gid 25059  df-ginv 25060  df-gdiv 25061  df-ablo 25149  df-vc 25304  df-nv 25350  df-va 25353  df-ba 25354  df-sm 25355  df-0v 25356  df-vs 25357  df-nmcv 25358  df-dip 25476
This theorem is referenced by:  4ipval3  25487  hhip  25959
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