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Theorem ipval3 21390
Description: Expansion of the inner product value ipval 21384. (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  =  ( .s OLD `  U
)
dipfval.6  |-  N  =  ( normCV `  U )
dipfval.7  |-  P  =  ( .i OLD `  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  =  ( .s OLD `  U
)
4 dipfval.6 . . 3  |-  N  =  ( normCV `  U )
5 dipfval.7 . . 3  |-  P  =  ( .i OLD `  U
)
61, 2, 3, 4, 5ipval2 21388 . 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 21308 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A G (
-u 1 S B ) ) )
98fveq2d 5609 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M B ) )  =  ( N `  ( A G ( -u 1 S B ) ) ) )
109oveq1d 5957 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  ( A M B ) ) ^ 2 )  =  ( ( N `  ( A G ( -u
1 S B ) ) ) ^ 2 ) )
1110oveq2d 5958 . . . 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 8883 . . . . . . . . . . . 12  |-  _i  e.  CC
131, 3nvscl 21292 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  _i  e.  CC  /\  B  e.  X )  ->  (
_i S B )  e.  X )
1412, 13mp3an2 1265 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
_i S B )  e.  X )
15143adant2 974 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
_i S B )  e.  X )
161, 2, 3, 7nvmval 21308 . . . . . . . . . 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 1227 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( _i S B ) )  =  ( A G (
-u 1 S ( _i S B ) ) ) )
1812mulm1i 9311 . . . . . . . . . . . . 13  |-  ( -u
1  x.  _i )  =  -u _i
1918oveq1i 5952 . . . . . . . . . . . 12  |-  ( (
-u 1  x.  _i ) S B )  =  ( -u _i S B )
20 neg1cn 9900 . . . . . . . . . . . . . 14  |-  -u 1  e.  CC
211, 3nvsass 21294 . . . . . . . . . . . . . 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 1274 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  (
_i  e.  CC  /\  B  e.  X )
)  ->  ( ( -u 1  x.  _i ) S B )  =  ( -u 1 S ( _i S B ) ) )
2312, 22mpanr1 664 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( -u 1  x.  _i ) S B )  =  ( -u 1 S ( _i S B ) ) )
2419, 23syl5reqr 2405 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 S ( _i S B ) )  =  ( -u _i S B ) )
25243adant2 974 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 S ( _i S B ) )  =  ( -u _i S B ) )
2625oveq2d 5958 . . . . . . . . 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 2390 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( _i S B ) )  =  ( A G (
-u _i S B ) ) )
2827fveq2d 5609 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M ( _i S B ) ) )  =  ( N `  ( A G ( -u _i S B ) ) ) )
2928oveq1d 5957 . . . . . 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 5958 . . . . 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 5958 . . . 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 5960 . . 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 5957 . 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 2393 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   ` cfv 5334  (class class class)co 5942   CCcc 8822   1c1 8825   _ici 8826    + caddc 8827    x. cmul 8829    - cmin 9124   -ucneg 9125    / cdiv 9510   2c2 9882   4c4 9884   ^cexp 11194   NrmCVeccnv 21248   +vcpv 21249   BaseSetcba 21250   .s
OLDcns 21251   -vcnsb 21253   normCVcnmcv 21254   .i OLDcdip 21381
This theorem is referenced by:  4ipval3  21393  hhip  21864
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-oi 7312  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-fz 10872  df-fzo 10960  df-seq 11136  df-exp 11195  df-hash 11428  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-clim 12052  df-sum 12250  df-grpo 20964  df-gid 20965  df-ginv 20966  df-gdiv 20967  df-ablo 21055  df-vc 21210  df-nv 21256  df-va 21259  df-ba 21260  df-sm 21261  df-0v 21262  df-vs 21263  df-nmcv 21264  df-dip 21382
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