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Theorem nvpi 22003
Description: The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvdif.1  |-  X  =  ( BaseSet `  U )
nvdif.2  |-  G  =  ( +v `  U
)
nvdif.4  |-  S  =  ( .s OLD `  U
)
nvdif.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvpi  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( _i S B ) ) )  =  ( N `  ( B G ( -u _i S A ) ) ) )

Proof of Theorem nvpi
StepHypRef Expression
1 simp1 957 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  U  e.  NrmCVec )
2 ax-icn 8982 . . . . . . . 8  |-  _i  e.  CC
3 nvdif.1 . . . . . . . . 9  |-  X  =  ( BaseSet `  U )
4 nvdif.4 . . . . . . . . 9  |-  S  =  ( .s OLD `  U
)
53, 4nvscl 21955 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  _i  e.  CC  /\  B  e.  X )  ->  (
_i S B )  e.  X )
62, 5mp3an2 1267 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
_i S B )  e.  X )
763adant2 976 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
_i S B )  e.  X )
8 nvdif.2 . . . . . . 7  |-  G  =  ( +v `  U
)
93, 8nvgcl 21947 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  (
_i S B )  e.  X )  -> 
( A G ( _i S B ) )  e.  X )
107, 9syld3an3 1229 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( _i S B ) )  e.  X )
11 nvdif.6 . . . . . 6  |-  N  =  ( normCV `  U )
123, 11nvcl 21996 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A G ( _i S B ) )  e.  X )  ->  ( N `  ( A G ( _i S B ) ) )  e.  RR )
131, 10, 12syl2anc 643 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( _i S B ) ) )  e.  RR )
1413recnd 9047 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( _i S B ) ) )  e.  CC )
1514mulid2d 9039 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
1  x.  ( N `
 ( A G ( _i S B ) ) ) )  =  ( N `  ( A G ( _i S B ) ) ) )
162absnegi 12130 . . . . 5  |-  ( abs `  -u _i )  =  ( abs `  _i )
17 absi 12018 . . . . 5  |-  ( abs `  _i )  =  1
1816, 17eqtri 2407 . . . 4  |-  ( abs `  -u _i )  =  1
1918oveq1i 6030 . . 3  |-  ( ( abs `  -u _i )  x.  ( N `  ( A G ( _i S B ) ) ) )  =  ( 1  x.  ( N `  ( A G ( _i S B ) ) ) )
202negcli 9300 . . . . . 6  |-  -u _i  e.  CC
213, 4, 11nvs 21999 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  -u _i  e.  CC  /\  ( A G ( _i S B ) )  e.  X )  ->  ( N `  ( -u _i S ( A G ( _i S B ) ) ) )  =  ( ( abs `  -u _i )  x.  ( N `  ( A G ( _i S B ) ) ) ) )
2220, 21mp3an2 1267 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A G ( _i S B ) )  e.  X )  ->  ( N `  ( -u _i S ( A G ( _i S B ) ) ) )  =  ( ( abs `  -u _i )  x.  ( N `  ( A G ( _i S B ) ) ) ) )
231, 10, 22syl2anc 643 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( -u _i S ( A G ( _i S B ) ) ) )  =  ( ( abs `  -u _i )  x.  ( N `  ( A G ( _i S B ) ) ) ) )
24 simp2 958 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
253, 8, 4nvdi 21959 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( -u _i  e.  CC  /\  A  e.  X  /\  ( _i S B )  e.  X ) )  ->  ( -u _i S ( A G ( _i S B ) ) )  =  ( ( -u _i S A ) G (
-u _i S ( _i S B ) ) ) )
2620, 25mp3anr1 1276 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  ( _i S B )  e.  X ) )  ->  ( -u _i S ( A G ( _i S B ) ) )  =  ( ( -u _i S A ) G (
-u _i S ( _i S B ) ) ) )
271, 24, 7, 26syl12anc 1182 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u _i S ( A G ( _i S B ) ) )  =  ( ( -u _i S A ) G ( -u _i S
( _i S B ) ) ) )
282, 2mulneg1i 9411 . . . . . . . . . . 11  |-  ( -u _i  x.  _i )  = 
-u ( _i  x.  _i )
29 ixi 9583 . . . . . . . . . . . . 13  |-  ( _i  x.  _i )  = 
-u 1
3029negeqi 9231 . . . . . . . . . . . 12  |-  -u (
_i  x.  _i )  =  -u -u 1
31 ax-1cn 8981 . . . . . . . . . . . . 13  |-  1  e.  CC
3231negnegi 9302 . . . . . . . . . . . 12  |-  -u -u 1  =  1
3330, 32eqtri 2407 . . . . . . . . . . 11  |-  -u (
_i  x.  _i )  =  1
3428, 33eqtri 2407 . . . . . . . . . 10  |-  ( -u _i  x.  _i )  =  1
3534oveq1i 6030 . . . . . . . . 9  |-  ( (
-u _i  x.  _i ) S B )  =  ( 1 S B )
363, 4nvsass 21957 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  ( -u _i  e.  CC  /\  _i  e.  CC  /\  B  e.  X ) )  -> 
( ( -u _i  x.  _i ) S B )  =  ( -u _i S ( _i S B ) ) )
3720, 36mp3anr1 1276 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  (
_i  e.  CC  /\  B  e.  X )
)  ->  ( ( -u _i  x.  _i ) S B )  =  ( -u _i S
( _i S B ) ) )
382, 37mpanr1 665 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( -u _i  x.  _i ) S B )  =  ( -u _i S
( _i S B ) ) )
393, 4nvsid 21956 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1 S B )  =  B )
4035, 38, 393eqtr3a 2443 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u _i S ( _i S B ) )  =  B )
41403adant2 976 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u _i S ( _i S B ) )  =  B )
4241oveq2d 6036 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( -u _i S A ) G ( -u _i S ( _i S B ) ) )  =  ( ( -u _i S A ) G B ) )
433, 4nvscl 21955 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  -u _i  e.  CC  /\  A  e.  X )  ->  ( -u _i S A )  e.  X )
4420, 43mp3an2 1267 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u _i S A )  e.  X )
45443adant3 977 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u _i S A )  e.  X )
463, 8nvcom 21948 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  ( -u _i S A )  e.  X  /\  B  e.  X )  ->  (
( -u _i S A ) G B )  =  ( B G ( -u _i S A ) ) )
4745, 46syld3an2 1231 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( -u _i S A ) G B )  =  ( B G ( -u _i S A ) ) )
4827, 42, 473eqtrd 2423 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u _i S ( A G ( _i S B ) ) )  =  ( B G ( -u _i S A ) ) )
4948fveq2d 5672 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( -u _i S ( A G ( _i S B ) ) ) )  =  ( N `  ( B G ( -u _i S A ) ) ) )
5023, 49eqtr3d 2421 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( abs `  -u _i )  x.  ( N `  ( A G ( _i S B ) ) ) )  =  ( N `  ( B G ( -u _i S A ) ) ) )
5119, 50syl5eqr 2433 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
1  x.  ( N `
 ( A G ( _i S B ) ) ) )  =  ( N `  ( B G ( -u _i S A ) ) ) )
5215, 51eqtr3d 2421 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( _i S B ) ) )  =  ( N `  ( B G ( -u _i S A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   1c1 8924   _ici 8925    x. cmul 8928   -ucneg 9224   abscabs 11966   NrmCVeccnv 21911   +vcpv 21912   BaseSetcba 21913   .s
OLDcns 21914   normCVcnmcv 21917
This theorem is referenced by:  dipcj  22061
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-grpo 21627  df-ablo 21718  df-vc 21873  df-nv 21919  df-va 21922  df-ba 21923  df-sm 21924  df-0v 21925  df-nmcv 21927
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