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Theorem nvpi 22147
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 9041 . . . . . . . 8  |-  _i  e.  CC
3 nvdif.1 . . . . . . . . 9  |-  X  =  ( BaseSet `  U )
4 nvdif.4 . . . . . . . . 9  |-  S  =  ( .s OLD `  U
)
53, 4nvscl 22099 . . . . . . . 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 22091 . . . . . 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 22140 . . . . 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 9106 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( _i S B ) ) )  e.  CC )
1514mulid2d 9098 . 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 12195 . . . . 5  |-  ( abs `  -u _i )  =  ( abs `  _i )
17 absi 12083 . . . . 5  |-  ( abs `  _i )  =  1
1816, 17eqtri 2455 . . . 4  |-  ( abs `  -u _i )  =  1
1918oveq1i 6083 . . 3  |-  ( ( abs `  -u _i )  x.  ( N `  ( A G ( _i S B ) ) ) )  =  ( 1  x.  ( N `  ( A G ( _i S B ) ) ) )
202negcli 9360 . . . . . 6  |-  -u _i  e.  CC
213, 4, 11nvs 22143 . . . . . 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 22103 . . . . . . . 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 9471 . . . . . . . . . . 11  |-  ( -u _i  x.  _i )  = 
-u ( _i  x.  _i )
29 ixi 9643 . . . . . . . . . . . . 13  |-  ( _i  x.  _i )  = 
-u 1
3029negeqi 9291 . . . . . . . . . . . 12  |-  -u (
_i  x.  _i )  =  -u -u 1
31 ax-1cn 9040 . . . . . . . . . . . . 13  |-  1  e.  CC
3231negnegi 9362 . . . . . . . . . . . 12  |-  -u -u 1  =  1
3330, 32eqtri 2455 . . . . . . . . . . 11  |-  -u (
_i  x.  _i )  =  1
3428, 33eqtri 2455 . . . . . . . . . 10  |-  ( -u _i  x.  _i )  =  1
3534oveq1i 6083 . . . . . . . . 9  |-  ( (
-u _i  x.  _i ) S B )  =  ( 1 S B )
363, 4nvsass 22101 . . . . . . . . . . 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 22100 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1 S B )  =  B )
4035, 38, 393eqtr3a 2491 . . . . . . . 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 6089 . . . . . 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 22099 . . . . . . . . 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 22092 . . . . . . 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 2471 . . . . 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 5724 . . . 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 2469 . . 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 2481 . 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 2469 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 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   1c1 8983   _ici 8984    x. cmul 8987   -ucneg 9284   abscabs 12031   NrmCVeccnv 22055   +vcpv 22056   BaseSetcba 22057   .s
OLDcns 22058   normCVcnmcv 22061
This theorem is referenced by:  dipcj  22205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-grpo 21771  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071
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