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Theorem nvpi 21232
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 955 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  U  e.  NrmCVec )
2 ax-icn 8796 . . . . . . . 8  |-  _i  e.  CC
3 nvdif.1 . . . . . . . . 9  |-  X  =  ( BaseSet `  U )
4 nvdif.4 . . . . . . . . 9  |-  S  =  ( .s OLD `  U
)
53, 4nvscl 21184 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  _i  e.  CC  /\  B  e.  X )  ->  (
_i S B )  e.  X )
62, 5mp3an2 1265 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
_i S B )  e.  X )
763adant2 974 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
_i S B )  e.  X )
8 nvdif.2 . . . . . . 7  |-  G  =  ( +v `  U
)
93, 8nvgcl 21176 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  (
_i S B )  e.  X )  -> 
( A G ( _i S B ) )  e.  X )
107, 9syld3an3 1227 . . . . 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 21225 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A G ( _i S B ) )  e.  X )  ->  ( N `  ( A G ( _i S B ) ) )  e.  RR )
131, 10, 12syl2anc 642 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( _i S B ) ) )  e.  RR )
1413recnd 8861 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( _i S B ) ) )  e.  CC )
1514mulid2d 8853 . 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 11883 . . . . 5  |-  ( abs `  -u _i )  =  ( abs `  _i )
17 absi 11771 . . . . 5  |-  ( abs `  _i )  =  1
1816, 17eqtri 2303 . . . 4  |-  ( abs `  -u _i )  =  1
1918oveq1i 5868 . . 3  |-  ( ( abs `  -u _i )  x.  ( N `  ( A G ( _i S B ) ) ) )  =  ( 1  x.  ( N `  ( A G ( _i S B ) ) ) )
202negcli 9114 . . . . . 6  |-  -u _i  e.  CC
213, 4, 11nvs 21228 . . . . . 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 1265 . . . . 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 642 . . . 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 956 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
253, 8, 4nvdi 21188 . . . . . . . 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 1274 . . . . . . 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 1180 . . . . . 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 9225 . . . . . . . . . . 11  |-  ( -u _i  x.  _i )  = 
-u ( _i  x.  _i )
29 ixi 9397 . . . . . . . . . . . . 13  |-  ( _i  x.  _i )  = 
-u 1
3029negeqi 9045 . . . . . . . . . . . 12  |-  -u (
_i  x.  _i )  =  -u -u 1
31 ax-1cn 8795 . . . . . . . . . . . . 13  |-  1  e.  CC
3231negnegi 9116 . . . . . . . . . . . 12  |-  -u -u 1  =  1
3330, 32eqtri 2303 . . . . . . . . . . 11  |-  -u (
_i  x.  _i )  =  1
3428, 33eqtri 2303 . . . . . . . . . 10  |-  ( -u _i  x.  _i )  =  1
3534oveq1i 5868 . . . . . . . . 9  |-  ( (
-u _i  x.  _i ) S B )  =  ( 1 S B )
363, 4nvsass 21186 . . . . . . . . . . 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 1274 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  (
_i  e.  CC  /\  B  e.  X )
)  ->  ( ( -u _i  x.  _i ) S B )  =  ( -u _i S
( _i S B ) ) )
382, 37mpanr1 664 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( -u _i  x.  _i ) S B )  =  ( -u _i S
( _i S B ) ) )
393, 4nvsid 21185 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1 S B )  =  B )
4035, 38, 393eqtr3a 2339 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u _i S ( _i S B ) )  =  B )
41403adant2 974 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u _i S ( _i S B ) )  =  B )
4241oveq2d 5874 . . . . . 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 21184 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  -u _i  e.  CC  /\  A  e.  X )  ->  ( -u _i S A )  e.  X )
4420, 43mp3an2 1265 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u _i S A )  e.  X )
45443adant3 975 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u _i S A )  e.  X )
463, 8nvcom 21177 . . . . . . 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 1229 . . . . . 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 2319 . . . . 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 5529 . . . 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 2317 . . 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 2329 . 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 2317 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   1c1 8738   _ici 8739    x. cmul 8742   -ucneg 9038   abscabs 11719   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   normCVcnmcv 21146
This theorem is referenced by:  dipcj  21290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-grpo 20858  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156
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