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Theorem nvpi 24233
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  =  ( .sOLD `  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 988 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  U  e.  NrmCVec )
2 ax-icn 9456 . . . . . . . 8  |-  _i  e.  CC
3 nvdif.1 . . . . . . . . 9  |-  X  =  ( BaseSet `  U )
4 nvdif.4 . . . . . . . . 9  |-  S  =  ( .sOLD `  U )
53, 4nvscl 24185 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  _i  e.  CC  /\  B  e.  X )  ->  (
_i S B )  e.  X )
62, 5mp3an2 1303 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
_i S B )  e.  X )
763adant2 1007 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
_i S B )  e.  X )
8 nvdif.2 . . . . . . 7  |-  G  =  ( +v `  U
)
93, 8nvgcl 24177 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  (
_i S B )  e.  X )  -> 
( A G ( _i S B ) )  e.  X )
107, 9syld3an3 1264 . . . . 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 24226 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A G ( _i S B ) )  e.  X )  ->  ( N `  ( A G ( _i S B ) ) )  e.  RR )
131, 10, 12syl2anc 661 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( _i S B ) ) )  e.  RR )
1413recnd 9527 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( _i S B ) ) )  e.  CC )
1514mulid2d 9519 . 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 13009 . . . . 5  |-  ( abs `  -u _i )  =  ( abs `  _i )
17 absi 12897 . . . . 5  |-  ( abs `  _i )  =  1
1816, 17eqtri 2483 . . . 4  |-  ( abs `  -u _i )  =  1
1918oveq1i 6213 . . 3  |-  ( ( abs `  -u _i )  x.  ( N `  ( A G ( _i S B ) ) ) )  =  ( 1  x.  ( N `  ( A G ( _i S B ) ) ) )
20 negicn 9726 . . . . . 6  |-  -u _i  e.  CC
213, 4, 11nvs 24229 . . . . . 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 1303 . . . . 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 661 . . . 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 989 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
253, 8, 4nvdi 24189 . . . . . . . 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 1312 . . . . . . 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 1217 . . . . . 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 9905 . . . . . . . . . . 11  |-  ( -u _i  x.  _i )  = 
-u ( _i  x.  _i )
29 ixi 10080 . . . . . . . . . . . . 13  |-  ( _i  x.  _i )  = 
-u 1
3029negeqi 9718 . . . . . . . . . . . 12  |-  -u (
_i  x.  _i )  =  -u -u 1
31 negneg1e1 10544 . . . . . . . . . . . 12  |-  -u -u 1  =  1
3230, 31eqtri 2483 . . . . . . . . . . 11  |-  -u (
_i  x.  _i )  =  1
3328, 32eqtri 2483 . . . . . . . . . 10  |-  ( -u _i  x.  _i )  =  1
3433oveq1i 6213 . . . . . . . . 9  |-  ( (
-u _i  x.  _i ) S B )  =  ( 1 S B )
353, 4nvsass 24187 . . . . . . . . . . 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 ) ) )
3620, 35mp3anr1 1312 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  (
_i  e.  CC  /\  B  e.  X )
)  ->  ( ( -u _i  x.  _i ) S B )  =  ( -u _i S
( _i S B ) ) )
372, 36mpanr1 683 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( -u _i  x.  _i ) S B )  =  ( -u _i S
( _i S B ) ) )
383, 4nvsid 24186 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1 S B )  =  B )
3934, 37, 383eqtr3a 2519 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u _i S ( _i S B ) )  =  B )
40393adant2 1007 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u _i S ( _i S B ) )  =  B )
4140oveq2d 6219 . . . . . 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 ) )
423, 4nvscl 24185 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  -u _i  e.  CC  /\  A  e.  X )  ->  ( -u _i S A )  e.  X )
4320, 42mp3an2 1303 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u _i S A )  e.  X )
44433adant3 1008 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u _i S A )  e.  X )
453, 8nvcom 24178 . . . . . . 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 ) ) )
4644, 45syld3an2 1266 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( -u _i S A ) G B )  =  ( B G ( -u _i S A ) ) )
4727, 41, 463eqtrd 2499 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u _i S ( A G ( _i S B ) ) )  =  ( B G ( -u _i S A ) ) )
4847fveq2d 5806 . . . 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 ) ) ) )
4923, 48eqtr3d 2497 . . 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 ) ) ) )
5019, 49syl5eqr 2509 . 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 ) ) ) )
5115, 50eqtr3d 2497 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5529  (class class class)co 6203   CCcc 9395   RRcr 9396   1c1 9398   _ici 9399    x. cmul 9402   -ucneg 9711   abscabs 12845   NrmCVeccnv 24141   +vcpv 24142   BaseSetcba 24143   .sOLDcns 24144   normCVcnmcv 24147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7806  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-seq 11928  df-exp 11987  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-grpo 23857  df-ablo 23948  df-vc 24103  df-nv 24149  df-va 24152  df-ba 24153  df-sm 24154  df-0v 24155  df-nmcv 24157
This theorem is referenced by:  dipcj  24291
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