MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvpi Structured version   Unicode version

Theorem nvpi 25434
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 995 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  U  e.  NrmCVec )
2 ax-icn 9549 . . . . . . . 8  |-  _i  e.  CC
3 nvdif.1 . . . . . . . . 9  |-  X  =  ( BaseSet `  U )
4 nvdif.4 . . . . . . . . 9  |-  S  =  ( .sOLD `  U )
53, 4nvscl 25386 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  _i  e.  CC  /\  B  e.  X )  ->  (
_i S B )  e.  X )
62, 5mp3an2 1311 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
_i S B )  e.  X )
763adant2 1014 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
_i S B )  e.  X )
8 nvdif.2 . . . . . . 7  |-  G  =  ( +v `  U
)
93, 8nvgcl 25378 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  (
_i S B )  e.  X )  -> 
( A G ( _i S B ) )  e.  X )
107, 9syld3an3 1272 . . . . 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 25427 . . . . 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 9620 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( _i S B ) ) )  e.  CC )
1514mulid2d 9612 . 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 13206 . . . . 5  |-  ( abs `  -u _i )  =  ( abs `  _i )
17 absi 13093 . . . . 5  |-  ( abs `  _i )  =  1
1816, 17eqtri 2470 . . . 4  |-  ( abs `  -u _i )  =  1
1918oveq1i 6287 . . 3  |-  ( ( abs `  -u _i )  x.  ( N `  ( A G ( _i S B ) ) ) )  =  ( 1  x.  ( N `  ( A G ( _i S B ) ) ) )
20 negicn 9821 . . . . . 6  |-  -u _i  e.  CC
213, 4, 11nvs 25430 . . . . . 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 1311 . . . . 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 996 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
253, 8, 4nvdi 25390 . . . . . . . 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 1320 . . . . . . 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 1225 . . . . . 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 10003 . . . . . . . . . . 11  |-  ( -u _i  x.  _i )  = 
-u ( _i  x.  _i )
29 ixi 10179 . . . . . . . . . . . . 13  |-  ( _i  x.  _i )  = 
-u 1
3029negeqi 9813 . . . . . . . . . . . 12  |-  -u (
_i  x.  _i )  =  -u -u 1
31 negneg1e1 10644 . . . . . . . . . . . 12  |-  -u -u 1  =  1
3230, 31eqtri 2470 . . . . . . . . . . 11  |-  -u (
_i  x.  _i )  =  1
3328, 32eqtri 2470 . . . . . . . . . 10  |-  ( -u _i  x.  _i )  =  1
3433oveq1i 6287 . . . . . . . . 9  |-  ( (
-u _i  x.  _i ) S B )  =  ( 1 S B )
353, 4nvsass 25388 . . . . . . . . . . 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 1320 . . . . . . . . . 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 25387 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1 S B )  =  B )
3934, 37, 383eqtr3a 2506 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u _i S ( _i S B ) )  =  B )
40393adant2 1014 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u _i S ( _i S B ) )  =  B )
4140oveq2d 6293 . . . . . 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 25386 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  -u _i  e.  CC  /\  A  e.  X )  ->  ( -u _i S A )  e.  X )
4320, 42mp3an2 1311 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u _i S A )  e.  X )
44433adant3 1015 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u _i S A )  e.  X )
453, 8nvcom 25379 . . . . . . 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 1274 . . . . . 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 2486 . . . . 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 5856 . . . 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 2484 . . 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 2496 . 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 2484 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 972    = wceq 1381    e. wcel 1802   ` cfv 5574  (class class class)co 6277   CCcc 9488   RRcr 9489   1c1 9491   _ici 9492    x. cmul 9495   -ucneg 9806   abscabs 13041   NrmCVeccnv 25342   +vcpv 25343   BaseSetcba 25344   .sOLDcns 25345   normCVcnmcv 25348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-seq 12082  df-exp 12141  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-grpo 25058  df-ablo 25149  df-vc 25304  df-nv 25350  df-va 25353  df-ba 25354  df-sm 25355  df-0v 25356  df-nmcv 25358
This theorem is referenced by:  dipcj  25492
  Copyright terms: Public domain W3C validator