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

Theorem nvdif 22146
Description: The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (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
nvdif  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( -u 1 S B ) ) )  =  ( N `  ( B G ( -u
1 S A ) ) ) )

Proof of Theorem nvdif
StepHypRef Expression
1 simp1 957 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  U  e.  NrmCVec )
2 neg1cn 10059 . . . . . 6  |-  -u 1  e.  CC
32a1i 11 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  -u 1  e.  CC )
4 simp3 959 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
5 nvdif.1 . . . . . . . 8  |-  X  =  ( BaseSet `  U )
6 nvdif.4 . . . . . . . 8  |-  S  =  ( .s OLD `  U
)
75, 6nvscl 22099 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X )
82, 7mp3an2 1267 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X )
983adant3 977 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 S A )  e.  X )
10 nvdif.2 . . . . . 6  |-  G  =  ( +v `  U
)
115, 10, 6nvdi 22103 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  B  e.  X  /\  ( -u 1 S A )  e.  X ) )  ->  ( -u 1 S ( B G ( -u 1 S A ) ) )  =  ( ( -u
1 S B ) G ( -u 1 S ( -u 1 S A ) ) ) )
121, 3, 4, 9, 11syl13anc 1186 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 S ( B G ( -u 1 S A ) ) )  =  ( ( -u
1 S B ) G ( -u 1 S ( -u 1 S A ) ) ) )
135, 6nvnegneg 22124 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 S ( -u
1 S A ) )  =  A )
14133adant3 977 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 S ( -u
1 S A ) )  =  A )
1514oveq2d 6089 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( -u 1 S B ) G ( -u
1 S ( -u
1 S A ) ) )  =  ( ( -u 1 S B ) G A ) )
165, 6nvscl 22099 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 S B )  e.  X )
172, 16mp3an2 1267 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 S B )  e.  X )
18173adant2 976 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 S B )  e.  X )
19 simp2 958 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
205, 10nvcom 22092 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( -u 1 S B )  e.  X  /\  A  e.  X )  ->  (
( -u 1 S B ) G A )  =  ( A G ( -u 1 S B ) ) )
211, 18, 19, 20syl3anc 1184 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( -u 1 S B ) G A )  =  ( A G ( -u 1 S B ) ) )
2212, 15, 213eqtrd 2471 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 S ( B G ( -u 1 S A ) ) )  =  ( A G ( -u 1 S B ) ) )
2322fveq2d 5724 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( -u 1 S ( B G ( -u 1 S A ) ) ) )  =  ( N `
 ( A G ( -u 1 S B ) ) ) )
245, 10nvgcl 22091 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  ( -u 1 S A )  e.  X )  -> 
( B G (
-u 1 S A ) )  e.  X
)
251, 4, 9, 24syl3anc 1184 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( B G ( -u 1 S A ) )  e.  X )
26 nvdif.6 . . . 4  |-  N  =  ( normCV `  U )
275, 6, 26nvm1 22145 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( B G ( -u 1 S A ) )  e.  X )  ->  ( N `  ( -u 1 S ( B G ( -u 1 S A ) ) ) )  =  ( N `
 ( B G ( -u 1 S A ) ) ) )
281, 25, 27syl2anc 643 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( -u 1 S ( B G ( -u 1 S A ) ) ) )  =  ( N `
 ( B G ( -u 1 S A ) ) ) )
2923, 28eqtr3d 2469 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( -u 1 S B ) ) )  =  ( N `  ( B G ( -u
1 S A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980   1c1 8983   -ucneg 9284   NrmCVeccnv 22055   +vcpv 22056   BaseSetcba 22057   .s
OLDcns 22058   normCVcnmcv 22061
This theorem is referenced by:  nvsub  22148  nvabs  22154  imsmetlem  22174  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-gid 21772  df-ginv 21773  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071
  Copyright terms: Public domain W3C validator