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Theorem nvdif 22003
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 10000 . . . . . 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 21956 . . . . . . 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 21960 . . . . 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 21981 . . . . . 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 6037 . . . 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 21956 . . . . . . 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 21949 . . . . 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 2424 . . 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 5673 . 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 21948 . . . 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 22002 . . 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 2422 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 1649    e. wcel 1717   ` cfv 5395  (class class class)co 6021   CCcc 8922   1c1 8925   -ucneg 9225   NrmCVeccnv 21912   +vcpv 21913   BaseSetcba 21914   .s
OLDcns 21915   normCVcnmcv 21918
This theorem is referenced by:  nvsub  22005  nvabs  22011  imsmetlem  22031  dipcj  22062
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-grpo 21628  df-gid 21629  df-ginv 21630  df-ablo 21719  df-vc 21874  df-nv 21920  df-va 21923  df-ba 21924  df-sm 21925  df-0v 21926  df-nmcv 21928
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