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Theorem nvdif 25766
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  =  ( .sOLD `  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 994 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  U  e.  NrmCVec )
2 neg1cn 10635 . . . . . 6  |-  -u 1  e.  CC
32a1i 11 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  -u 1  e.  CC )
4 simp3 996 . . . . 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  =  ( .sOLD `  U )
75, 6nvscl 25719 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X )
82, 7mp3an2 1310 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X )
983adant3 1014 . . . . 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 25723 . . . . 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 1228 . . . 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 25744 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 S ( -u
1 S A ) )  =  A )
14133adant3 1014 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 S ( -u
1 S A ) )  =  A )
1514oveq2d 6286 . . . 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 25719 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 S B )  e.  X )
172, 16mp3an2 1310 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 S B )  e.  X )
18173adant2 1013 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 S B )  e.  X )
19 simp2 995 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
205, 10nvcom 25712 . . . . 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 1226 . . . 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 2499 . . 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 5852 . 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 25711 . . . 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 1226 . . 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 25765 . . 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 659 . 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 2497 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 setvar class
Syntax hints:    -> wi 4    /\ w3a 971    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   CCcc 9479   1c1 9482   -ucneg 9797   NrmCVeccnv 25675   +vcpv 25676   BaseSetcba 25677   .sOLDcns 25678   normCVcnmcv 25681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-grpo 25391  df-gid 25392  df-ginv 25393  df-ablo 25482  df-vc 25637  df-nv 25683  df-va 25686  df-ba 25687  df-sm 25688  df-0v 25689  df-nmcv 25691
This theorem is referenced by:  nvsub  25768  nvabs  25774  imsmetlem  25794  dipcj  25825
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