Theorem nvdif 24204

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

Step	Hyp	Ref	Expression
1		simp1 988	. . . . 5 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> U e. NrmCVec )
2		neg1cn 10535	. . . . . 6 |- -u 1 e. CC
3	2	a1i 11	. . . . 5 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> -u 1 e. CC )
4		simp3 990	. . . . 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 )
7	5, 6	nvscl 24157	. . . . . . 7 |- ( ( U e. NrmCVec /\ -u 1 e. CC /\ A e. X ) -> ( -u 1 S A ) e. X )
8	2, 7	mp3an2 1303	. . . . . 6 |- ( ( U e. NrmCVec /\ A e. X ) -> ( -u 1 S A ) e. X )
9	8	3adant3 1008	. . . . 5 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u 1 S A ) e. X )
10		nvdif.2	. . . . . 6 |- G = ( +v ` U )
11	5, 10, 6	nvdi 24161	. . . . 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 ) ) ) )
12	1, 3, 4, 9, 11	syl13anc 1221	. . . 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 ) ) ) )
13	5, 6	nvnegneg 24182	. . . . . 6 |- ( ( U e. NrmCVec /\ A e. X ) -> ( -u 1 S ( -u 1 S A ) ) = A )
14	13	3adant3 1008	. . . . 5 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u 1 S ( -u 1 S A ) ) = A )
15	14	oveq2d 6215	. . . 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 ) )
16	5, 6	nvscl 24157	. . . . . . 7 |- ( ( U e. NrmCVec /\ -u 1 e. CC /\ B e. X ) -> ( -u 1 S B ) e. X )
17	2, 16	mp3an2 1303	. . . . . 6 |- ( ( U e. NrmCVec /\ B e. X ) -> ( -u 1 S B ) e. X )
18	17	3adant2 1007	. . . . 5 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u 1 S B ) e. X )
19		simp2 989	. . . . 5 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> A e. X )
20	5, 10	nvcom 24150	. . . . 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 ) ) )
21	1, 18, 19, 20	syl3anc 1219	. . . 4 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( -u 1 S B ) G A ) = ( A G ( -u 1 S B ) ) )
22	12, 15, 21	3eqtrd 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 ) ) )
23	22	fveq2d 5802	. 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 ) ) ) )
24	5, 10	nvgcl 24149	. . . 4 |- ( ( U e. NrmCVec /\ B e. X /\ ( -u 1 S A ) e. X ) -> ( B G ( -u 1 S A ) ) e. X )
25	1, 4, 9, 24	syl3anc 1219	. . 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 )
27	5, 6, 26	nvm1 24203	. . 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 ) ) ) )
28	1, 25, 27	syl2anc 661	. 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 ) ) ) )
29	23, 28	eqtr3d 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 ) ) ) )

Syntax hints:   -> wi 4   /\ w3a 965   = wceq 1370   e. wcel 1758   ` cfv 5525  (class class class)co 6199   CCcc 9390   1c1 9393   -ucneg 9706   NrmCVeccnv 24113   +vcpv 24114   BaseSetcba 24115   .sOLDcns 24116   norm_CVcnmcv 24119

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 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470

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 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-sup 7801  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-n0 10690  df-z 10757  df-uz 10972  df-rp 11102  df-seq 11923  df-exp 11982  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-grpo 23829  df-gid 23830  df-ginv 23831  df-ablo 23920  df-vc 24075  df-nv 24121  df-va 24124  df-ba 24125  df-sm 24126  df-0v 24127  df-nmcv 24129

This theorem is referenced by:  nvsub  24206  nvabs  24212  imsmetlem  24232  dipcj  24263