Theorem nvdif 27521

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 1061	. . . . 5 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> U e. NrmCVec )
2		neg1cn 11124	. . . . . 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 1063	. . . . 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 27481	. . . . . . 7 |- ( ( U e. NrmCVec /\ -u 1 e. CC /\ A e. X ) -> ( -u 1 S A ) e. X )
8	2, 7	mp3an2 1412	. . . . . 6 |- ( ( U e. NrmCVec /\ A e. X ) -> ( -u 1 S A ) e. X )
9	8	3adant3 1081	. . . . 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 27485	. . . . 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 1328	. . . 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 27504	. . . . . 6 |- ( ( U e. NrmCVec /\ A e. X ) -> ( -u 1 S ( -u 1 S A ) ) = A )
14	13	3adant3 1081	. . . . 5 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u 1 S ( -u 1 S A ) ) = A )
15	14	oveq2d 6666	. . . 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 27481	. . . . . . 7 |- ( ( U e. NrmCVec /\ -u 1 e. CC /\ B e. X ) -> ( -u 1 S B ) e. X )
17	2, 16	mp3an2 1412	. . . . . 6 |- ( ( U e. NrmCVec /\ B e. X ) -> ( -u 1 S B ) e. X )
18	17	3adant2 1080	. . . . 5 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u 1 S B ) e. X )
19		simp2 1062	. . . . 5 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> A e. X )
20	5, 10	nvcom 27476	. . . . 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 1326	. . . 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 2660	. . 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 6195	. 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 27475	. . . 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 1326	. . 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 27520	. . 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 693	. 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 2658	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 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   -ucneg 10267   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442   norm_CVcnmcv 27445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455

This theorem is referenced by:  nvabs  27527  imsmetlem  27545  dipcj  27569