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Metamath Proof Explorer

Theorem nvdif

Description: The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006) (New usage is discouraged.)

Ref Expression
Hypotheses nvdif.1 
|- X = ( BaseSet ` U )
nvdif.2 
|- G = ( +v ` U )
nvdif.4 
|- S = ( .sOLD ` U )
nvdif.6 
|- N = ( normCV ` U )
Assertion 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

Step Hyp Ref Expression
1 nvdif.1 
 |-  X = ( BaseSet ` U )
2 nvdif.2 
 |-  G = ( +v ` U )
3 nvdif.4 
 |-  S = ( .sOLD ` U )
4 nvdif.6 
 |-  N = ( normCV ` U )
5 simp1 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> U e. NrmCVec )
6 neg1cn 
 |-  -u 1 e. CC
7 6 a1i 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> -u 1 e. CC )
8 simp3 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> B e. X )
9 1 3 nvscl 
 |-  ( ( U e. NrmCVec /\ -u 1 e. CC /\ A e. X ) -> ( -u 1 S A ) e. X )
10 6 9 mp3an2 
 |-  ( ( U e. NrmCVec /\ A e. X ) -> ( -u 1 S A ) e. X )
11 10 3adant3 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u 1 S A ) e. X )
12 1 2 3 nvdi 
 |-  ( ( 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 ) ) ) )
13 5 7 8 11 12 syl13anc 
 |-  ( ( 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 ) ) ) )
14 1 3 nvnegneg 
 |-  ( ( U e. NrmCVec /\ A e. X ) -> ( -u 1 S ( -u 1 S A ) ) = A )
15 14 3adant3 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u 1 S ( -u 1 S A ) ) = A )
16 15 oveq2d 
 |-  ( ( 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 ) )
17 1 3 nvscl 
 |-  ( ( U e. NrmCVec /\ -u 1 e. CC /\ B e. X ) -> ( -u 1 S B ) e. X )
18 6 17 mp3an2 
 |-  ( ( U e. NrmCVec /\ B e. X ) -> ( -u 1 S B ) e. X )
19 18 3adant2 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u 1 S B ) e. X )
20 simp2 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> A e. X )
21 1 2 nvcom 
 |-  ( ( U e. NrmCVec /\ ( -u 1 S B ) e. X /\ A e. X ) -> ( ( -u 1 S B ) G A ) = ( A G ( -u 1 S B ) ) )
22 5 19 20 21 syl3anc 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( -u 1 S B ) G A ) = ( A G ( -u 1 S B ) ) )
23 13 16 22 3eqtrd 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u 1 S ( B G ( -u 1 S A ) ) ) = ( A G ( -u 1 S B ) ) )
24 23 fveq2d 
 |-  ( ( 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 ) ) ) )
25 1 2 nvgcl 
 |-  ( ( U e. NrmCVec /\ B e. X /\ ( -u 1 S A ) e. X ) -> ( B G ( -u 1 S A ) ) e. X )
26 5 8 11 25 syl3anc 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( B G ( -u 1 S A ) ) e. X )
27 1 3 4 nvm1 
 |-  ( ( 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 5 26 27 syl2anc 
 |-  ( ( 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 24 28 eqtr3d 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( N ` ( A G ( -u 1 S B ) ) ) = ( N ` ( B G ( -u 1 S A ) ) ) )