Metamath Proof Explorer

Theorem nvpi

Description: The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007) (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 nvpi 
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( N ` ( A G ( _i S B ) ) ) = ( N ` ( B G ( -u _i 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 ax-icn 
 |-  _i e. CC
7 1 3 nvscl 
 |-  ( ( U e. NrmCVec /\ _i e. CC /\ B e. X ) -> ( _i S B ) e. X )
8 6 7 mp3an2 
 |-  ( ( U e. NrmCVec /\ B e. X ) -> ( _i S B ) e. X )
9 8 3adant2 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( _i S B ) e. X )
10 1 2 nvgcl 
 |-  ( ( U e. NrmCVec /\ A e. X /\ ( _i S B ) e. X ) -> ( A G ( _i S B ) ) e. X )
11 9 10 syld3an3 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A G ( _i S B ) ) e. X )
12 1 4 nvcl 
 |-  ( ( U e. NrmCVec /\ ( A G ( _i S B ) ) e. X ) -> ( N ` ( A G ( _i S B ) ) ) e. RR )
13 5 11 12 syl2anc 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( N ` ( A G ( _i S B ) ) ) e. RR )
14 13 recnd 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( N ` ( A G ( _i S B ) ) ) e. CC )
15 14 mulid2d 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( 1 x. ( N ` ( A G ( _i S B ) ) ) ) = ( N ` ( A G ( _i S B ) ) ) )
16 6 absnegi 
 |-  ( abs ` -u _i ) = ( abs ` _i )
17 absi 
 |-  ( abs ` _i ) = 1
18 16 17 eqtri 
 |-  ( abs ` -u _i ) = 1
19 18 oveq1i 
 |-  ( ( abs ` -u _i ) x. ( N ` ( A G ( _i S B ) ) ) ) = ( 1 x. ( N ` ( A G ( _i S B ) ) ) )
20 negicn 
 |-  -u _i e. CC
21 1 3 4 nvs 
 |-  ( ( U e. NrmCVec /\ -u _i e. CC /\ ( A G ( _i S B ) ) e. X ) -> ( N ` ( -u _i S ( A G ( _i S B ) ) ) ) = ( ( abs ` -u _i ) x. ( N ` ( A G ( _i S B ) ) ) ) )
22 20 21 mp3an2 
 |-  ( ( U e. NrmCVec /\ ( A G ( _i S B ) ) e. X ) -> ( N ` ( -u _i S ( A G ( _i S B ) ) ) ) = ( ( abs ` -u _i ) x. ( N ` ( A G ( _i S B ) ) ) ) )
23 5 11 22 syl2anc 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( N ` ( -u _i S ( A G ( _i S B ) ) ) ) = ( ( abs ` -u _i ) x. ( N ` ( A G ( _i S B ) ) ) ) )
24 simp2 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> A e. X )
25 1 2 3 nvdi 
 |-  ( ( U e. NrmCVec /\ ( -u _i e. CC /\ A e. X /\ ( _i S B ) e. X ) ) -> ( -u _i S ( A G ( _i S B ) ) ) = ( ( -u _i S A ) G ( -u _i S ( _i S B ) ) ) )
26 20 25 mp3anr1 
 |-  ( ( U e. NrmCVec /\ ( A e. X /\ ( _i S B ) e. X ) ) -> ( -u _i S ( A G ( _i S B ) ) ) = ( ( -u _i S A ) G ( -u _i S ( _i S B ) ) ) )
27 5 24 9 26 syl12anc 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u _i S ( A G ( _i S B ) ) ) = ( ( -u _i S A ) G ( -u _i S ( _i S B ) ) ) )
28 6 6 mulneg1i 
 |-  ( -u _i x. _i ) = -u ( _i x. _i )
29 ixi 
 |-  ( _i x. _i ) = -u 1
30 29 negeqi 
 |-  -u ( _i x. _i ) = -u -u 1
31 negneg1e1 
 |-  -u -u 1 = 1
32 30 31 eqtri 
 |-  -u ( _i x. _i ) = 1
33 28 32 eqtri 
 |-  ( -u _i x. _i ) = 1
34 33 oveq1i 
 |-  ( ( -u _i x. _i ) S B ) = ( 1 S B )
35 1 3 nvsass 
 |-  ( ( U e. NrmCVec /\ ( -u _i e. CC /\ _i e. CC /\ B e. X ) ) -> ( ( -u _i x. _i ) S B ) = ( -u _i S ( _i S B ) ) )
36 20 35 mp3anr1 
 |-  ( ( U e. NrmCVec /\ ( _i e. CC /\ B e. X ) ) -> ( ( -u _i x. _i ) S B ) = ( -u _i S ( _i S B ) ) )
37 6 36 mpanr1 
 |-  ( ( U e. NrmCVec /\ B e. X ) -> ( ( -u _i x. _i ) S B ) = ( -u _i S ( _i S B ) ) )
38 1 3 nvsid 
 |-  ( ( U e. NrmCVec /\ B e. X ) -> ( 1 S B ) = B )
39 34 37 38 3eqtr3a 
 |-  ( ( U e. NrmCVec /\ B e. X ) -> ( -u _i S ( _i S B ) ) = B )
40 39 3adant2 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u _i S ( _i S B ) ) = B )
41 40 oveq2d 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( -u _i S A ) G ( -u _i S ( _i S B ) ) ) = ( ( -u _i S A ) G B ) )
42 1 3 nvscl 
 |-  ( ( U e. NrmCVec /\ -u _i e. CC /\ A e. X ) -> ( -u _i S A ) e. X )
43 20 42 mp3an2 
 |-  ( ( U e. NrmCVec /\ A e. X ) -> ( -u _i S A ) e. X )
44 43 3adant3 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u _i S A ) e. X )
45 1 2 nvcom 
 |-  ( ( U e. NrmCVec /\ ( -u _i S A ) e. X /\ B e. X ) -> ( ( -u _i S A ) G B ) = ( B G ( -u _i S A ) ) )
46 44 45 syld3an2 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( -u _i S A ) G B ) = ( B G ( -u _i S A ) ) )
47 27 41 46 3eqtrd 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u _i S ( A G ( _i S B ) ) ) = ( B G ( -u _i S A ) ) )
48 47 fveq2d 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( N ` ( -u _i S ( A G ( _i S B ) ) ) ) = ( N ` ( B G ( -u _i S A ) ) ) )
49 23 48 eqtr3d 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( abs ` -u _i ) x. ( N ` ( A G ( _i S B ) ) ) ) = ( N ` ( B G ( -u _i S A ) ) ) )
50 19 49 syl5eqr 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( 1 x. ( N ` ( A G ( _i S B ) ) ) ) = ( N ` ( B G ( -u _i S A ) ) ) )
51 15 50 eqtr3d 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( N ` ( A G ( _i S B ) ) ) = ( N ` ( B G ( -u _i S A ) ) ) )