Metamath Proof Explorer


Theorem hdmaplns1

Description: Subtraction property of first (inner product) argument. (Contributed by NM, 12-Jun-2015)

Ref Expression
Hypotheses hdmaplns1.h H = LHyp K
hdmaplns1.u U = DVecH K W
hdmaplns1.v V = Base U
hdmaplns1.m - ˙ = - U
hdmaplns1.r R = Scalar U
hdmaplns1.n N = - R
hdmaplns1.s S = HDMap K W
hdmaplns1.k φ K HL W H
hdmaplns1.x φ X V
hdmaplns1.y φ Y V
hdmaplns1.z φ Z V
Assertion hdmaplns1 φ S Z X - ˙ Y = S Z X N S Z Y

Proof

Step Hyp Ref Expression
1 hdmaplns1.h H = LHyp K
2 hdmaplns1.u U = DVecH K W
3 hdmaplns1.v V = Base U
4 hdmaplns1.m - ˙ = - U
5 hdmaplns1.r R = Scalar U
6 hdmaplns1.n N = - R
7 hdmaplns1.s S = HDMap K W
8 hdmaplns1.k φ K HL W H
9 hdmaplns1.x φ X V
10 hdmaplns1.y φ Y V
11 hdmaplns1.z φ Z V
12 1 2 8 dvhlmod φ U LMod
13 eqid LCDual K W = LCDual K W
14 eqid Base LCDual K W = Base LCDual K W
15 eqid LFnl U = LFnl U
16 1 2 3 13 14 7 8 11 hdmapcl φ S Z Base LCDual K W
17 1 13 14 2 15 8 16 lcdvbaselfl φ S Z LFnl U
18 5 6 3 4 15 lflsub U LMod S Z LFnl U X V Y V S Z X - ˙ Y = S Z X N S Z Y
19 12 17 9 10 18 syl112anc φ S Z X - ˙ Y = S Z X N S Z Y