Metamath Proof Explorer


Theorem 0prjspnlem

Description: Lemma for 0prjspn . The given unit vector is a nonzero vector. (Contributed by Steven Nguyen, 16-Jul-2023)

Ref Expression
Hypotheses 0prjspnlem.b B = Base W 0 W
0prjspnlem.w W = K freeLMod 0 0
0prjspnlem.1 1 ˙ = K unitVec 0 0 0
Assertion 0prjspnlem K DivRing 1 ˙ B

Proof

Step Hyp Ref Expression
1 0prjspnlem.b B = Base W 0 W
2 0prjspnlem.w W = K freeLMod 0 0
3 0prjspnlem.1 1 ˙ = K unitVec 0 0 0
4 drngnzr K DivRing K NzRing
5 ovex 0 0 V
6 c0ex 0 V
7 6 snid 0 0
8 fz0sn 0 0 = 0
9 7 8 eleqtrri 0 0 0
10 nzrring K NzRing K Ring
11 eqid K unitVec 0 0 = K unitVec 0 0
12 eqid Base W = Base W
13 11 2 12 uvccl K Ring 0 0 V 0 0 0 K unitVec 0 0 0 Base W
14 10 13 syl3an1 K NzRing 0 0 V 0 0 0 K unitVec 0 0 0 Base W
15 eqid 0 W = 0 W
16 11 2 12 15 uvcn0 K NzRing 0 0 V 0 0 0 K unitVec 0 0 0 0 W
17 eldifsn K unitVec 0 0 0 Base W 0 W K unitVec 0 0 0 Base W K unitVec 0 0 0 0 W
18 14 16 17 sylanbrc K NzRing 0 0 V 0 0 0 K unitVec 0 0 0 Base W 0 W
19 5 9 18 mp3an23 K NzRing K unitVec 0 0 0 Base W 0 W
20 4 19 syl K DivRing K unitVec 0 0 0 Base W 0 W
21 20 3 1 3eltr4g K DivRing 1 ˙ B