This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem uvcn0

Description: A unit vector is nonzero. (Contributed by Steven Nguyen, 16-Jul-2023)

Ref Expression
Hypotheses uvcn0.u ⊢ 𝑈 = ( 𝑅 unitVec 𝐼 )
uvcn0.y ⊢ 𝑌 = ( 𝑅 freeLMod 𝐼 )
uvcn0.b ⊢ 𝐵 = ( Base ‘ 𝑌 )
uvcn0.0 ⊢ 0 = ( 0g ‘ 𝑌 )
Assertion uvcn0 ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼 ) → ( 𝑈 ‘ 𝐽 ) ≠ 0 )

Proof

Step Hyp Ref Expression
1 uvcn0.u ⊢ 𝑈 = ( 𝑅 unitVec 𝐼 )
2 uvcn0.y ⊢ 𝑌 = ( 𝑅 freeLMod 𝐼 )
3 uvcn0.b ⊢ 𝐵 = ( Base ‘ 𝑌 )
4 uvcn0.0 ⊢ 0 = ( 0g ‘ 𝑌 )
5 eqid ⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
6 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
7 5 6 nzrnz ⊢ ( 𝑅 ∈ NzRing → ( 1r ‘ 𝑅 ) ≠ ( 0g ‘ 𝑅 ) )
8 7 3ad2ant1 ⊢ ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼 ) → ( 1r ‘ 𝑅 ) ≠ ( 0g ‘ 𝑅 ) )
9 simp1 ⊢ ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼 ) → 𝑅 ∈ NzRing )
10 simp2 ⊢ ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼 ) → 𝐼 ∈ 𝑊 )
11 simp3 ⊢ ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼 ) → 𝐽 ∈ 𝐼 )
12 1 9 10 11 5 uvcvv1 ⊢ ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼 ) → ( ( 𝑈 ‘ 𝐽 ) ‘ 𝐽 ) = ( 1r ‘ 𝑅 ) )
13 nzrring ⊢ ( 𝑅 ∈ NzRing → 𝑅 ∈ Ring )
14 13 3ad2ant1 ⊢ ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼 ) → 𝑅 ∈ Ring )
15 2 6 14 10 11 frlm0vald ⊢ ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼 ) → ( ( 0g ‘ 𝑌 ) ‘ 𝐽 ) = ( 0g ‘ 𝑅 ) )
16 8 12 15 3netr4d ⊢ ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼 ) → ( ( 𝑈 ‘ 𝐽 ) ‘ 𝐽 ) ≠ ( ( 0g ‘ 𝑌 ) ‘ 𝐽 ) )
17 fveq1 ⊢ ( ( 𝑈 ‘ 𝐽 ) = ( 0g ‘ 𝑌 ) → ( ( 𝑈 ‘ 𝐽 ) ‘ 𝐽 ) = ( ( 0g ‘ 𝑌 ) ‘ 𝐽 ) )
18 17 necon3i ⊢ ( ( ( 𝑈 ‘ 𝐽 ) ‘ 𝐽 ) ≠ ( ( 0g ‘ 𝑌 ) ‘ 𝐽 ) → ( 𝑈 ‘ 𝐽 ) ≠ ( 0g ‘ 𝑌 ) )
19 16 18 syl ⊢ ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼 ) → ( 𝑈 ‘ 𝐽 ) ≠ ( 0g ‘ 𝑌 ) )
20 4 a1i ⊢ ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼 ) → 0 = ( 0g ‘ 𝑌 ) )
21 19 20 neeqtrrd ⊢ ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼 ) → ( 𝑈 ‘ 𝐽 ) ≠ 0 )