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 )