Theorem uvcvvcl2

Description: A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015)

	Ref	Expression
Hypotheses	uvcvvcl2.u	⊢ 𝑈 = ( 𝑅 unitVec 𝐼 )
	uvcvvcl2.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	uvcvvcl2.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
	uvcvvcl2.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑊 )
	uvcvvcl2.j	⊢ ( 𝜑 → 𝐽 ∈ 𝐼 )
	uvcvvcl2.k	⊢ ( 𝜑 → 𝐾 ∈ 𝐼 )
Assertion	uvcvvcl2	⊢ ( 𝜑 → ( ( 𝑈 ‘ 𝐽 ) ‘ 𝐾 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		uvcvvcl2.u	⊢ 𝑈 = ( 𝑅 unitVec 𝐼 )
2		uvcvvcl2.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
3		uvcvvcl2.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
4		uvcvvcl2.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑊 )
5		uvcvvcl2.j	⊢ ( 𝜑 → 𝐽 ∈ 𝐼 )
6		uvcvvcl2.k	⊢ ( 𝜑 → 𝐾 ∈ 𝐼 )
7		eqid	⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
8		eqid	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
9	1 7 8	uvcvval	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼 ) ∧ 𝐾 ∈ 𝐼 ) → ( ( 𝑈 ‘ 𝐽 ) ‘ 𝐾 ) = if ( 𝐾 = 𝐽 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) )
10	3 4 5 6 9	syl31anc	⊢ ( 𝜑 → ( ( 𝑈 ‘ 𝐽 ) ‘ 𝐾 ) = if ( 𝐾 = 𝐽 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) )
11	2 7	ringidcl	⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ 𝐵 )
12	2 8	ring0cl	⊢ ( 𝑅 ∈ Ring → ( 0g ‘ 𝑅 ) ∈ 𝐵 )
13	11 12	ifcld	⊢ ( 𝑅 ∈ Ring → if ( 𝐾 = 𝐽 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ∈ 𝐵 )
14	3 13	syl	⊢ ( 𝜑 → if ( 𝐾 = 𝐽 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ∈ 𝐵 )
15	10 14	eqeltrd	⊢ ( 𝜑 → ( ( 𝑈 ‘ 𝐽 ) ‘ 𝐾 ) ∈ 𝐵 )