Metamath Proof Explorer


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 ( 𝜑𝑅 ∈ Ring )
uvcvvcl2.i ( 𝜑𝐼𝑊 )
uvcvvcl2.j ( 𝜑𝐽𝐼 )
uvcvvcl2.k ( 𝜑𝐾𝐼 )
Assertion uvcvvcl2 ( 𝜑 → ( ( 𝑈𝐽 ) ‘ 𝐾 ) ∈ 𝐵 )

Proof

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