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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 ⊢ ( 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 )