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Metamath Proof Explorer

Theorem evl1scad

Description: Polynomial evaluation builder for scalars. (Contributed by Mario Carneiro, 4-Jul-2015)

Ref Expression
Hypotheses evl1sca.o ⊢ 𝑂 = ( eval1 ‘ 𝑅 )
evl1sca.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
evl1sca.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
evl1sca.a ⊢ 𝐴 = ( algSc ‘ 𝑃 )
evl1scad.u ⊢ 𝑈 = ( Base ‘ 𝑃 )
evl1scad.1 ⊢ ( 𝜑 → 𝑅 ∈ CRing )
evl1scad.2 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
evl1scad.3 ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion evl1scad ( 𝜑 → ( ( 𝐴 ‘ 𝑋 ) ∈ 𝑈 ∧ ( ( 𝑂 ‘ ( 𝐴 ‘ 𝑋 ) ) ‘ 𝑌 ) = 𝑋 ) )

Proof

Step Hyp Ref Expression
1 evl1sca.o ⊢ 𝑂 = ( eval1 ‘ 𝑅 )
2 evl1sca.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
3 evl1sca.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
4 evl1sca.a ⊢ 𝐴 = ( algSc ‘ 𝑃 )
5 evl1scad.u ⊢ 𝑈 = ( Base ‘ 𝑃 )
6 evl1scad.1 ⊢ ( 𝜑 → 𝑅 ∈ CRing )
7 evl1scad.2 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
8 evl1scad.3 ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
9 crngring ⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring )
10 2 4 3 5 ply1sclf ⊢ ( 𝑅 ∈ Ring → 𝐴 : 𝐵 ⟶ 𝑈 )
11 6 9 10 3syl ⊢ ( 𝜑 → 𝐴 : 𝐵 ⟶ 𝑈 )
12 11 7 ffvelcdmd ⊢ ( 𝜑 → ( 𝐴 ‘ 𝑋 ) ∈ 𝑈 )
13 1 2 3 4 evl1sca ⊢ ( ( 𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ) → ( 𝑂 ‘ ( 𝐴 ‘ 𝑋 ) ) = ( 𝐵 × { 𝑋 } ) )
14 6 7 13 syl2anc ⊢ ( 𝜑 → ( 𝑂 ‘ ( 𝐴 ‘ 𝑋 ) ) = ( 𝐵 × { 𝑋 } ) )
15 14 fveq1d ⊢ ( 𝜑 → ( ( 𝑂 ‘ ( 𝐴 ‘ 𝑋 ) ) ‘ 𝑌 ) = ( ( 𝐵 × { 𝑋 } ) ‘ 𝑌 ) )
16 fvconst2g ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐵 × { 𝑋 } ) ‘ 𝑌 ) = 𝑋 )
17 7 8 16 syl2anc ⊢ ( 𝜑 → ( ( 𝐵 × { 𝑋 } ) ‘ 𝑌 ) = 𝑋 )
18 15 17 eqtrd ⊢ ( 𝜑 → ( ( 𝑂 ‘ ( 𝐴 ‘ 𝑋 ) ) ‘ 𝑌 ) = 𝑋 )
19 12 18 jca ⊢ ( 𝜑 → ( ( 𝐴 ‘ 𝑋 ) ∈ 𝑈 ∧ ( ( 𝑂 ‘ ( 𝐴 ‘ 𝑋 ) ) ‘ 𝑌 ) = 𝑋 ) )