Metamath Proof Explorer


Theorem coe1sclmulfv

Description: A single coefficient of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015)

Ref Expression
Hypotheses coe1sclmul.p 𝑃 = ( Poly1𝑅 )
coe1sclmul.b 𝐵 = ( Base ‘ 𝑃 )
coe1sclmul.k 𝐾 = ( Base ‘ 𝑅 )
coe1sclmul.a 𝐴 = ( algSc ‘ 𝑃 )
coe1sclmul.t = ( .r𝑃 )
coe1sclmul.u · = ( .r𝑅 )
Assertion coe1sclmulfv ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐾𝑌𝐵 ) ∧ 0 ∈ ℕ0 ) → ( ( coe1 ‘ ( ( 𝐴𝑋 ) 𝑌 ) ) ‘ 0 ) = ( 𝑋 · ( ( coe1𝑌 ) ‘ 0 ) ) )

Proof

Step Hyp Ref Expression
1 coe1sclmul.p 𝑃 = ( Poly1𝑅 )
2 coe1sclmul.b 𝐵 = ( Base ‘ 𝑃 )
3 coe1sclmul.k 𝐾 = ( Base ‘ 𝑅 )
4 coe1sclmul.a 𝐴 = ( algSc ‘ 𝑃 )
5 coe1sclmul.t = ( .r𝑃 )
6 coe1sclmul.u · = ( .r𝑅 )
7 1 2 3 4 5 6 coe1sclmul ( ( 𝑅 ∈ Ring ∧ 𝑋𝐾𝑌𝐵 ) → ( coe1 ‘ ( ( 𝐴𝑋 ) 𝑌 ) ) = ( ( ℕ0 × { 𝑋 } ) ∘f · ( coe1𝑌 ) ) )
8 7 3expb ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐾𝑌𝐵 ) ) → ( coe1 ‘ ( ( 𝐴𝑋 ) 𝑌 ) ) = ( ( ℕ0 × { 𝑋 } ) ∘f · ( coe1𝑌 ) ) )
9 8 3adant3 ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐾𝑌𝐵 ) ∧ 0 ∈ ℕ0 ) → ( coe1 ‘ ( ( 𝐴𝑋 ) 𝑌 ) ) = ( ( ℕ0 × { 𝑋 } ) ∘f · ( coe1𝑌 ) ) )
10 9 fveq1d ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐾𝑌𝐵 ) ∧ 0 ∈ ℕ0 ) → ( ( coe1 ‘ ( ( 𝐴𝑋 ) 𝑌 ) ) ‘ 0 ) = ( ( ( ℕ0 × { 𝑋 } ) ∘f · ( coe1𝑌 ) ) ‘ 0 ) )
11 simp3 ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐾𝑌𝐵 ) ∧ 0 ∈ ℕ0 ) → 0 ∈ ℕ0 )
12 nn0ex 0 ∈ V
13 12 a1i ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐾𝑌𝐵 ) ∧ 0 ∈ ℕ0 ) → ℕ0 ∈ V )
14 simp2l ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐾𝑌𝐵 ) ∧ 0 ∈ ℕ0 ) → 𝑋𝐾 )
15 simp2r ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐾𝑌𝐵 ) ∧ 0 ∈ ℕ0 ) → 𝑌𝐵 )
16 eqid ( coe1𝑌 ) = ( coe1𝑌 )
17 eqid ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
18 16 2 1 17 coe1f ( 𝑌𝐵 → ( coe1𝑌 ) : ℕ0 ⟶ ( Base ‘ 𝑅 ) )
19 ffn ( ( coe1𝑌 ) : ℕ0 ⟶ ( Base ‘ 𝑅 ) → ( coe1𝑌 ) Fn ℕ0 )
20 15 18 19 3syl ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐾𝑌𝐵 ) ∧ 0 ∈ ℕ0 ) → ( coe1𝑌 ) Fn ℕ0 )
21 eqidd ( ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐾𝑌𝐵 ) ∧ 0 ∈ ℕ0 ) ∧ 0 ∈ ℕ0 ) → ( ( coe1𝑌 ) ‘ 0 ) = ( ( coe1𝑌 ) ‘ 0 ) )
22 13 14 20 21 ofc1 ( ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐾𝑌𝐵 ) ∧ 0 ∈ ℕ0 ) ∧ 0 ∈ ℕ0 ) → ( ( ( ℕ0 × { 𝑋 } ) ∘f · ( coe1𝑌 ) ) ‘ 0 ) = ( 𝑋 · ( ( coe1𝑌 ) ‘ 0 ) ) )
23 11 22 mpdan ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐾𝑌𝐵 ) ∧ 0 ∈ ℕ0 ) → ( ( ( ℕ0 × { 𝑋 } ) ∘f · ( coe1𝑌 ) ) ‘ 0 ) = ( 𝑋 · ( ( coe1𝑌 ) ‘ 0 ) ) )
24 10 23 eqtrd ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐾𝑌𝐵 ) ∧ 0 ∈ ℕ0 ) → ( ( coe1 ‘ ( ( 𝐴𝑋 ) 𝑌 ) ) ‘ 0 ) = ( 𝑋 · ( ( coe1𝑌 ) ‘ 0 ) ) )