Metamath Proof Explorer


Theorem coe1scl

Description: Coefficient vector of a scalar. (Contributed by Stefan O'Rear, 28-Mar-2015)

Ref Expression
Hypotheses ply1scl.p 𝑃 = ( Poly1𝑅 )
ply1scl.a 𝐴 = ( algSc ‘ 𝑃 )
coe1scl.k 𝐾 = ( Base ‘ 𝑅 )
coe1scl.z 0 = ( 0g𝑅 )
Assertion coe1scl ( ( 𝑅 ∈ Ring ∧ 𝑋𝐾 ) → ( coe1 ‘ ( 𝐴𝑋 ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝑥 = 0 , 𝑋 , 0 ) ) )

Proof

Step Hyp Ref Expression
1 ply1scl.p 𝑃 = ( Poly1𝑅 )
2 ply1scl.a 𝐴 = ( algSc ‘ 𝑃 )
3 coe1scl.k 𝐾 = ( Base ‘ 𝑅 )
4 coe1scl.z 0 = ( 0g𝑅 )
5 eqid ( var1𝑅 ) = ( var1𝑅 )
6 eqid ( ·𝑠𝑃 ) = ( ·𝑠𝑃 )
7 eqid ( mulGrp ‘ 𝑃 ) = ( mulGrp ‘ 𝑃 )
8 eqid ( .g ‘ ( mulGrp ‘ 𝑃 ) ) = ( .g ‘ ( mulGrp ‘ 𝑃 ) )
9 3 1 5 6 7 8 2 ply1scltm ( ( 𝑅 ∈ Ring ∧ 𝑋𝐾 ) → ( 𝐴𝑋 ) = ( 𝑋 ( ·𝑠𝑃 ) ( 0 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1𝑅 ) ) ) )
10 9 fveq2d ( ( 𝑅 ∈ Ring ∧ 𝑋𝐾 ) → ( coe1 ‘ ( 𝐴𝑋 ) ) = ( coe1 ‘ ( 𝑋 ( ·𝑠𝑃 ) ( 0 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1𝑅 ) ) ) ) )
11 0nn0 0 ∈ ℕ0
12 4 3 1 5 6 7 8 coe1tm ( ( 𝑅 ∈ Ring ∧ 𝑋𝐾 ∧ 0 ∈ ℕ0 ) → ( coe1 ‘ ( 𝑋 ( ·𝑠𝑃 ) ( 0 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1𝑅 ) ) ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝑥 = 0 , 𝑋 , 0 ) ) )
13 11 12 mp3an3 ( ( 𝑅 ∈ Ring ∧ 𝑋𝐾 ) → ( coe1 ‘ ( 𝑋 ( ·𝑠𝑃 ) ( 0 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1𝑅 ) ) ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝑥 = 0 , 𝑋 , 0 ) ) )
14 10 13 eqtrd ( ( 𝑅 ∈ Ring ∧ 𝑋𝐾 ) → ( coe1 ‘ ( 𝐴𝑋 ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝑥 = 0 , 𝑋 , 0 ) ) )