Metamath Proof Explorer


Theorem ply1idvr1

Description: The identity of a polynomial ring expressed as power of the polynomial variable. (Contributed by AV, 14-Aug-2019)

Ref Expression
Hypotheses ply1idvr1.p 𝑃 = ( Poly1𝑅 )
ply1idvr1.x 𝑋 = ( var1𝑅 )
ply1idvr1.n 𝑁 = ( mulGrp ‘ 𝑃 )
ply1idvr1.e = ( .g𝑁 )
Assertion ply1idvr1 ( 𝑅 ∈ Ring → ( 0 𝑋 ) = ( 1r𝑃 ) )

Proof

Step Hyp Ref Expression
1 ply1idvr1.p 𝑃 = ( Poly1𝑅 )
2 ply1idvr1.x 𝑋 = ( var1𝑅 )
3 ply1idvr1.n 𝑁 = ( mulGrp ‘ 𝑃 )
4 ply1idvr1.e = ( .g𝑁 )
5 eqid ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
6 eqid ( 1r𝑅 ) = ( 1r𝑅 )
7 5 6 ringidcl ( 𝑅 ∈ Ring → ( 1r𝑅 ) ∈ ( Base ‘ 𝑅 ) )
8 eqid ( ·𝑠𝑃 ) = ( ·𝑠𝑃 )
9 eqid ( algSc ‘ 𝑃 ) = ( algSc ‘ 𝑃 )
10 5 1 2 8 3 4 9 ply1scltm ( ( 𝑅 ∈ Ring ∧ ( 1r𝑅 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( algSc ‘ 𝑃 ) ‘ ( 1r𝑅 ) ) = ( ( 1r𝑅 ) ( ·𝑠𝑃 ) ( 0 𝑋 ) ) )
11 7 10 mpdan ( 𝑅 ∈ Ring → ( ( algSc ‘ 𝑃 ) ‘ ( 1r𝑅 ) ) = ( ( 1r𝑅 ) ( ·𝑠𝑃 ) ( 0 𝑋 ) ) )
12 1 ply1sca ( 𝑅 ∈ Ring → 𝑅 = ( Scalar ‘ 𝑃 ) )
13 12 fveq2d ( 𝑅 ∈ Ring → ( 1r𝑅 ) = ( 1r ‘ ( Scalar ‘ 𝑃 ) ) )
14 13 oveq1d ( 𝑅 ∈ Ring → ( ( 1r𝑅 ) ( ·𝑠𝑃 ) ( 0 𝑋 ) ) = ( ( 1r ‘ ( Scalar ‘ 𝑃 ) ) ( ·𝑠𝑃 ) ( 0 𝑋 ) ) )
15 1 ply1lmod ( 𝑅 ∈ Ring → 𝑃 ∈ LMod )
16 0nn0 0 ∈ ℕ0
17 eqid ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 )
18 1 2 3 4 17 ply1moncl ( ( 𝑅 ∈ Ring ∧ 0 ∈ ℕ0 ) → ( 0 𝑋 ) ∈ ( Base ‘ 𝑃 ) )
19 16 18 mpan2 ( 𝑅 ∈ Ring → ( 0 𝑋 ) ∈ ( Base ‘ 𝑃 ) )
20 eqid ( Scalar ‘ 𝑃 ) = ( Scalar ‘ 𝑃 )
21 eqid ( 1r ‘ ( Scalar ‘ 𝑃 ) ) = ( 1r ‘ ( Scalar ‘ 𝑃 ) )
22 17 20 8 21 lmodvs1 ( ( 𝑃 ∈ LMod ∧ ( 0 𝑋 ) ∈ ( Base ‘ 𝑃 ) ) → ( ( 1r ‘ ( Scalar ‘ 𝑃 ) ) ( ·𝑠𝑃 ) ( 0 𝑋 ) ) = ( 0 𝑋 ) )
23 15 19 22 syl2anc ( 𝑅 ∈ Ring → ( ( 1r ‘ ( Scalar ‘ 𝑃 ) ) ( ·𝑠𝑃 ) ( 0 𝑋 ) ) = ( 0 𝑋 ) )
24 11 14 23 3eqtrrd ( 𝑅 ∈ Ring → ( 0 𝑋 ) = ( ( algSc ‘ 𝑃 ) ‘ ( 1r𝑅 ) ) )
25 eqid ( 1r𝑃 ) = ( 1r𝑃 )
26 1 9 6 25 ply1scl1 ( 𝑅 ∈ Ring → ( ( algSc ‘ 𝑃 ) ‘ ( 1r𝑅 ) ) = ( 1r𝑃 ) )
27 24 26 eqtrd ( 𝑅 ∈ Ring → ( 0 𝑋 ) = ( 1r𝑃 ) )