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Metamath Proof Explorer

Theorem ply1scl1OLD

Description: Obsolete version of ply1scl1 as of 12-Mar-2025. (Contributed by Stefan O'Rear, 1-Apr-2015) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses ply1scl.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
ply1scl.a ⊢ 𝐴 = ( algSc ‘ 𝑃 )
ply1scl1.o ⊢ 1 = ( 1r ‘ 𝑅 )
ply1scl1.n ⊢ 𝑁 = ( 1r ‘ 𝑃 )
Assertion ply1scl1OLD ( 𝑅 ∈ Ring → ( 𝐴 ‘ 1 ) = 𝑁 )

Proof

Step Hyp Ref Expression
1 ply1scl.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
2 ply1scl.a ⊢ 𝐴 = ( algSc ‘ 𝑃 )
3 ply1scl1.o ⊢ 1 = ( 1r ‘ 𝑅 )
4 ply1scl1.n ⊢ 𝑁 = ( 1r ‘ 𝑃 )
5 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
6 5 3 ringidcl ⊢ ( 𝑅 ∈ Ring → 1 ∈ ( Base ‘ 𝑅 ) )
7 1 ply1sca2 ⊢ ( I ‘ 𝑅 ) = ( Scalar ‘ 𝑃 )
8 baseid ⊢ Base = Slot ( Base ‘ ndx )
9 8 5 strfvi ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ ( I ‘ 𝑅 ) )
10 eqid ⊢ ( ·𝑠 ‘ 𝑃 ) = ( ·𝑠 ‘ 𝑃 )
11 2 7 9 10 4 asclval ⊢ ( 1 ∈ ( Base ‘ 𝑅 ) → ( 𝐴 ‘ 1 ) = ( 1 ( ·𝑠 ‘ 𝑃 ) 𝑁 ) )
12 6 11 syl ⊢ ( 𝑅 ∈ Ring → ( 𝐴 ‘ 1 ) = ( 1 ( ·𝑠 ‘ 𝑃 ) 𝑁 ) )
13 fvi ⊢ ( 𝑅 ∈ Ring → ( I ‘ 𝑅 ) = 𝑅 )
14 13 fveq2d ⊢ ( 𝑅 ∈ Ring → ( 1r ‘ ( I ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) )
15 14 3 eqtr4di ⊢ ( 𝑅 ∈ Ring → ( 1r ‘ ( I ‘ 𝑅 ) ) = 1 )
16 15 oveq1d ⊢ ( 𝑅 ∈ Ring → ( ( 1r ‘ ( I ‘ 𝑅 ) ) ( ·𝑠 ‘ 𝑃 ) 𝑁 ) = ( 1 ( ·𝑠 ‘ 𝑃 ) 𝑁 ) )
17 1 ply1lmod ⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ LMod )
18 1 ply1ring ⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring )
19 eqid ⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 )
20 19 4 ringidcl ⊢ ( 𝑃 ∈ Ring → 𝑁 ∈ ( Base ‘ 𝑃 ) )
21 18 20 syl ⊢ ( 𝑅 ∈ Ring → 𝑁 ∈ ( Base ‘ 𝑃 ) )
22 eqid ⊢ ( 1r ‘ ( I ‘ 𝑅 ) ) = ( 1r ‘ ( I ‘ 𝑅 ) )
23 19 7 10 22 lmodvs1 ⊢ ( ( 𝑃 ∈ LMod ∧ 𝑁 ∈ ( Base ‘ 𝑃 ) ) → ( ( 1r ‘ ( I ‘ 𝑅 ) ) ( ·𝑠 ‘ 𝑃 ) 𝑁 ) = 𝑁 )
24 17 21 23 syl2anc ⊢ ( 𝑅 ∈ Ring → ( ( 1r ‘ ( I ‘ 𝑅 ) ) ( ·𝑠 ‘ 𝑃 ) 𝑁 ) = 𝑁 )
25 12 16 24 3eqtr2d ⊢ ( 𝑅 ∈ Ring → ( 𝐴 ‘ 1 ) = 𝑁 )