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Metamath Proof Explorer

Theorem mplascl

Description: Value of the scalar injection into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015)

Ref Expression
Hypotheses mplascl.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
mplascl.d ⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
mplascl.z ⊢ 0 = ( 0g ‘ 𝑅 )
mplascl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
mplascl.a ⊢ 𝐴 = ( algSc ‘ 𝑃 )
mplascl.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑊 )
mplascl.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
mplascl.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion mplascl ( 𝜑 → ( 𝐴 ‘ 𝑋 ) = ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = ( 𝐼 × { 0 } ) , 𝑋 , 0 ) ) )

Proof

Step Hyp Ref Expression
1 mplascl.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
2 mplascl.d ⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
3 mplascl.z ⊢ 0 = ( 0g ‘ 𝑅 )
4 mplascl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
5 mplascl.a ⊢ 𝐴 = ( algSc ‘ 𝑃 )
6 mplascl.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑊 )
7 mplascl.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
8 mplascl.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
9 1 6 7 mplsca ⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑃 ) )
10 9 fveq2d ⊢ ( 𝜑 → ( Base ‘ 𝑅 ) = ( Base ‘ ( Scalar ‘ 𝑃 ) ) )
11 4 10 eqtrid ⊢ ( 𝜑 → 𝐵 = ( Base ‘ ( Scalar ‘ 𝑃 ) ) )
12 8 11 eleqtrd ⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ ( Scalar ‘ 𝑃 ) ) )
13 eqid ⊢ ( Scalar ‘ 𝑃 ) = ( Scalar ‘ 𝑃 )
14 eqid ⊢ ( Base ‘ ( Scalar ‘ 𝑃 ) ) = ( Base ‘ ( Scalar ‘ 𝑃 ) )
15 eqid ⊢ ( ·𝑠 ‘ 𝑃 ) = ( ·𝑠 ‘ 𝑃 )
16 eqid ⊢ ( 1r ‘ 𝑃 ) = ( 1r ‘ 𝑃 )
17 5 13 14 15 16 asclval ⊢ ( 𝑋 ∈ ( Base ‘ ( Scalar ‘ 𝑃 ) ) → ( 𝐴 ‘ 𝑋 ) = ( 𝑋 ( ·𝑠 ‘ 𝑃 ) ( 1r ‘ 𝑃 ) ) )
18 12 17 syl ⊢ ( 𝜑 → ( 𝐴 ‘ 𝑋 ) = ( 𝑋 ( ·𝑠 ‘ 𝑃 ) ( 1r ‘ 𝑃 ) ) )
19 eqid ⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
20 1 2 3 19 16 6 7 mpl1 ⊢ ( 𝜑 → ( 1r ‘ 𝑃 ) = ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = ( 𝐼 × { 0 } ) , ( 1r ‘ 𝑅 ) , 0 ) ) )
21 20 oveq2d ⊢ ( 𝜑 → ( 𝑋 ( ·𝑠 ‘ 𝑃 ) ( 1r ‘ 𝑃 ) ) = ( 𝑋 ( ·𝑠 ‘ 𝑃 ) ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = ( 𝐼 × { 0 } ) , ( 1r ‘ 𝑅 ) , 0 ) ) ) )
22 2 psrbag0 ⊢ ( 𝐼 ∈ 𝑊 → ( 𝐼 × { 0 } ) ∈ 𝐷 )
23 6 22 syl ⊢ ( 𝜑 → ( 𝐼 × { 0 } ) ∈ 𝐷 )
24 1 15 2 19 3 4 6 7 23 8 mplmon2 ⊢ ( 𝜑 → ( 𝑋 ( ·𝑠 ‘ 𝑃 ) ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = ( 𝐼 × { 0 } ) , ( 1r ‘ 𝑅 ) , 0 ) ) ) = ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = ( 𝐼 × { 0 } ) , 𝑋 , 0 ) ) )
25 18 21 24 3eqtrd ⊢ ( 𝜑 → ( 𝐴 ‘ 𝑋 ) = ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = ( 𝐼 × { 0 } ) , 𝑋 , 0 ) ) )