Metamath Proof Explorer


Theorem subrgmvr

Description: The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 4-Jul-2015)

Ref Expression
Hypotheses subrgmvr.v 𝑉 = ( 𝐼 mVar 𝑅 )
subrgmvr.i ( 𝜑𝐼𝑊 )
subrgmvr.r ( 𝜑𝑇 ∈ ( SubRing ‘ 𝑅 ) )
subrgmvr.h 𝐻 = ( 𝑅s 𝑇 )
Assertion subrgmvr ( 𝜑𝑉 = ( 𝐼 mVar 𝐻 ) )

Proof

Step Hyp Ref Expression
1 subrgmvr.v 𝑉 = ( 𝐼 mVar 𝑅 )
2 subrgmvr.i ( 𝜑𝐼𝑊 )
3 subrgmvr.r ( 𝜑𝑇 ∈ ( SubRing ‘ 𝑅 ) )
4 subrgmvr.h 𝐻 = ( 𝑅s 𝑇 )
5 eqid ( 1r𝑅 ) = ( 1r𝑅 )
6 4 5 subrg1 ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → ( 1r𝑅 ) = ( 1r𝐻 ) )
7 3 6 syl ( 𝜑 → ( 1r𝑅 ) = ( 1r𝐻 ) )
8 eqid ( 0g𝑅 ) = ( 0g𝑅 )
9 4 8 subrg0 ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → ( 0g𝑅 ) = ( 0g𝐻 ) )
10 3 9 syl ( 𝜑 → ( 0g𝑅 ) = ( 0g𝐻 ) )
11 7 10 ifeq12d ( 𝜑 → if ( 𝑦 = ( 𝑧𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r𝑅 ) , ( 0g𝑅 ) ) = if ( 𝑦 = ( 𝑧𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r𝐻 ) , ( 0g𝐻 ) ) )
12 11 mpteq2dv ( 𝜑 → ( 𝑦 ∈ { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑦 = ( 𝑧𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r𝑅 ) , ( 0g𝑅 ) ) ) = ( 𝑦 ∈ { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑦 = ( 𝑧𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r𝐻 ) , ( 0g𝐻 ) ) ) )
13 12 mpteq2dv ( 𝜑 → ( 𝑥𝐼 ↦ ( 𝑦 ∈ { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑦 = ( 𝑧𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r𝑅 ) , ( 0g𝑅 ) ) ) ) = ( 𝑥𝐼 ↦ ( 𝑦 ∈ { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑦 = ( 𝑧𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r𝐻 ) , ( 0g𝐻 ) ) ) ) )
14 eqid { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin } = { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin }
15 subrgrcl ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → 𝑅 ∈ Ring )
16 3 15 syl ( 𝜑𝑅 ∈ Ring )
17 1 14 8 5 2 16 mvrfval ( 𝜑𝑉 = ( 𝑥𝐼 ↦ ( 𝑦 ∈ { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑦 = ( 𝑧𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r𝑅 ) , ( 0g𝑅 ) ) ) ) )
18 eqid ( 𝐼 mVar 𝐻 ) = ( 𝐼 mVar 𝐻 )
19 eqid ( 0g𝐻 ) = ( 0g𝐻 )
20 eqid ( 1r𝐻 ) = ( 1r𝐻 )
21 4 ovexi 𝐻 ∈ V
22 21 a1i ( 𝜑𝐻 ∈ V )
23 18 14 19 20 2 22 mvrfval ( 𝜑 → ( 𝐼 mVar 𝐻 ) = ( 𝑥𝐼 ↦ ( 𝑦 ∈ { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑦 = ( 𝑧𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r𝐻 ) , ( 0g𝐻 ) ) ) ) )
24 13 17 23 3eqtr4d ( 𝜑𝑉 = ( 𝐼 mVar 𝐻 ) )