Metamath Proof Explorer


Theorem ressply1add

Description: A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015)

Ref Expression
Hypotheses ressply1.s 𝑆 = ( Poly1𝑅 )
ressply1.h 𝐻 = ( 𝑅s 𝑇 )
ressply1.u 𝑈 = ( Poly1𝐻 )
ressply1.b 𝐵 = ( Base ‘ 𝑈 )
ressply1.2 ( 𝜑𝑇 ∈ ( SubRing ‘ 𝑅 ) )
ressply1.p 𝑃 = ( 𝑆s 𝐵 )
Assertion ressply1add ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 ( +g𝑈 ) 𝑌 ) = ( 𝑋 ( +g𝑃 ) 𝑌 ) )

Proof

Step Hyp Ref Expression
1 ressply1.s 𝑆 = ( Poly1𝑅 )
2 ressply1.h 𝐻 = ( 𝑅s 𝑇 )
3 ressply1.u 𝑈 = ( Poly1𝐻 )
4 ressply1.b 𝐵 = ( Base ‘ 𝑈 )
5 ressply1.2 ( 𝜑𝑇 ∈ ( SubRing ‘ 𝑅 ) )
6 ressply1.p 𝑃 = ( 𝑆s 𝐵 )
7 eqid ( 1o mPoly 𝑅 ) = ( 1o mPoly 𝑅 )
8 eqid ( 1o mPoly 𝐻 ) = ( 1o mPoly 𝐻 )
9 3 4 ply1bas 𝐵 = ( Base ‘ ( 1o mPoly 𝐻 ) )
10 1on 1o ∈ On
11 10 a1i ( 𝜑 → 1o ∈ On )
12 eqid ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) = ( ( 1o mPoly 𝑅 ) ↾s 𝐵 )
13 7 2 8 9 11 5 12 ressmpladd ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 ( +g ‘ ( 1o mPoly 𝐻 ) ) 𝑌 ) = ( 𝑋 ( +g ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) 𝑌 ) )
14 eqid ( +g𝑈 ) = ( +g𝑈 )
15 3 8 14 ply1plusg ( +g𝑈 ) = ( +g ‘ ( 1o mPoly 𝐻 ) )
16 15 oveqi ( 𝑋 ( +g𝑈 ) 𝑌 ) = ( 𝑋 ( +g ‘ ( 1o mPoly 𝐻 ) ) 𝑌 )
17 eqid ( +g𝑆 ) = ( +g𝑆 )
18 1 7 17 ply1plusg ( +g𝑆 ) = ( +g ‘ ( 1o mPoly 𝑅 ) )
19 4 fvexi 𝐵 ∈ V
20 6 17 ressplusg ( 𝐵 ∈ V → ( +g𝑆 ) = ( +g𝑃 ) )
21 19 20 ax-mp ( +g𝑆 ) = ( +g𝑃 )
22 eqid ( +g ‘ ( 1o mPoly 𝑅 ) ) = ( +g ‘ ( 1o mPoly 𝑅 ) )
23 12 22 ressplusg ( 𝐵 ∈ V → ( +g ‘ ( 1o mPoly 𝑅 ) ) = ( +g ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) )
24 19 23 ax-mp ( +g ‘ ( 1o mPoly 𝑅 ) ) = ( +g ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) )
25 18 21 24 3eqtr3i ( +g𝑃 ) = ( +g ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) )
26 25 oveqi ( 𝑋 ( +g𝑃 ) 𝑌 ) = ( 𝑋 ( +g ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) 𝑌 )
27 13 16 26 3eqtr4g ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 ( +g𝑈 ) 𝑌 ) = ( 𝑋 ( +g𝑃 ) 𝑌 ) )