This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 = ( Poly1 ` R )
ressply1.h 
|- H = ( R |`s T )
ressply1.u 
|- U = ( Poly1 ` H )
ressply1.b 
|- B = ( Base ` U )
ressply1.2 
|- ( ph -> T e. ( SubRing ` R ) )
ressply1.p 
|- P = ( S |`s B )
Assertion ressply1add 
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( +g ` U ) Y ) = ( X ( +g ` P ) Y ) )

Proof

Step Hyp Ref Expression
1 ressply1.s 
 |-  S = ( Poly1 ` R )
2 ressply1.h 
 |-  H = ( R |`s T )
3 ressply1.u 
 |-  U = ( Poly1 ` H )
4 ressply1.b 
 |-  B = ( Base ` U )
5 ressply1.2 
 |-  ( ph -> T e. ( SubRing ` R ) )
6 ressply1.p 
 |-  P = ( S |`s 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 e. On
11 10 a1i 
 |-  ( ph -> 1o e. On )
12 eqid 
 |-  ( ( 1o mPoly R ) |`s B ) = ( ( 1o mPoly R ) |`s B )
13 7 2 8 9 11 5 12 ressmpladd 
 |-  ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( +g ` ( 1o mPoly H ) ) Y ) = ( X ( +g ` ( ( 1o mPoly R ) |`s B ) ) Y ) )
14 eqid 
 |-  ( +g ` U ) = ( +g ` U )
15 3 8 14 ply1plusg 
 |-  ( +g ` U ) = ( +g ` ( 1o mPoly H ) )
16 15 oveqi 
 |-  ( X ( +g ` U ) Y ) = ( X ( +g ` ( 1o mPoly H ) ) Y )
17 eqid 
 |-  ( +g ` S ) = ( +g ` S )
18 1 7 17 ply1plusg 
 |-  ( +g ` S ) = ( +g ` ( 1o mPoly R ) )
19 4 fvexi 
 |-  B e. _V
20 6 17 ressplusg 
 |-  ( B e. _V -> ( +g ` S ) = ( +g ` P ) )
21 19 20 ax-mp 
 |-  ( +g ` S ) = ( +g ` P )
22 eqid 
 |-  ( +g ` ( 1o mPoly R ) ) = ( +g ` ( 1o mPoly R ) )
23 12 22 ressplusg 
 |-  ( B e. _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 
 |-  ( +g ` P ) = ( +g ` ( ( 1o mPoly R ) |`s B ) )
26 25 oveqi 
 |-  ( X ( +g ` P ) Y ) = ( X ( +g ` ( ( 1o mPoly R ) |`s B ) ) Y )
27 13 16 26 3eqtr4g 
 |-  ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( +g ` U ) Y ) = ( X ( +g ` P ) Y ) )