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Metamath Proof Explorer

Theorem rhmply1

Description: Provide a ring homomorphism between two univariate polynomial algebras over their respective base rings given a ring homomorphism between the two base rings. (Contributed by SN, 20-May-2025)

Ref Expression
Hypotheses rhmply1.p 
|- P = ( Poly1 ` R )
rhmply1.q 
|- Q = ( Poly1 ` S )
rhmply1.b 
|- B = ( Base ` P )
rhmply1.f 
|- F = ( p e. B |-> ( H o. p ) )
rhmply1.h 
|- ( ph -> H e. ( R RingHom S ) )
Assertion rhmply1 
|- ( ph -> F e. ( P RingHom Q ) )

Proof

Step Hyp Ref Expression
1 rhmply1.p 
 |-  P = ( Poly1 ` R )
2 rhmply1.q 
 |-  Q = ( Poly1 ` S )
3 rhmply1.b 
 |-  B = ( Base ` P )
4 rhmply1.f 
 |-  F = ( p e. B |-> ( H o. p ) )
5 rhmply1.h 
 |-  ( ph -> H e. ( R RingHom S ) )
6 eqid 
 |-  ( 1o mPoly R ) = ( 1o mPoly R )
7 eqid 
 |-  ( 1o mPoly S ) = ( 1o mPoly S )
8 1 3 ply1bas 
 |-  B = ( Base ` ( 1o mPoly R ) )
9 1oex 
 |-  1o e. _V
10 9 a1i 
 |-  ( ph -> 1o e. _V )
11 6 7 8 4 10 5 rhmmpl 
 |-  ( ph -> F e. ( ( 1o mPoly R ) RingHom ( 1o mPoly S ) ) )
12 3 a1i 
 |-  ( ph -> B = ( Base ` P ) )
13 eqid 
 |-  ( Base ` Q ) = ( Base ` Q )
14 13 a1i 
 |-  ( ph -> ( Base ` Q ) = ( Base ` Q ) )
15 8 a1i 
 |-  ( ph -> B = ( Base ` ( 1o mPoly R ) ) )
16 2 13 ply1bas 
 |-  ( Base ` Q ) = ( Base ` ( 1o mPoly S ) )
17 16 a1i 
 |-  ( ph -> ( Base ` Q ) = ( Base ` ( 1o mPoly S ) ) )
18 eqid 
 |-  ( +g ` P ) = ( +g ` P )
19 1 6 18 ply1plusg 
 |-  ( +g ` P ) = ( +g ` ( 1o mPoly R ) )
20 19 oveqi 
 |-  ( x ( +g ` P ) y ) = ( x ( +g ` ( 1o mPoly R ) ) y )
21 20 a1i 
 |-  ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` P ) y ) = ( x ( +g ` ( 1o mPoly R ) ) y ) )
22 eqid 
 |-  ( +g ` Q ) = ( +g ` Q )
23 2 7 22 ply1plusg 
 |-  ( +g ` Q ) = ( +g ` ( 1o mPoly S ) )
24 23 oveqi 
 |-  ( x ( +g ` Q ) y ) = ( x ( +g ` ( 1o mPoly S ) ) y )
25 24 a1i 
 |-  ( ( ph /\ ( x e. ( Base ` Q ) /\ y e. ( Base ` Q ) ) ) -> ( x ( +g ` Q ) y ) = ( x ( +g ` ( 1o mPoly S ) ) y ) )
26 eqid 
 |-  ( .r ` P ) = ( .r ` P )
27 1 6 26 ply1mulr 
 |-  ( .r ` P ) = ( .r ` ( 1o mPoly R ) )
28 27 oveqi 
 |-  ( x ( .r ` P ) y ) = ( x ( .r ` ( 1o mPoly R ) ) y )
29 28 a1i 
 |-  ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` P ) y ) = ( x ( .r ` ( 1o mPoly R ) ) y ) )
30 eqid 
 |-  ( .r ` Q ) = ( .r ` Q )
31 2 7 30 ply1mulr 
 |-  ( .r ` Q ) = ( .r ` ( 1o mPoly S ) )
32 31 oveqi 
 |-  ( x ( .r ` Q ) y ) = ( x ( .r ` ( 1o mPoly S ) ) y )
33 32 a1i 
 |-  ( ( ph /\ ( x e. ( Base ` Q ) /\ y e. ( Base ` Q ) ) ) -> ( x ( .r ` Q ) y ) = ( x ( .r ` ( 1o mPoly S ) ) y ) )
34 12 14 15 17 21 25 29 33 rhmpropd 
 |-  ( ph -> ( P RingHom Q ) = ( ( 1o mPoly R ) RingHom ( 1o mPoly S ) ) )
35 11 34 eleqtrrd 
 |-  ( ph -> F e. ( P RingHom Q ) )