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Metamath Proof Explorer

Theorem rhmpsr1

Description: Provide a ring homomorphism between two univariate power series algebras over their respective base rings given a ring homomorphism between the two base rings. (Contributed by SN, 8-Feb-2025)

Ref Expression
Hypotheses rhmpsr1.p 
|- P = ( PwSer1 ` R )
rhmpsr1.q 
|- Q = ( PwSer1 ` S )
rhmpsr1.b 
|- B = ( Base ` P )
rhmpsr1.f 
|- F = ( p e. B |-> ( H o. p ) )
rhmpsr1.h 
|- ( ph -> H e. ( R RingHom S ) )
Assertion rhmpsr1 
|- ( ph -> F e. ( P RingHom Q ) )

Proof

Step Hyp Ref Expression
1 rhmpsr1.p 
 |-  P = ( PwSer1 ` R )
2 rhmpsr1.q 
 |-  Q = ( PwSer1 ` S )
3 rhmpsr1.b 
 |-  B = ( Base ` P )
4 rhmpsr1.f 
 |-  F = ( p e. B |-> ( H o. p ) )
5 rhmpsr1.h 
 |-  ( ph -> H e. ( R RingHom S ) )
6 eqid 
 |-  ( 1o mPwSer R ) = ( 1o mPwSer R )
7 eqid 
 |-  ( 1o mPwSer S ) = ( 1o mPwSer S )
8 1 3 6 psr1bas2 
 |-  B = ( Base ` ( 1o mPwSer R ) )
9 1oex 
 |-  1o e. _V
10 9 a1i 
 |-  ( ph -> 1o e. _V )
11 6 7 8 4 10 5 rhmpsr 
 |-  ( ph -> F e. ( ( 1o mPwSer R ) RingHom ( 1o mPwSer S ) ) )
12 eqid 
 |-  ( Base ` P ) = ( Base ` P )
13 1 12 6 psr1bas2 
 |-  ( Base ` P ) = ( Base ` ( 1o mPwSer R ) )
14 13 a1i 
 |-  ( ph -> ( Base ` P ) = ( Base ` ( 1o mPwSer R ) ) )
15 eqid 
 |-  ( Base ` Q ) = ( Base ` Q )
16 2 15 7 psr1bas2 
 |-  ( Base ` Q ) = ( Base ` ( 1o mPwSer S ) )
17 16 a1i 
 |-  ( ph -> ( Base ` Q ) = ( Base ` ( 1o mPwSer S ) ) )
18 eqidd 
 |-  ( ph -> ( Base ` P ) = ( Base ` P ) )
19 eqidd 
 |-  ( ph -> ( Base ` Q ) = ( Base ` Q ) )
20 eqid 
 |-  ( +g ` P ) = ( +g ` P )
21 1 6 20 psr1plusg 
 |-  ( +g ` P ) = ( +g ` ( 1o mPwSer R ) )
22 21 eqcomi 
 |-  ( +g ` ( 1o mPwSer R ) ) = ( +g ` P )
23 22 a1i 
 |-  ( ( ph /\ ( x e. ( Base ` P ) /\ y e. ( Base ` P ) ) ) -> ( +g ` ( 1o mPwSer R ) ) = ( +g ` P ) )
24 23 oveqd 
 |-  ( ( ph /\ ( x e. ( Base ` P ) /\ y e. ( Base ` P ) ) ) -> ( x ( +g ` ( 1o mPwSer R ) ) y ) = ( x ( +g ` P ) y ) )
25 eqid 
 |-  ( +g ` Q ) = ( +g ` Q )
26 2 7 25 psr1plusg 
 |-  ( +g ` Q ) = ( +g ` ( 1o mPwSer S ) )
27 26 eqcomi 
 |-  ( +g ` ( 1o mPwSer S ) ) = ( +g ` Q )
28 27 a1i 
 |-  ( ( ph /\ ( x e. ( Base ` Q ) /\ y e. ( Base ` Q ) ) ) -> ( +g ` ( 1o mPwSer S ) ) = ( +g ` Q ) )
29 28 oveqd 
 |-  ( ( ph /\ ( x e. ( Base ` Q ) /\ y e. ( Base ` Q ) ) ) -> ( x ( +g ` ( 1o mPwSer S ) ) y ) = ( x ( +g ` Q ) y ) )
30 eqid 
 |-  ( .r ` P ) = ( .r ` P )
31 1 6 30 psr1mulr 
 |-  ( .r ` P ) = ( .r ` ( 1o mPwSer R ) )
32 31 eqcomi 
 |-  ( .r ` ( 1o mPwSer R ) ) = ( .r ` P )
33 32 a1i 
 |-  ( ( ph /\ ( x e. ( Base ` P ) /\ y e. ( Base ` P ) ) ) -> ( .r ` ( 1o mPwSer R ) ) = ( .r ` P ) )
34 33 oveqd 
 |-  ( ( ph /\ ( x e. ( Base ` P ) /\ y e. ( Base ` P ) ) ) -> ( x ( .r ` ( 1o mPwSer R ) ) y ) = ( x ( .r ` P ) y ) )
35 eqid 
 |-  ( .r ` Q ) = ( .r ` Q )
36 2 7 35 psr1mulr 
 |-  ( .r ` Q ) = ( .r ` ( 1o mPwSer S ) )
37 36 eqcomi 
 |-  ( .r ` ( 1o mPwSer S ) ) = ( .r ` Q )
38 37 a1i 
 |-  ( ( ph /\ ( x e. ( Base ` Q ) /\ y e. ( Base ` Q ) ) ) -> ( .r ` ( 1o mPwSer S ) ) = ( .r ` Q ) )
39 38 oveqd 
 |-  ( ( ph /\ ( x e. ( Base ` Q ) /\ y e. ( Base ` Q ) ) ) -> ( x ( .r ` ( 1o mPwSer S ) ) y ) = ( x ( .r ` Q ) y ) )
40 14 17 18 19 24 29 34 39 rhmpropd 
 |-  ( ph -> ( ( 1o mPwSer R ) RingHom ( 1o mPwSer S ) ) = ( P RingHom Q ) )
41 11 40 eleqtrd 
 |-  ( ph -> F e. ( P RingHom Q ) )