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Metamath Proof Explorer

Theorem evls1rhmlem

Description: Lemma for evl1rhm and evls1rhm (formerly part of the proof of evl1rhm ): The first function of the composition forming the univariate polynomial evaluation map function for a (sub)ring is a ring homomorphism. (Contributed by AV, 11-Sep-2019)

Ref Expression
Hypotheses evl1rhmlem.b 
|- B = ( Base ` R )
evl1rhmlem.t 
|- T = ( R ^s B )
evl1rhmlem.f 
|- F = ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) )
Assertion evls1rhmlem 
|- ( R e. CRing -> F e. ( ( R ^s ( B ^m 1o ) ) RingHom T ) )

Proof

Step Hyp Ref Expression
1 evl1rhmlem.b 
 |-  B = ( Base ` R )
2 evl1rhmlem.t 
 |-  T = ( R ^s B )
3 evl1rhmlem.f 
 |-  F = ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) )
4 ovex 
 |-  ( B ^m 1o ) e. _V
5 eqid 
 |-  ( R ^s ( B ^m 1o ) ) = ( R ^s ( B ^m 1o ) )
6 5 1 pwsbas 
 |-  ( ( R e. CRing /\ ( B ^m 1o ) e. _V ) -> ( B ^m ( B ^m 1o ) ) = ( Base ` ( R ^s ( B ^m 1o ) ) ) )
7 4 6 mpan2 
 |-  ( R e. CRing -> ( B ^m ( B ^m 1o ) ) = ( Base ` ( R ^s ( B ^m 1o ) ) ) )
8 7 mpteq1d 
 |-  ( R e. CRing -> ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) = ( x e. ( Base ` ( R ^s ( B ^m 1o ) ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) )
9 3 8 eqtrid 
 |-  ( R e. CRing -> F = ( x e. ( Base ` ( R ^s ( B ^m 1o ) ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) )
10 eqid 
 |-  ( Base ` ( R ^s ( B ^m 1o ) ) ) = ( Base ` ( R ^s ( B ^m 1o ) ) )
11 crngring 
 |-  ( R e. CRing -> R e. Ring )
12 1 fvexi 
 |-  B e. _V
13 12 a1i 
 |-  ( R e. CRing -> B e. _V )
14 4 a1i 
 |-  ( R e. CRing -> ( B ^m 1o ) e. _V )
15 df1o2 
 |-  1o = { (/) }
16 0ex 
 |-  (/) e. _V
17 eqid 
 |-  ( y e. B |-> ( 1o X. { y } ) ) = ( y e. B |-> ( 1o X. { y } ) )
18 15 12 16 17 mapsnf1o3 
 |-  ( y e. B |-> ( 1o X. { y } ) ) : B -1-1-onto-> ( B ^m 1o )
19 f1of 
 |-  ( ( y e. B |-> ( 1o X. { y } ) ) : B -1-1-onto-> ( B ^m 1o ) -> ( y e. B |-> ( 1o X. { y } ) ) : B --> ( B ^m 1o ) )
20 18 19 mp1i 
 |-  ( R e. CRing -> ( y e. B |-> ( 1o X. { y } ) ) : B --> ( B ^m 1o ) )
21 2 5 10 11 13 14 20 pwsco1rhm 
 |-  ( R e. CRing -> ( x e. ( Base ` ( R ^s ( B ^m 1o ) ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) e. ( ( R ^s ( B ^m 1o ) ) RingHom T ) )
22 9 21 eqeltrd 
 |-  ( R e. CRing -> F e. ( ( R ^s ( B ^m 1o ) ) RingHom T ) )