Metamath Proof Explorer

Theorem lmod1lem5

Description: Lemma 5 for lmod1 . (Contributed by AV, 28-Apr-2019)

Ref Expression
Hypothesis lmod1.m 
|- M = ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. ( Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( x e. ( Base ` R ) , y e. { I } |-> y ) >. } )
Assertion lmod1lem5 
|- ( ( I e. V /\ R e. Ring ) -> ( ( 1r ` ( Scalar ` M ) ) ( .s ` M ) I ) = I )

Proof

Step Hyp Ref Expression
1 lmod1.m 
 |-  M = ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. ( Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( x e. ( Base ` R ) , y e. { I } |-> y ) >. } )
2 fvex 
 |-  ( Base ` R ) e. _V
3 snex 
 |-  { I } e. _V
4 2 3 pm3.2i 
 |-  ( ( Base ` R ) e. _V /\ { I } e. _V )
5 4 a1i 
 |-  ( ( I e. V /\ R e. Ring ) -> ( ( Base ` R ) e. _V /\ { I } e. _V ) )
6 mpoexga 
 |-  ( ( ( Base ` R ) e. _V /\ { I } e. _V ) -> ( x e. ( Base ` R ) , y e. { I } |-> y ) e. _V )
7 1 lmodvsca 
 |-  ( ( x e. ( Base ` R ) , y e. { I } |-> y ) e. _V -> ( x e. ( Base ` R ) , y e. { I } |-> y ) = ( .s ` M ) )
8 5 6 7 3syl 
 |-  ( ( I e. V /\ R e. Ring ) -> ( x e. ( Base ` R ) , y e. { I } |-> y ) = ( .s ` M ) )
9 8 eqcomd 
 |-  ( ( I e. V /\ R e. Ring ) -> ( .s ` M ) = ( x e. ( Base ` R ) , y e. { I } |-> y ) )
10 simprr 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( x = ( 1r ` ( Scalar ` M ) ) /\ y = I ) ) -> y = I )
11 1 lmodsca 
 |-  ( R e. Ring -> R = ( Scalar ` M ) )
12 11 adantl 
 |-  ( ( I e. V /\ R e. Ring ) -> R = ( Scalar ` M ) )
13 12 eqcomd 
 |-  ( ( I e. V /\ R e. Ring ) -> ( Scalar ` M ) = R )
14 13 fveq2d 
 |-  ( ( I e. V /\ R e. Ring ) -> ( 1r ` ( Scalar ` M ) ) = ( 1r ` R ) )
15 eqid 
 |-  ( Base ` R ) = ( Base ` R )
16 eqid 
 |-  ( 1r ` R ) = ( 1r ` R )
17 15 16 ringidcl 
 |-  ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) )
18 17 adantl 
 |-  ( ( I e. V /\ R e. Ring ) -> ( 1r ` R ) e. ( Base ` R ) )
19 14 18 eqeltrd 
 |-  ( ( I e. V /\ R e. Ring ) -> ( 1r ` ( Scalar ` M ) ) e. ( Base ` R ) )
20 snidg 
 |-  ( I e. V -> I e. { I } )
21 20 adantr 
 |-  ( ( I e. V /\ R e. Ring ) -> I e. { I } )
22 9 10 19 21 21 ovmpod 
 |-  ( ( I e. V /\ R e. Ring ) -> ( ( 1r ` ( Scalar ` M ) ) ( .s ` M ) I ) = I )