Metamath Proof Explorer

Theorem lmod1lem4

Description: Lemma 4 for lmod1 . (Contributed by AV, 29-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 lmod1lem4 
|- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( ( q ( .r ` ( Scalar ` M ) ) r ) ( .s ` M ) I ) = ( q ( .s ` M ) ( r ( .s ` M ) 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 ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( ( 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 ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( x e. ( Base ` R ) , y e. { I } |-> y ) = ( .s ` M ) )
9 8 eqcomd 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( .s ` M ) = ( x e. ( Base ` R ) , y e. { I } |-> y ) )
10 simprr 
 |-  ( ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) /\ ( x = q /\ y = I ) ) -> y = I )
11 simprl 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> q e. ( Base ` R ) )
12 snidg 
 |-  ( I e. V -> I e. { I } )
13 12 ad2antrr 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> I e. { I } )
14 9 10 11 13 13 ovmpod 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( q ( .s ` M ) I ) = I )
15 simprr 
 |-  ( ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) /\ ( x = r /\ y = I ) ) -> y = I )
16 simprr 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> r e. ( Base ` R ) )
17 9 15 16 13 13 ovmpod 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( r ( .s ` M ) I ) = I )
18 17 oveq2d 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( q ( .s ` M ) ( r ( .s ` M ) I ) ) = ( q ( .s ` M ) I ) )
19 simprr 
 |-  ( ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) /\ ( x = ( q ( .r ` ( Scalar ` M ) ) r ) /\ y = I ) ) -> y = I )
20 simplr 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> R e. Ring )
21 1 lmodsca 
 |-  ( R e. Ring -> R = ( Scalar ` M ) )
22 21 fveq2d 
 |-  ( R e. Ring -> ( .r ` R ) = ( .r ` ( Scalar ` M ) ) )
23 20 22 syl 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( .r ` R ) = ( .r ` ( Scalar ` M ) ) )
24 23 eqcomd 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( .r ` ( Scalar ` M ) ) = ( .r ` R ) )
25 24 oveqd 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( q ( .r ` ( Scalar ` M ) ) r ) = ( q ( .r ` R ) r ) )
26 eqid 
 |-  ( Base ` R ) = ( Base ` R )
27 eqid 
 |-  ( .r ` R ) = ( .r ` R )
28 26 27 ringcl 
 |-  ( ( R e. Ring /\ q e. ( Base ` R ) /\ r e. ( Base ` R ) ) -> ( q ( .r ` R ) r ) e. ( Base ` R ) )
29 20 11 16 28 syl3anc 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( q ( .r ` R ) r ) e. ( Base ` R ) )
30 25 29 eqeltrd 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( q ( .r ` ( Scalar ` M ) ) r ) e. ( Base ` R ) )
31 9 19 30 13 13 ovmpod 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( ( q ( .r ` ( Scalar ` M ) ) r ) ( .s ` M ) I ) = I )
32 14 18 31 3eqtr4rd 
 |-  ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( ( q ( .r ` ( Scalar ` M ) ) r ) ( .s ` M ) I ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) )