Metamath Proof Explorer

Theorem pws0g

Description: Zero in a structure power of a monoid. (Contributed by Mario Carneiro, 11-Jan-2015)

Ref Expression
Hypotheses pwsmnd.y 
|- Y = ( R ^s I )
pws0g.z 
|- .0. = ( 0g ` R )
Assertion pws0g 
|- ( ( R e. Mnd /\ I e. V ) -> ( I X. { .0. } ) = ( 0g ` Y ) )

Proof

Step Hyp Ref Expression
1 pwsmnd.y 
 |-  Y = ( R ^s I )
2 pws0g.z 
 |-  .0. = ( 0g ` R )
3 eqid 
 |-  ( ( Scalar ` R ) Xs_ ( I X. { R } ) ) = ( ( Scalar ` R ) Xs_ ( I X. { R } ) )
4 simpr 
 |-  ( ( R e. Mnd /\ I e. V ) -> I e. V )
5 fvexd 
 |-  ( ( R e. Mnd /\ I e. V ) -> ( Scalar ` R ) e. _V )
6 fconst6g 
 |-  ( R e. Mnd -> ( I X. { R } ) : I --> Mnd )
7 6 adantr 
 |-  ( ( R e. Mnd /\ I e. V ) -> ( I X. { R } ) : I --> Mnd )
8 3 4 5 7 prds0g 
 |-  ( ( R e. Mnd /\ I e. V ) -> ( 0g o. ( I X. { R } ) ) = ( 0g ` ( ( Scalar ` R ) Xs_ ( I X. { R } ) ) ) )
9 fconstmpt 
 |-  ( I X. { .0. } ) = ( x e. I |-> .0. )
10 elex 
 |-  ( R e. Mnd -> R e. _V )
11 10 ad2antrr 
 |-  ( ( ( R e. Mnd /\ I e. V ) /\ x e. I ) -> R e. _V )
12 fconstmpt 
 |-  ( I X. { R } ) = ( x e. I |-> R )
13 12 a1i 
 |-  ( ( R e. Mnd /\ I e. V ) -> ( I X. { R } ) = ( x e. I |-> R ) )
14 fn0g 
 |-  0g Fn _V
15 14 a1i 
 |-  ( ( R e. Mnd /\ I e. V ) -> 0g Fn _V )
16 dffn5 
 |-  ( 0g Fn _V <-> 0g = ( r e. _V |-> ( 0g ` r ) ) )
17 15 16 sylib 
 |-  ( ( R e. Mnd /\ I e. V ) -> 0g = ( r e. _V |-> ( 0g ` r ) ) )
18 fveq2 
 |-  ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
19 18 2 eqtr4di 
 |-  ( r = R -> ( 0g ` r ) = .0. )
20 11 13 17 19 fmptco 
 |-  ( ( R e. Mnd /\ I e. V ) -> ( 0g o. ( I X. { R } ) ) = ( x e. I |-> .0. ) )
21 9 20 eqtr4id 
 |-  ( ( R e. Mnd /\ I e. V ) -> ( I X. { .0. } ) = ( 0g o. ( I X. { R } ) ) )
22 eqid 
 |-  ( Scalar ` R ) = ( Scalar ` R )
23 1 22 pwsval 
 |-  ( ( R e. Mnd /\ I e. V ) -> Y = ( ( Scalar ` R ) Xs_ ( I X. { R } ) ) )
24 23 fveq2d 
 |-  ( ( R e. Mnd /\ I e. V ) -> ( 0g ` Y ) = ( 0g ` ( ( Scalar ` R ) Xs_ ( I X. { R } ) ) ) )
25 8 21 24 3eqtr4d 
 |-  ( ( R e. Mnd /\ I e. V ) -> ( I X. { .0. } ) = ( 0g ` Y ) )