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Metamath Proof Explorer

Theorem dsmmbase

Description: Base set of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015)

Ref Expression
Hypothesis dsmmval.b 
|- B = { f e. ( Base ` ( S Xs_ R ) ) | { x e. dom R | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin }
Assertion dsmmbase 
|- ( R e. V -> B = ( Base ` ( S (+)m R ) ) )

Proof

Step Hyp Ref Expression
1 dsmmval.b 
 |-  B = { f e. ( Base ` ( S Xs_ R ) ) | { x e. dom R | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin }
2 elex 
 |-  ( R e. V -> R e. _V )
3 1 ssrab3 
 |-  B C_ ( Base ` ( S Xs_ R ) )
4 eqid 
 |-  ( ( S Xs_ R ) |`s B ) = ( ( S Xs_ R ) |`s B )
5 eqid 
 |-  ( Base ` ( S Xs_ R ) ) = ( Base ` ( S Xs_ R ) )
6 4 5 ressbas2 
 |-  ( B C_ ( Base ` ( S Xs_ R ) ) -> B = ( Base ` ( ( S Xs_ R ) |`s B ) ) )
7 3 6 ax-mp 
 |-  B = ( Base ` ( ( S Xs_ R ) |`s B ) )
8 1 dsmmval 
 |-  ( R e. _V -> ( S (+)m R ) = ( ( S Xs_ R ) |`s B ) )
9 8 fveq2d 
 |-  ( R e. _V -> ( Base ` ( S (+)m R ) ) = ( Base ` ( ( S Xs_ R ) |`s B ) ) )
10 7 9 eqtr4id 
 |-  ( R e. _V -> B = ( Base ` ( S (+)m R ) ) )
11 2 10 syl 
 |-  ( R e. V -> B = ( Base ` ( S (+)m R ) ) )