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Metamath Proof Explorer

Theorem frlm0vald

Description: All coordinates of the zero vector are zero. (Contributed by SN, 14-Aug-2024)

Ref Expression
Hypotheses frlm0vald.f 
|- F = ( R freeLMod I )
frlm0vald.0 
|- .0. = ( 0g ` R )
frlm0vald.r 
|- ( ph -> R e. Ring )
frlm0vald.i 
|- ( ph -> I e. W )
frlm0vald.j 
|- ( ph -> J e. I )
Assertion frlm0vald 
|- ( ph -> ( ( 0g ` F ) ` J ) = .0. )

Proof

Step Hyp Ref Expression
1 frlm0vald.f 
 |-  F = ( R freeLMod I )
2 frlm0vald.0 
 |-  .0. = ( 0g ` R )
3 frlm0vald.r 
 |-  ( ph -> R e. Ring )
4 frlm0vald.i 
 |-  ( ph -> I e. W )
5 frlm0vald.j 
 |-  ( ph -> J e. I )
6 1 2 frlm0 
 |-  ( ( R e. Ring /\ I e. W ) -> ( I X. { .0. } ) = ( 0g ` F ) )
7 3 4 6 syl2anc 
 |-  ( ph -> ( I X. { .0. } ) = ( 0g ` F ) )
8 7 fveq1d 
 |-  ( ph -> ( ( I X. { .0. } ) ` J ) = ( ( 0g ` F ) ` J ) )
9 2 fvexi 
 |-  .0. e. _V
10 9 fvconst2 
 |-  ( J e. I -> ( ( I X. { .0. } ) ` J ) = .0. )
11 5 10 syl 
 |-  ( ph -> ( ( I X. { .0. } ) ` J ) = .0. )
12 8 11 eqtr3d 
 |-  ( ph -> ( ( 0g ` F ) ` J ) = .0. )