This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lmodvs0

Description: Anything times the zero vector is the zero vector. Equation 1b of Kreyszig p. 51. ( hvmul0 analog.) (Contributed by NM, 12-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014)

Ref Expression
Hypotheses lmodvs0.f 
|- F = ( Scalar ` W )
lmodvs0.s 
|- .x. = ( .s ` W )
lmodvs0.k 
|- K = ( Base ` F )
lmodvs0.z 
|- .0. = ( 0g ` W )
Assertion lmodvs0 
|- ( ( W e. LMod /\ X e. K ) -> ( X .x. .0. ) = .0. )

Proof

Step Hyp Ref Expression
1 lmodvs0.f 
 |-  F = ( Scalar ` W )
2 lmodvs0.s 
 |-  .x. = ( .s ` W )
3 lmodvs0.k 
 |-  K = ( Base ` F )
4 lmodvs0.z 
 |-  .0. = ( 0g ` W )
5 1 lmodring 
 |-  ( W e. LMod -> F e. Ring )
6 eqid 
 |-  ( .r ` F ) = ( .r ` F )
7 eqid 
 |-  ( 0g ` F ) = ( 0g ` F )
8 3 6 7 ringrz 
 |-  ( ( F e. Ring /\ X e. K ) -> ( X ( .r ` F ) ( 0g ` F ) ) = ( 0g ` F ) )
9 5 8 sylan 
 |-  ( ( W e. LMod /\ X e. K ) -> ( X ( .r ` F ) ( 0g ` F ) ) = ( 0g ` F ) )
10 9 oveq1d 
 |-  ( ( W e. LMod /\ X e. K ) -> ( ( X ( .r ` F ) ( 0g ` F ) ) .x. .0. ) = ( ( 0g ` F ) .x. .0. ) )
11 simpl 
 |-  ( ( W e. LMod /\ X e. K ) -> W e. LMod )
12 simpr 
 |-  ( ( W e. LMod /\ X e. K ) -> X e. K )
13 5 adantr 
 |-  ( ( W e. LMod /\ X e. K ) -> F e. Ring )
14 3 7 ring0cl 
 |-  ( F e. Ring -> ( 0g ` F ) e. K )
15 13 14 syl 
 |-  ( ( W e. LMod /\ X e. K ) -> ( 0g ` F ) e. K )
16 eqid 
 |-  ( Base ` W ) = ( Base ` W )
17 16 4 lmod0vcl 
 |-  ( W e. LMod -> .0. e. ( Base ` W ) )
18 17 adantr 
 |-  ( ( W e. LMod /\ X e. K ) -> .0. e. ( Base ` W ) )
19 16 1 2 3 6 lmodvsass 
 |-  ( ( W e. LMod /\ ( X e. K /\ ( 0g ` F ) e. K /\ .0. e. ( Base ` W ) ) ) -> ( ( X ( .r ` F ) ( 0g ` F ) ) .x. .0. ) = ( X .x. ( ( 0g ` F ) .x. .0. ) ) )
20 11 12 15 18 19 syl13anc 
 |-  ( ( W e. LMod /\ X e. K ) -> ( ( X ( .r ` F ) ( 0g ` F ) ) .x. .0. ) = ( X .x. ( ( 0g ` F ) .x. .0. ) ) )
21 16 1 2 7 4 lmod0vs 
 |-  ( ( W e. LMod /\ .0. e. ( Base ` W ) ) -> ( ( 0g ` F ) .x. .0. ) = .0. )
22 18 21 syldan 
 |-  ( ( W e. LMod /\ X e. K ) -> ( ( 0g ` F ) .x. .0. ) = .0. )
23 22 oveq2d 
 |-  ( ( W e. LMod /\ X e. K ) -> ( X .x. ( ( 0g ` F ) .x. .0. ) ) = ( X .x. .0. ) )
24 20 23 eqtrd 
 |-  ( ( W e. LMod /\ X e. K ) -> ( ( X ( .r ` F ) ( 0g ` F ) ) .x. .0. ) = ( X .x. .0. ) )
25 10 24 22 3eqtr3d 
 |-  ( ( W e. LMod /\ X e. K ) -> ( X .x. .0. ) = .0. )