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. )