slmdvs0

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) (Revised by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	slmdvs0.f	\|- F = ( Scalar ` W )
	slmdvs0.s	\|- .x. = ( .s ` W )
	slmdvs0.k	\|- K = ( Base ` F )
	slmdvs0.z	\|- .0. = ( 0g ` W )
Assertion	slmdvs0	\|- ( ( W e. SLMod /\ X e. K ) -> ( X .x. .0. ) = .0. )

Proof

Step	Hyp	Ref	Expression
1		slmdvs0.f	\|- F = ( Scalar ` W )
2		slmdvs0.s	\|- .x. = ( .s ` W )
3		slmdvs0.k	\|- K = ( Base ` F )
4		slmdvs0.z	\|- .0. = ( 0g ` W )
5	1	slmdsrg	\|- ( W e. SLMod -> F e. SRing )
6		eqid	\|- ( .r ` F ) = ( .r ` F )
7		eqid	\|- ( 0g ` F ) = ( 0g ` F )
8	3 6 7	srgrz	\|- ( ( F e. SRing /\ X e. K ) -> ( X ( .r ` F ) ( 0g ` F ) ) = ( 0g ` F ) )
9	5 8	sylan	\|- ( ( W e. SLMod /\ X e. K ) -> ( X ( .r ` F ) ( 0g ` F ) ) = ( 0g ` F ) )
10	9	oveq1d	\|- ( ( W e. SLMod /\ X e. K ) -> ( ( X ( .r ` F ) ( 0g ` F ) ) .x. .0. ) = ( ( 0g ` F ) .x. .0. ) )
11		simpl	\|- ( ( W e. SLMod /\ X e. K ) -> W e. SLMod )
12		simpr	\|- ( ( W e. SLMod /\ X e. K ) -> X e. K )
13	5	adantr	\|- ( ( W e. SLMod /\ X e. K ) -> F e. SRing )
14	3 7	srg0cl	\|- ( F e. SRing -> ( 0g ` F ) e. K )
15	13 14	syl	\|- ( ( W e. SLMod /\ X e. K ) -> ( 0g ` F ) e. K )
16		eqid	\|- ( Base ` W ) = ( Base ` W )
17	16 4	slmd0vcl	\|- ( W e. SLMod -> .0. e. ( Base ` W ) )
18	17	adantr	\|- ( ( W e. SLMod /\ X e. K ) -> .0. e. ( Base ` W ) )
19	16 1 2 3 6	slmdvsass	\|- ( ( W e. SLMod /\ ( 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. SLMod /\ X e. K ) -> ( ( X ( .r ` F ) ( 0g ` F ) ) .x. .0. ) = ( X .x. ( ( 0g ` F ) .x. .0. ) ) )
21	16 1 2 7 4	slmd0vs	\|- ( ( W e. SLMod /\ .0. e. ( Base ` W ) ) -> ( ( 0g ` F ) .x. .0. ) = .0. )
22	18 21	syldan	\|- ( ( W e. SLMod /\ X e. K ) -> ( ( 0g ` F ) .x. .0. ) = .0. )
23	22	oveq2d	\|- ( ( W e. SLMod /\ X e. K ) -> ( X .x. ( ( 0g ` F ) .x. .0. ) ) = ( X .x. .0. ) )
24	20 23	eqtrd	\|- ( ( W e. SLMod /\ X e. K ) -> ( ( X ( .r ` F ) ( 0g ` F ) ) .x. .0. ) = ( X .x. .0. ) )
25	10 24 22	3eqtr3d	\|- ( ( W e. SLMod /\ X e. K ) -> ( X .x. .0. ) = .0. )

Metamath Proof Explorer

Theorem slmdvs0

Proof