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	⊢ · = ( ·𝑠 ‘ 𝑊 )
	lmodvs0.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
	lmodvs0.z	⊢ 0 = ( 0g ‘ 𝑊 )
Assertion	lmodvs0	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ) → ( 𝑋 · 0 ) = 0 )

Proof

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