Theorem slmd0vs

Description: Zero times a vector is the zero vector. Equation 1a of Kreyszig p. 51. ( ax-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	slmd0vs.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	slmd0vs.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	slmd0vs.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
	slmd0vs.o	⊢ 𝑂 = ( 0g ‘ 𝐹 )
	slmd0vs.z	⊢ 0 = ( 0g ‘ 𝑊 )
Assertion	slmd0vs	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑂 · 𝑋 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		slmd0vs.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		slmd0vs.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
3		slmd0vs.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
4		slmd0vs.o	⊢ 𝑂 = ( 0g ‘ 𝐹 )
5		slmd0vs.z	⊢ 0 = ( 0g ‘ 𝑊 )
6		simpl	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → 𝑊 ∈ SLMod )
7		eqid	⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 )
8	2 7 4	slmd0cl	⊢ ( 𝑊 ∈ SLMod → 𝑂 ∈ ( Base ‘ 𝐹 ) )
9	8	adantr	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → 𝑂 ∈ ( Base ‘ 𝐹 ) )
10		simpr	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ 𝑉 )
11		eqid	⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 )
12		eqid	⊢ ( +g ‘ 𝐹 ) = ( +g ‘ 𝐹 )
13		eqid	⊢ ( .r ‘ 𝐹 ) = ( .r ‘ 𝐹 )
14		eqid	⊢ ( 1r ‘ 𝐹 ) = ( 1r ‘ 𝐹 )
15	1 11 3 5 2 7 12 13 14 4	slmdlema	⊢ ( ( 𝑊 ∈ SLMod ∧ ( 𝑂 ∈ ( Base ‘ 𝐹 ) ∧ 𝑂 ∈ ( Base ‘ 𝐹 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( ( 𝑂 · 𝑋 ) ∈ 𝑉 ∧ ( 𝑂 · ( 𝑋 ( +g ‘ 𝑊 ) 𝑋 ) ) = ( ( 𝑂 · 𝑋 ) ( +g ‘ 𝑊 ) ( 𝑂 · 𝑋 ) ) ∧ ( ( 𝑂 ( +g ‘ 𝐹 ) 𝑂 ) · 𝑋 ) = ( ( 𝑂 · 𝑋 ) ( +g ‘ 𝑊 ) ( 𝑂 · 𝑋 ) ) ) ∧ ( ( ( 𝑂 ( .r ‘ 𝐹 ) 𝑂 ) · 𝑋 ) = ( 𝑂 · ( 𝑂 · 𝑋 ) ) ∧ ( ( 1r ‘ 𝐹 ) · 𝑋 ) = 𝑋 ∧ ( 𝑂 · 𝑋 ) = 0 ) ) )
16	6 9 9 10 10 15	syl122anc	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → ( ( ( 𝑂 · 𝑋 ) ∈ 𝑉 ∧ ( 𝑂 · ( 𝑋 ( +g ‘ 𝑊 ) 𝑋 ) ) = ( ( 𝑂 · 𝑋 ) ( +g ‘ 𝑊 ) ( 𝑂 · 𝑋 ) ) ∧ ( ( 𝑂 ( +g ‘ 𝐹 ) 𝑂 ) · 𝑋 ) = ( ( 𝑂 · 𝑋 ) ( +g ‘ 𝑊 ) ( 𝑂 · 𝑋 ) ) ) ∧ ( ( ( 𝑂 ( .r ‘ 𝐹 ) 𝑂 ) · 𝑋 ) = ( 𝑂 · ( 𝑂 · 𝑋 ) ) ∧ ( ( 1r ‘ 𝐹 ) · 𝑋 ) = 𝑋 ∧ ( 𝑂 · 𝑋 ) = 0 ) ) )
17	16	simprd	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → ( ( ( 𝑂 ( .r ‘ 𝐹 ) 𝑂 ) · 𝑋 ) = ( 𝑂 · ( 𝑂 · 𝑋 ) ) ∧ ( ( 1r ‘ 𝐹 ) · 𝑋 ) = 𝑋 ∧ ( 𝑂 · 𝑋 ) = 0 ) )
18	17	simp3d	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑂 · 𝑋 ) = 0 )