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Metamath Proof Explorer

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 )