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Metamath Proof Explorer

Theorem spansnmul

Description: A scalar product with a vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion spansnmul ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ) → ( 𝐵 ·ℎ 𝐴 ) ∈ ( span ‘ { 𝐴 } ) )

Proof

Step Hyp Ref Expression
1 spansnsh ⊢ ( 𝐴 ∈ ℋ → ( span ‘ { 𝐴 } ) ∈ Sℋ )
2 spansnid ⊢ ( 𝐴 ∈ ℋ → 𝐴 ∈ ( span ‘ { 𝐴 } ) )
3 1 2 jca ⊢ ( 𝐴 ∈ ℋ → ( ( span ‘ { 𝐴 } ) ∈ Sℋ ∧ 𝐴 ∈ ( span ‘ { 𝐴 } ) ) )
4 shmulcl ⊢ ( ( ( span ‘ { 𝐴 } ) ∈ Sℋ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ( span ‘ { 𝐴 } ) ) → ( 𝐵 ·ℎ 𝐴 ) ∈ ( span ‘ { 𝐴 } ) )
5 4 3com12 ⊢ ( ( 𝐵 ∈ ℂ ∧ ( span ‘ { 𝐴 } ) ∈ Sℋ ∧ 𝐴 ∈ ( span ‘ { 𝐴 } ) ) → ( 𝐵 ·ℎ 𝐴 ) ∈ ( span ‘ { 𝐴 } ) )
6 5 3expb ⊢ ( ( 𝐵 ∈ ℂ ∧ ( ( span ‘ { 𝐴 } ) ∈ Sℋ ∧ 𝐴 ∈ ( span ‘ { 𝐴 } ) ) ) → ( 𝐵 ·ℎ 𝐴 ) ∈ ( span ‘ { 𝐴 } ) )
7 3 6 sylan2 ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ ) → ( 𝐵 ·ℎ 𝐴 ) ∈ ( span ‘ { 𝐴 } ) )
8 7 ancoms ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ) → ( 𝐵 ·ℎ 𝐴 ) ∈ ( span ‘ { 𝐴 } ) )