Description: Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypotheses | hlmul0.1 | ⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) | |
hlmul0.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 ) | ||
hlmul0.5 | ⊢ 𝑍 = ( 0vec ‘ 𝑈 ) | ||
Assertion | hlmul0 | ⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ) → ( 0 𝑆 𝐴 ) = 𝑍 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlmul0.1 | ⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) | |
2 | hlmul0.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 ) | |
3 | hlmul0.5 | ⊢ 𝑍 = ( 0vec ‘ 𝑈 ) | |
4 | hlnv | ⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec ) | |
5 | 1 2 3 | nv0 | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 0 𝑆 𝐴 ) = 𝑍 ) |
6 | 4 5 | sylan | ⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ) → ( 0 𝑆 𝐴 ) = 𝑍 ) |