Description: A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypothesis | ch0le.1 | ⊢ 𝐴 ∈ Cℋ | |
Assertion | chne0i | ⊢ ( 𝐴 ≠ 0ℋ ↔ ∃ 𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ch0le.1 | ⊢ 𝐴 ∈ Cℋ | |
2 | 1 | chshii | ⊢ 𝐴 ∈ Sℋ |
3 | 2 | shne0i | ⊢ ( 𝐴 ≠ 0ℋ ↔ ∃ 𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ ) |