Description: Membership in the span of a singleton. (Contributed by NM, 3-Jun-2004) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypothesis | spansn.1 | ⊢ 𝐴 ∈ ℋ | |
Assertion | elspansni | ⊢ ( 𝐵 ∈ ( span ‘ { 𝐴 } ) ↔ ∃ 𝑥 ∈ ℂ 𝐵 = ( 𝑥 ·ℎ 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spansn.1 | ⊢ 𝐴 ∈ ℋ | |
2 | 1 | spansni | ⊢ ( span ‘ { 𝐴 } ) = ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) |
3 | 2 | eleq2i | ⊢ ( 𝐵 ∈ ( span ‘ { 𝐴 } ) ↔ 𝐵 ∈ ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ) |
4 | 1 | h1de2ci | ⊢ ( 𝐵 ∈ ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ↔ ∃ 𝑥 ∈ ℂ 𝐵 = ( 𝑥 ·ℎ 𝐴 ) ) |
5 | 3 4 | bitri | ⊢ ( 𝐵 ∈ ( span ‘ { 𝐴 } ) ↔ ∃ 𝑥 ∈ ℂ 𝐵 = ( 𝑥 ·ℎ 𝐴 ) ) |