This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem elspansn

Description: Membership in the span of a singleton. (Contributed by NM, 5-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion elspansn ( 𝐴 ∈ ℋ → ( 𝐵 ∈ ( span ‘ { 𝐴 } ) ↔ ∃ 𝑥 ∈ ℂ 𝐵 = ( 𝑥 ·ℎ 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 sneq ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → { 𝐴 } = { if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) } )
2 1 fveq2d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( span ‘ { 𝐴 } ) = ( span ‘ { if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) } ) )
3 2 eleq2d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐵 ∈ ( span ‘ { 𝐴 } ) ↔ 𝐵 ∈ ( span ‘ { if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) } ) ) )
4 oveq2 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝑥 ·ℎ 𝐴 ) = ( 𝑥 ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
5 4 eqeq2d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐵 = ( 𝑥 ·ℎ 𝐴 ) ↔ 𝐵 = ( 𝑥 ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
6 5 rexbidv ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ∃ 𝑥 ∈ ℂ 𝐵 = ( 𝑥 ·ℎ 𝐴 ) ↔ ∃ 𝑥 ∈ ℂ 𝐵 = ( 𝑥 ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
7 3 6 bibi12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( 𝐵 ∈ ( span ‘ { 𝐴 } ) ↔ ∃ 𝑥 ∈ ℂ 𝐵 = ( 𝑥 ·ℎ 𝐴 ) ) ↔ ( 𝐵 ∈ ( span ‘ { if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) } ) ↔ ∃ 𝑥 ∈ ℂ 𝐵 = ( 𝑥 ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) ) )
8 ifhvhv0 ⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
9 8 elspansni ⊢ ( 𝐵 ∈ ( span ‘ { if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) } ) ↔ ∃ 𝑥 ∈ ℂ 𝐵 = ( 𝑥 ·ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
10 7 9 dedth ⊢ ( 𝐴 ∈ ℋ → ( 𝐵 ∈ ( span ‘ { 𝐴 } ) ↔ ∃ 𝑥 ∈ ℂ 𝐵 = ( 𝑥 ·ℎ 𝐴 ) ) )