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

Metamath Proof Explorer

Theorem lsatlspsn

Description: The span of a nonzero singleton is an atom. (Contributed by NM, 16-Jan-2015)

Ref Expression
Hypotheses lsatset.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
lsatset.n ⊢ 𝑁 = ( LSpan ‘ 𝑊 )
lsatset.z ⊢ 0 = ( 0g ‘ 𝑊 )
lsatset.a ⊢ 𝐴 = ( LSAtoms ‘ 𝑊 )
lsatlspsn.w ⊢ ( 𝜑 → 𝑊 ∈ LMod )
lsatlspsn.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion lsatlspsn ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ 𝐴 )

Proof

Step Hyp Ref Expression
1 lsatset.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 lsatset.n ⊢ 𝑁 = ( LSpan ‘ 𝑊 )
3 lsatset.z ⊢ 0 = ( 0g ‘ 𝑊 )
4 lsatset.a ⊢ 𝐴 = ( LSAtoms ‘ 𝑊 )
5 lsatlspsn.w ⊢ ( 𝜑 → 𝑊 ∈ LMod )
6 lsatlspsn.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
7 eqid ⊢ ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑋 } )
8 sneq ⊢ ( 𝑣 = 𝑋 → { 𝑣 } = { 𝑋 } )
9 8 fveq2d ⊢ ( 𝑣 = 𝑋 → ( 𝑁 ‘ { 𝑣 } ) = ( 𝑁 ‘ { 𝑋 } ) )
10 9 rspceeqv ⊢ ( ( 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑋 } ) ) → ∃ 𝑣 ∈ ( 𝑉 ∖ { 0 } ) ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑣 } ) )
11 6 7 10 sylancl ⊢ ( 𝜑 → ∃ 𝑣 ∈ ( 𝑉 ∖ { 0 } ) ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑣 } ) )
12 1 2 3 4 islsat ⊢ ( 𝑊 ∈ LMod → ( ( 𝑁 ‘ { 𝑋 } ) ∈ 𝐴 ↔ ∃ 𝑣 ∈ ( 𝑉 ∖ { 0 } ) ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑣 } ) ) )
13 5 12 syl ⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ∈ 𝐴 ↔ ∃ 𝑣 ∈ ( 𝑉 ∖ { 0 } ) ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑣 } ) ) )
14 11 13 mpbird ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ 𝐴 )