Theorem lsatlspsn2

Description: The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn instead? (Contributed by NM, 9-Apr-2014) (Revised by Mario Carneiro, 24-Jun-2014)

	Ref	Expression
Hypotheses	lsatset.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lsatset.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
	lsatset.z	⊢ 0 = ( 0g ‘ 𝑊 )
	lsatset.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑊 )
Assertion	lsatlspsn2	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → ( 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		3simpc	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → ( 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) )
6		eldifsn	⊢ ( 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ↔ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) )
7	5 6	sylibr	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
8		eqid	⊢ ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑋 } )
9		sneq	⊢ ( 𝑣 = 𝑋 → { 𝑣 } = { 𝑋 } )
10	9	fveq2d	⊢ ( 𝑣 = 𝑋 → ( 𝑁 ‘ { 𝑣 } ) = ( 𝑁 ‘ { 𝑋 } ) )
11	10	rspceeqv	⊢ ( ( 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑋 } ) ) → ∃ 𝑣 ∈ ( 𝑉 ∖ { 0 } ) ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑣 } ) )
12	7 8 11	sylancl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → ∃ 𝑣 ∈ ( 𝑉 ∖ { 0 } ) ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑣 } ) )
13	1 2 3 4	islsat	⊢ ( 𝑊 ∈ LMod → ( ( 𝑁 ‘ { 𝑋 } ) ∈ 𝐴 ↔ ∃ 𝑣 ∈ ( 𝑉 ∖ { 0 } ) ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑣 } ) ) )
14	13	3ad2ant1	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → ( ( 𝑁 ‘ { 𝑋 } ) ∈ 𝐴 ↔ ∃ 𝑣 ∈ ( 𝑉 ∖ { 0 } ) ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑣 } ) ) )
15	12 14	mpbird	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ 𝐴 )