Theorem mrclsp

Description: Moore closure generalizes module span. (Contributed by Stefan O'Rear, 31-Jan-2015)

	Ref	Expression
Hypotheses	mrclsp.u	⊢ 𝑈 = ( LSubSp ‘ 𝑊 )
	mrclsp.k	⊢ 𝐾 = ( LSpan ‘ 𝑊 )
	mrclsp.f	⊢ 𝐹 = ( mrCls ‘ 𝑈 )
Assertion	mrclsp	⊢ ( 𝑊 ∈ LMod → 𝐾 = 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		mrclsp.u	⊢ 𝑈 = ( LSubSp ‘ 𝑊 )
2		mrclsp.k	⊢ 𝐾 = ( LSpan ‘ 𝑊 )
3		mrclsp.f	⊢ 𝐹 = ( mrCls ‘ 𝑈 )
4		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
5	4 1 2	lspfval	⊢ ( 𝑊 ∈ LMod → 𝐾 = ( 𝑎 ∈ 𝒫 ( Base ‘ 𝑊 ) ↦ ∩ { 𝑏 ∈ 𝑈 ∣ 𝑎 ⊆ 𝑏 } ) )
6	4 1	lssmre	⊢ ( 𝑊 ∈ LMod → 𝑈 ∈ ( Moore ‘ ( Base ‘ 𝑊 ) ) )
7	3	mrcfval	⊢ ( 𝑈 ∈ ( Moore ‘ ( Base ‘ 𝑊 ) ) → 𝐹 = ( 𝑎 ∈ 𝒫 ( Base ‘ 𝑊 ) ↦ ∩ { 𝑏 ∈ 𝑈 ∣ 𝑎 ⊆ 𝑏 } ) )
8	6 7	syl	⊢ ( 𝑊 ∈ LMod → 𝐹 = ( 𝑎 ∈ 𝒫 ( Base ‘ 𝑊 ) ↦ ∩ { 𝑏 ∈ 𝑈 ∣ 𝑎 ⊆ 𝑏 } ) )
9	5 8	eqtr4d	⊢ ( 𝑊 ∈ LMod → 𝐾 = 𝐹 )