Theorem mapd1dim2lem1N

Description: Value of the map defined by df-mapd at an atom. (Contributed by NM, 10-Feb-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	mapd1dim2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapd1dim2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapd1dim2.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
	mapd1dim2.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	mapd1dim2.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	mapd1dim2.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	mapd1dim2.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapd1dim2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapd1dim2.t	⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
Assertion	mapd1dim2lem1N	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑄 ) = { 𝑓 ∈ 𝐹 ∣ ∃ 𝑣 ∈ 𝑄 ( 𝑂 ‘ { 𝑣 } ) = ( 𝐿 ‘ 𝑓 ) } )

Proof

Step	Hyp	Ref	Expression
1		mapd1dim2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapd1dim2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		mapd1dim2.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
4		mapd1dim2.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
5		mapd1dim2.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
6		mapd1dim2.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
7		mapd1dim2.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
8		mapd1dim2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		mapd1dim2.t	⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
10		eqid	⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
11	1 2 8	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
12	10 3 11 9	lsatlssel	⊢ ( 𝜑 → 𝑄 ∈ ( LSubSp ‘ 𝑈 ) )
13	1 2 10 4 5 6 7 8 12	mapdval4N	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑄 ) = { 𝑓 ∈ 𝐹 ∣ ∃ 𝑣 ∈ 𝑄 ( 𝑂 ‘ { 𝑣 } ) = ( 𝐿 ‘ 𝑓 ) } )