Theorem mapdrn

Description: Range of the map defined by df-mapd . (Contributed by NM, 12-Mar-2015)

	Ref	Expression
Hypotheses	mapdrn.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdrn.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	mapdrn.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdrn.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdrn.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	mapdrn.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	mapdrn.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
	mapdrn.t	⊢ 𝑇 = ( LSubSp ‘ 𝐷 )
	mapdrn.c	⊢ 𝐶 = { 𝑔 ∈ 𝐹 ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑔 ) ) ) = ( 𝐿 ‘ 𝑔 ) }
	mapdrn.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion	mapdrn	⊢ ( 𝜑 → ran 𝑀 = ( 𝑇 ∩ 𝒫 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		mapdrn.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdrn.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		mapdrn.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
4		mapdrn.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5		mapdrn.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
6		mapdrn.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
7		mapdrn.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
8		mapdrn.t	⊢ 𝑇 = ( LSubSp ‘ 𝐷 )
9		mapdrn.c	⊢ 𝐶 = { 𝑔 ∈ 𝐹 ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑔 ) ) ) = ( 𝐿 ‘ 𝑔 ) }
10		mapdrn.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11		eqid	⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
12	1 2 3 4 11 5 6 7 8 9 10	mapd1o	⊢ ( 𝜑 → 𝑀 : ( LSubSp ‘ 𝑈 ) –1-1-onto→ ( 𝑇 ∩ 𝒫 𝐶 ) )
13		f1ofo	⊢ ( 𝑀 : ( LSubSp ‘ 𝑈 ) –1-1-onto→ ( 𝑇 ∩ 𝒫 𝐶 ) → 𝑀 : ( LSubSp ‘ 𝑈 ) –onto→ ( 𝑇 ∩ 𝒫 𝐶 ) )
14		forn	⊢ ( 𝑀 : ( LSubSp ‘ 𝑈 ) –onto→ ( 𝑇 ∩ 𝒫 𝐶 ) → ran 𝑀 = ( 𝑇 ∩ 𝒫 𝐶 ) )
15	12 13 14	3syl	⊢ ( 𝜑 → ran 𝑀 = ( 𝑇 ∩ 𝒫 𝐶 ) )