Theorem mapdrn2

Description: Range of the map defined by df-mapd . TODO: this seems to be needed a lot in hdmaprnlem3eN etc. Would it be better to change df-mapd theorems to use LSubSpC instead of ran M ? (Contributed by NM, 13-Mar-2015)

	Ref	Expression
Hypotheses	mapdrn2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdrn2.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdrn2.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	mapdrn2.t	⊢ 𝑇 = ( LSubSp ‘ 𝐶 )
	mapdrn2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion	mapdrn2	⊢ ( 𝜑 → ran 𝑀 = 𝑇 )

Proof

Step	Hyp	Ref	Expression
1		mapdrn2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdrn2.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
3		mapdrn2.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
4		mapdrn2.t	⊢ 𝑇 = ( LSubSp ‘ 𝐶 )
5		mapdrn2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6		eqid	⊢ ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
7		eqid	⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
8		eqid	⊢ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
9		eqid	⊢ ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
10		eqid	⊢ ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
11		eqid	⊢ ( LSubSp ‘ ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( LSubSp ‘ ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
12		eqid	⊢ { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑓 ) } = { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑓 ) }
13	1 6 2 7 8 9 10 11 12 5	mapdrn	⊢ ( 𝜑 → ran 𝑀 = ( ( LSubSp ‘ ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∩ 𝒫 { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑓 ) } ) )
14	1 6 3 4 7 8 9 10 11 12 5	lcdlss	⊢ ( 𝜑 → 𝑇 = ( ( LSubSp ‘ ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∩ 𝒫 { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑓 ) } ) )
15	13 14	eqtr4d	⊢ ( 𝜑 → ran 𝑀 = 𝑇 )