Theorem lcdvbasess

Description: The vector base set of the closed kernel dual space is a set of functionals. (Contributed by NM, 15-Mar-2015)

	Ref	Expression
Hypotheses	lcdvbasess.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcdvbasess.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	lcdvbasess.v	⊢ 𝑉 = ( Base ‘ 𝐶 )
	lcdvbasess.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcdvbasess.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	lcdvbasess.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion	lcdvbasess	⊢ ( 𝜑 → 𝑉 ⊆ 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		lcdvbasess.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcdvbasess.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
3		lcdvbasess.v	⊢ 𝑉 = ( Base ‘ 𝐶 )
4		lcdvbasess.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5		lcdvbasess.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
6		lcdvbasess.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7		eqid	⊢ ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
8		eqid	⊢ ( LKer ‘ 𝑈 ) = ( LKer ‘ 𝑈 )
9		eqid	⊢ { 𝑓 ∈ 𝐹 ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) } = { 𝑓 ∈ 𝐹 ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) }
10	1 7 2 3 4 5 8 9 6	lcdvbase	⊢ ( 𝜑 → 𝑉 = { 𝑓 ∈ 𝐹 ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) } )
11		ssrab2	⊢ { 𝑓 ∈ 𝐹 ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) } ⊆ 𝐹
12	10 11	eqsstrdi	⊢ ( 𝜑 → 𝑉 ⊆ 𝐹 )