Theorem lcd0v2

Description: The zero functional in the set of functionals with closed kernels. (Contributed by NM, 27-Mar-2015)

	Ref	Expression
Hypotheses	lcd0v2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcd0v2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcd0v2.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
	lcd0v2.z	⊢ 0 = ( 0g ‘ 𝐷 )
	lcd0v2.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	lcd0v2.o	⊢ 𝑂 = ( 0g ‘ 𝐶 )
	lcd0v2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion	lcd0v2	⊢ ( 𝜑 → 𝑂 = 0 )

Proof

Step	Hyp	Ref	Expression
1		lcd0v2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcd0v2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		lcd0v2.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
4		lcd0v2.z	⊢ 0 = ( 0g ‘ 𝐷 )
5		lcd0v2.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
6		lcd0v2.o	⊢ 𝑂 = ( 0g ‘ 𝐶 )
7		lcd0v2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
8		eqid	⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
9		eqid	⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 )
10		eqid	⊢ ( 0g ‘ ( Scalar ‘ 𝑈 ) ) = ( 0g ‘ ( Scalar ‘ 𝑈 ) )
11	1 2 8 9 10 5 6 7	lcd0v	⊢ ( 𝜑 → 𝑂 = ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) )
12	1 2 7	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
13	8 9 10 3 4 12	ldual0v	⊢ ( 𝜑 → 0 = ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) )
14	11 13	eqtr4d	⊢ ( 𝜑 → 𝑂 = 0 )