Theorem lcd0vvalN

Description: Value of the zero functional at any vector. (Contributed by NM, 28-Mar-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	lcd0vval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcd0vval.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcd0vval.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	lcd0vval.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
	lcd0vval.z	⊢ 0 = ( 0g ‘ 𝑆 )
	lcd0vval.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	lcd0vval.o	⊢ 𝑂 = ( 0g ‘ 𝐶 )
	lcd0vval.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	lcd0vval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
Assertion	lcd0vvalN	⊢ ( 𝜑 → ( 𝑂 ‘ 𝑋 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		lcd0vval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcd0vval.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		lcd0vval.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		lcd0vval.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
5		lcd0vval.z	⊢ 0 = ( 0g ‘ 𝑆 )
6		lcd0vval.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
7		lcd0vval.o	⊢ 𝑂 = ( 0g ‘ 𝐶 )
8		lcd0vval.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		lcd0vval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
10	1 2 3 4 5 6 7 8	lcd0v	⊢ ( 𝜑 → 𝑂 = ( 𝑉 × { 0 } ) )
11	10	fveq1d	⊢ ( 𝜑 → ( 𝑂 ‘ 𝑋 ) = ( ( 𝑉 × { 0 } ) ‘ 𝑋 ) )
12	5	fvexi	⊢ 0 ∈ V
13	12	fvconst2	⊢ ( 𝑋 ∈ 𝑉 → ( ( 𝑉 × { 0 } ) ‘ 𝑋 ) = 0 )
14	9 13	syl	⊢ ( 𝜑 → ( ( 𝑉 × { 0 } ) ‘ 𝑋 ) = 0 )
15	11 14	eqtrd	⊢ ( 𝜑 → ( 𝑂 ‘ 𝑋 ) = 0 )