Theorem lcdsmul

Description: Scalar multiplication for the closed kernel vector space dual. (Contributed by NM, 20-Mar-2015)

	Ref	Expression
Hypotheses	lcdsmul.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcdsmul.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcdsmul.f	⊢ 𝐹 = ( Scalar ‘ 𝑈 )
	lcdsmul.l	⊢ 𝐿 = ( Base ‘ 𝐹 )
	lcdsmul.t	⊢ · = ( .r ‘ 𝐹 )
	lcdsmul.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	lcdsmul.s	⊢ 𝑆 = ( Scalar ‘ 𝐶 )
	lcdsmul.m	⊢ ∙ = ( .r ‘ 𝑆 )
	lcdsmul.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	lcdsmul.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐿 )
	lcdsmul.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐿 )
Assertion	lcdsmul	⊢ ( 𝜑 → ( 𝑋 ∙ 𝑌 ) = ( 𝑌 · 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		lcdsmul.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcdsmul.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		lcdsmul.f	⊢ 𝐹 = ( Scalar ‘ 𝑈 )
4		lcdsmul.l	⊢ 𝐿 = ( Base ‘ 𝐹 )
5		lcdsmul.t	⊢ · = ( .r ‘ 𝐹 )
6		lcdsmul.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
7		lcdsmul.s	⊢ 𝑆 = ( Scalar ‘ 𝐶 )
8		lcdsmul.m	⊢ ∙ = ( .r ‘ 𝑆 )
9		lcdsmul.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10		lcdsmul.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐿 )
11		lcdsmul.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐿 )
12		eqid	⊢ ( oppr ‘ 𝐹 ) = ( oppr ‘ 𝐹 )
13	1 2 3 12 6 7 9	lcdsca	⊢ ( 𝜑 → 𝑆 = ( oppr ‘ 𝐹 ) )
14	13	fveq2d	⊢ ( 𝜑 → ( .r ‘ 𝑆 ) = ( .r ‘ ( oppr ‘ 𝐹 ) ) )
15	8 14	syl5eq	⊢ ( 𝜑 → ∙ = ( .r ‘ ( oppr ‘ 𝐹 ) ) )
16	15	oveqd	⊢ ( 𝜑 → ( 𝑋 ∙ 𝑌 ) = ( 𝑋 ( .r ‘ ( oppr ‘ 𝐹 ) ) 𝑌 ) )
17		eqid	⊢ ( .r ‘ ( oppr ‘ 𝐹 ) ) = ( .r ‘ ( oppr ‘ 𝐹 ) )
18	4 5 12 17	opprmul	⊢ ( 𝑋 ( .r ‘ ( oppr ‘ 𝐹 ) ) 𝑌 ) = ( 𝑌 · 𝑋 )
19	16 18	eqtrdi	⊢ ( 𝜑 → ( 𝑋 ∙ 𝑌 ) = ( 𝑌 · 𝑋 ) )