Theorem hdmaplnm1

Description: Multiplicative property of first (inner product) argument. (Contributed by NM, 11-Jun-2015)

	Ref	Expression
Hypotheses	hdmaplnm1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hdmaplnm1.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hdmaplnm1.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hdmaplnm1.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
	hdmaplnm1.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
	hdmaplnm1.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	hdmaplnm1.m	⊢ × = ( .r ‘ 𝑅 )
	hdmaplnm1.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
	hdmaplnm1.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hdmaplnm1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	hdmaplnm1.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
	hdmaplnm1.a	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
Assertion	hdmaplnm1	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑌 ) ‘ ( 𝐴 · 𝑋 ) ) = ( 𝐴 × ( ( 𝑆 ‘ 𝑌 ) ‘ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hdmaplnm1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hdmaplnm1.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hdmaplnm1.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hdmaplnm1.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
5		hdmaplnm1.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
6		hdmaplnm1.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
7		hdmaplnm1.m	⊢ × = ( .r ‘ 𝑅 )
8		hdmaplnm1.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
9		hdmaplnm1.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10		hdmaplnm1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
11		hdmaplnm1.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
12		hdmaplnm1.a	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
13	1 2 9	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
14		eqid	⊢ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
15		eqid	⊢ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
16		eqid	⊢ ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 )
17	1 2 3 14 15 8 9 11	hdmapcl	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑌 ) ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) )
18	1 14 15 2 16 9 17	lcdvbaselfl	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑌 ) ∈ ( LFnl ‘ 𝑈 ) )
19	5 6 7 3 4 16	lflmul	⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑆 ‘ 𝑌 ) ∈ ( LFnl ‘ 𝑈 ) ∧ ( 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑆 ‘ 𝑌 ) ‘ ( 𝐴 · 𝑋 ) ) = ( 𝐴 × ( ( 𝑆 ‘ 𝑌 ) ‘ 𝑋 ) ) )
20	13 18 12 10 19	syl112anc	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑌 ) ‘ ( 𝐴 · 𝑋 ) ) = ( 𝐴 × ( ( 𝑆 ‘ 𝑌 ) ‘ 𝑋 ) ) )