Theorem hdmapln1

Description: Linearity property that will be used for inner product. TODO: try to combine hypotheses in hdmap*ln* series. (Contributed by NM, 7-Jun-2015)

	Ref	Expression
Hypotheses	hdmapln1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hdmapln1.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hdmapln1.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hdmapln1.p	⊢ + = ( +g ‘ 𝑈 )
	hdmapln1.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
	hdmapln1.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
	hdmapln1.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	hdmapln1.q	⊢ ⨣ = ( +g ‘ 𝑅 )
	hdmapln1.m	⊢ × = ( .r ‘ 𝑅 )
	hdmapln1.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
	hdmapln1.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hdmapln1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	hdmapln1.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
	hdmapln1.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
	hdmapln1.a	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
Assertion	hdmapln1	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑍 ) ‘ ( ( 𝐴 · 𝑋 ) + 𝑌 ) ) = ( ( 𝐴 × ( ( 𝑆 ‘ 𝑍 ) ‘ 𝑋 ) ) ⨣ ( ( 𝑆 ‘ 𝑍 ) ‘ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hdmapln1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hdmapln1.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hdmapln1.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hdmapln1.p	⊢ + = ( +g ‘ 𝑈 )
5		hdmapln1.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
6		hdmapln1.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
7		hdmapln1.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
8		hdmapln1.q	⊢ ⨣ = ( +g ‘ 𝑅 )
9		hdmapln1.m	⊢ × = ( .r ‘ 𝑅 )
10		hdmapln1.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
11		hdmapln1.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12		hdmapln1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
13		hdmapln1.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
14		hdmapln1.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
15		hdmapln1.a	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
16	1 2 11	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
17		eqid	⊢ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
18		eqid	⊢ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
19		eqid	⊢ ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 )
20	1 2 3 17 18 10 11 14	hdmapcl	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑍 ) ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) )
21	1 17 18 2 19 11 20	lcdvbaselfl	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑍 ) ∈ ( LFnl ‘ 𝑈 ) )
22	3 4 6 5 7 8 9 19	lfli	⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑆 ‘ 𝑍 ) ∈ ( LFnl ‘ 𝑈 ) ∧ ( 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( 𝑆 ‘ 𝑍 ) ‘ ( ( 𝐴 · 𝑋 ) + 𝑌 ) ) = ( ( 𝐴 × ( ( 𝑆 ‘ 𝑍 ) ‘ 𝑋 ) ) ⨣ ( ( 𝑆 ‘ 𝑍 ) ‘ 𝑌 ) ) )
23	16 21 15 12 13 22	syl113anc	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑍 ) ‘ ( ( 𝐴 · 𝑋 ) + 𝑌 ) ) = ( ( 𝐴 × ( ( 𝑆 ‘ 𝑍 ) ‘ 𝑋 ) ) ⨣ ( ( 𝑆 ‘ 𝑍 ) ‘ 𝑌 ) ) )