Theorem hdmap14lem1a

Description: Prior to part 14 in Baer p. 49, line 25. (Contributed by NM, 31-May-2015)

	Ref	Expression
Hypotheses	hdmap14lem1a.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hdmap14lem1a.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hdmap14lem1a.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hdmap14lem1a.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
	hdmap14lem1a.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
	hdmap14lem1a.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	hdmap14lem1a.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	hdmap14lem2a.e	⊢ ∙ = ( ·𝑠 ‘ 𝐶 )
	hdmap14lem1a.l	⊢ 𝐿 = ( LSpan ‘ 𝐶 )
	hdmap14lem2a.p	⊢ 𝑃 = ( Scalar ‘ 𝐶 )
	hdmap14lem2a.a	⊢ 𝐴 = ( Base ‘ 𝑃 )
	hdmap14lem1a.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
	hdmap14lem1a.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hdmap14lem3a.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	hdmap14lem1a.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
	hdmap14lem1a.z	⊢ 0 = ( 0g ‘ 𝑅 )
	hdmap14lem1a.fn	⊢ ( 𝜑 → 𝐹 ≠ 0 )
Assertion	hdmap14lem1a	⊢ ( 𝜑 → ( 𝐿 ‘ { ( 𝑆 ‘ 𝑋 ) } ) = ( 𝐿 ‘ { ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) } ) )

Proof

Step	Hyp	Ref	Expression
1		hdmap14lem1a.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hdmap14lem1a.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hdmap14lem1a.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hdmap14lem1a.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
5		hdmap14lem1a.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
6		hdmap14lem1a.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
7		hdmap14lem1a.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8		hdmap14lem2a.e	⊢ ∙ = ( ·𝑠 ‘ 𝐶 )
9		hdmap14lem1a.l	⊢ 𝐿 = ( LSpan ‘ 𝐶 )
10		hdmap14lem2a.p	⊢ 𝑃 = ( Scalar ‘ 𝐶 )
11		hdmap14lem2a.a	⊢ 𝐴 = ( Base ‘ 𝑃 )
12		hdmap14lem1a.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
13		hdmap14lem1a.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
14		hdmap14lem3a.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
15		hdmap14lem1a.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
16		hdmap14lem1a.z	⊢ 0 = ( 0g ‘ 𝑅 )
17		hdmap14lem1a.fn	⊢ ( 𝜑 → 𝐹 ≠ 0 )
18	1 2 13	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
19		eqid	⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 )
20	3 5 4 6 16 19	lspsnvs	⊢ ( ( 𝑈 ∈ LVec ∧ ( 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) ∧ 𝑋 ∈ 𝑉 ) → ( ( LSpan ‘ 𝑈 ) ‘ { ( 𝐹 · 𝑋 ) } ) = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) )
21	18 15 17 14 20	syl121anc	⊢ ( 𝜑 → ( ( LSpan ‘ 𝑈 ) ‘ { ( 𝐹 · 𝑋 ) } ) = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) )
22	21	fveq2d	⊢ ( 𝜑 → ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LSpan ‘ 𝑈 ) ‘ { ( 𝐹 · 𝑋 ) } ) ) = ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ) )
23		eqid	⊢ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
24	1 2 13	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
25	3 5 4 6	lmodvscl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ) → ( 𝐹 · 𝑋 ) ∈ 𝑉 )
26	24 15 14 25	syl3anc	⊢ ( 𝜑 → ( 𝐹 · 𝑋 ) ∈ 𝑉 )
27	1 2 3 19 7 9 23 12 13 26	hdmap10	⊢ ( 𝜑 → ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LSpan ‘ 𝑈 ) ‘ { ( 𝐹 · 𝑋 ) } ) ) = ( 𝐿 ‘ { ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) } ) )
28	1 2 3 19 7 9 23 12 13 14	hdmap10	⊢ ( 𝜑 → ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { ( 𝑆 ‘ 𝑋 ) } ) )
29	22 27 28	3eqtr3rd	⊢ ( 𝜑 → ( 𝐿 ‘ { ( 𝑆 ‘ 𝑋 ) } ) = ( 𝐿 ‘ { ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) } ) )