Theorem hdmap14lem3

Description: Prior to part 14 in Baer p. 49, line 26. (Contributed by NM, 31-May-2015)

	Ref	Expression
Hypotheses	hdmap14lem1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hdmap14lem1.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hdmap14lem1.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hdmap14lem1.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
	hdmap14lem3.o	⊢ 0 = ( 0g ‘ 𝑈 )
	hdmap14lem1.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
	hdmap14lem1.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	hdmap14lem1.z	⊢ 𝑍 = ( 0g ‘ 𝑅 )
	hdmap14lem1.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	hdmap14lem2.e	⊢ ∙ = ( ·𝑠 ‘ 𝐶 )
	hdmap14lem1.l	⊢ 𝐿 = ( LSpan ‘ 𝐶 )
	hdmap14lem2.p	⊢ 𝑃 = ( Scalar ‘ 𝐶 )
	hdmap14lem2.a	⊢ 𝐴 = ( Base ‘ 𝑃 )
	hdmap14lem2.q	⊢ 𝑄 = ( 0g ‘ 𝑃 )
	hdmap14lem1.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
	hdmap14lem1.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hdmap14lem3.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	hdmap14lem1.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐵 ∖ { 𝑍 } ) )
Assertion	hdmap14lem3	⊢ ( 𝜑 → ∃! 𝑔 ∈ ( 𝐴 ∖ { 𝑄 } ) ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hdmap14lem1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hdmap14lem1.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hdmap14lem1.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hdmap14lem1.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
5		hdmap14lem3.o	⊢ 0 = ( 0g ‘ 𝑈 )
6		hdmap14lem1.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
7		hdmap14lem1.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
8		hdmap14lem1.z	⊢ 𝑍 = ( 0g ‘ 𝑅 )
9		hdmap14lem1.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
10		hdmap14lem2.e	⊢ ∙ = ( ·𝑠 ‘ 𝐶 )
11		hdmap14lem1.l	⊢ 𝐿 = ( LSpan ‘ 𝐶 )
12		hdmap14lem2.p	⊢ 𝑃 = ( Scalar ‘ 𝐶 )
13		hdmap14lem2.a	⊢ 𝐴 = ( Base ‘ 𝑃 )
14		hdmap14lem2.q	⊢ 𝑄 = ( 0g ‘ 𝑃 )
15		hdmap14lem1.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
16		hdmap14lem1.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
17		hdmap14lem3.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
18		hdmap14lem1.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐵 ∖ { 𝑍 } ) )
19	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18	hdmap14lem1	⊢ ( 𝜑 → ( 𝐿 ‘ { ( 𝑆 ‘ 𝑋 ) } ) = ( 𝐿 ‘ { ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) } ) )
20	19	eqcomd	⊢ ( 𝜑 → ( 𝐿 ‘ { ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) } ) = ( 𝐿 ‘ { ( 𝑆 ‘ 𝑋 ) } ) )
21		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
22		eqid	⊢ ( 0g ‘ 𝐶 ) = ( 0g ‘ 𝐶 )
23	1 9 16	lcdlvec	⊢ ( 𝜑 → 𝐶 ∈ LVec )
24	1 2 16	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
25	18	eldifad	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
26	17	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
27	3 6 4 7	lmodvscl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ) → ( 𝐹 · 𝑋 ) ∈ 𝑉 )
28	24 25 26 27	syl3anc	⊢ ( 𝜑 → ( 𝐹 · 𝑋 ) ∈ 𝑉 )
29	1 2 3 9 21 15 16 28	hdmapcl	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) ∈ ( Base ‘ 𝐶 ) )
30	1 2 3 5 9 22 21 15 16 17	hdmapnzcl	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑋 ) ∈ ( ( Base ‘ 𝐶 ) ∖ { ( 0g ‘ 𝐶 ) } ) )
31	21 12 13 14 10 22 11 23 29 30	lspsneu	⊢ ( 𝜑 → ( ( 𝐿 ‘ { ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) } ) = ( 𝐿 ‘ { ( 𝑆 ‘ 𝑋 ) } ) ↔ ∃! 𝑔 ∈ ( 𝐴 ∖ { 𝑄 } ) ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ) )
32	20 31	mpbid	⊢ ( 𝜑 → ∃! 𝑔 ∈ ( 𝐴 ∖ { 𝑄 } ) ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) )