Theorem hdmap14lem4a

Description: Simplify ( A \ { Q } ) in hdmap14lem3 to provide a slightly simpler definition later. (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	hdmap14lem4a	⊢ ( 𝜑 → ( ∃! 𝑔 ∈ ( 𝐴 ∖ { 𝑄 } ) ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ↔ ∃! 𝑔 ∈ 𝐴 ( 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		eqid	⊢ ( 0g ‘ 𝐶 ) = ( 0g ‘ 𝐶 )
20		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
21	1 2 16	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
22	18	eldifad	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
23	17	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
24	3 6 4 7	lmodvscl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ) → ( 𝐹 · 𝑋 ) ∈ 𝑉 )
25	21 22 23 24	syl3anc	⊢ ( 𝜑 → ( 𝐹 · 𝑋 ) ∈ 𝑉 )
26		eldifsni	⊢ ( 𝐹 ∈ ( 𝐵 ∖ { 𝑍 } ) → 𝐹 ≠ 𝑍 )
27	18 26	syl	⊢ ( 𝜑 → 𝐹 ≠ 𝑍 )
28		eldifsni	⊢ ( 𝑋 ∈ ( 𝑉 ∖ { 0 } ) → 𝑋 ≠ 0 )
29	17 28	syl	⊢ ( 𝜑 → 𝑋 ≠ 0 )
30	1 2 16	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
31	3 4 6 7 8 5 30 22 23	lvecvsn0	⊢ ( 𝜑 → ( ( 𝐹 · 𝑋 ) ≠ 0 ↔ ( 𝐹 ≠ 𝑍 ∧ 𝑋 ≠ 0 ) ) )
32	27 29 31	mpbir2and	⊢ ( 𝜑 → ( 𝐹 · 𝑋 ) ≠ 0 )
33		eldifsn	⊢ ( ( 𝐹 · 𝑋 ) ∈ ( 𝑉 ∖ { 0 } ) ↔ ( ( 𝐹 · 𝑋 ) ∈ 𝑉 ∧ ( 𝐹 · 𝑋 ) ≠ 0 ) )
34	25 32 33	sylanbrc	⊢ ( 𝜑 → ( 𝐹 · 𝑋 ) ∈ ( 𝑉 ∖ { 0 } ) )
35	1 2 3 5 9 19 20 15 16 34	hdmapnzcl	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) ∈ ( ( Base ‘ 𝐶 ) ∖ { ( 0g ‘ 𝐶 ) } ) )
36		eldifsni	⊢ ( ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) ∈ ( ( Base ‘ 𝐶 ) ∖ { ( 0g ‘ 𝐶 ) } ) → ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) ≠ ( 0g ‘ 𝐶 ) )
37	35 36	syl	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) ≠ ( 0g ‘ 𝐶 ) )
38	37	adantr	⊢ ( ( 𝜑 ∧ 𝑔 ∈ { 𝑄 } ) → ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) ≠ ( 0g ‘ 𝐶 ) )
39		elsni	⊢ ( 𝑔 ∈ { 𝑄 } → 𝑔 = 𝑄 )
40	39	oveq1d	⊢ ( 𝑔 ∈ { 𝑄 } → ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) = ( 𝑄 ∙ ( 𝑆 ‘ 𝑋 ) ) )
41	1 9 16	lcdlmod	⊢ ( 𝜑 → 𝐶 ∈ LMod )
42	1 2 3 9 20 15 16 23	hdmapcl	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑋 ) ∈ ( Base ‘ 𝐶 ) )
43	20 12 10 14 19	lmod0vs	⊢ ( ( 𝐶 ∈ LMod ∧ ( 𝑆 ‘ 𝑋 ) ∈ ( Base ‘ 𝐶 ) ) → ( 𝑄 ∙ ( 𝑆 ‘ 𝑋 ) ) = ( 0g ‘ 𝐶 ) )
44	41 42 43	syl2anc	⊢ ( 𝜑 → ( 𝑄 ∙ ( 𝑆 ‘ 𝑋 ) ) = ( 0g ‘ 𝐶 ) )
45	40 44	sylan9eqr	⊢ ( ( 𝜑 ∧ 𝑔 ∈ { 𝑄 } ) → ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) = ( 0g ‘ 𝐶 ) )
46	38 45	neeqtrrd	⊢ ( ( 𝜑 ∧ 𝑔 ∈ { 𝑄 } ) → ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) ≠ ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) )
47	46	neneqd	⊢ ( ( 𝜑 ∧ 𝑔 ∈ { 𝑄 } ) → ¬ ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) )
48	47	nrexdv	⊢ ( 𝜑 → ¬ ∃ 𝑔 ∈ { 𝑄 } ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) )
49		reuun2	⊢ ( ¬ ∃ 𝑔 ∈ { 𝑄 } ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) → ( ∃! 𝑔 ∈ ( ( 𝐴 ∖ { 𝑄 } ) ∪ { 𝑄 } ) ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ↔ ∃! 𝑔 ∈ ( 𝐴 ∖ { 𝑄 } ) ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ) )
50	48 49	syl	⊢ ( 𝜑 → ( ∃! 𝑔 ∈ ( ( 𝐴 ∖ { 𝑄 } ) ∪ { 𝑄 } ) ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ↔ ∃! 𝑔 ∈ ( 𝐴 ∖ { 𝑄 } ) ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ) )
51	12 13 14	lmod0cl	⊢ ( 𝐶 ∈ LMod → 𝑄 ∈ 𝐴 )
52		difsnid	⊢ ( 𝑄 ∈ 𝐴 → ( ( 𝐴 ∖ { 𝑄 } ) ∪ { 𝑄 } ) = 𝐴 )
53		reueq1	⊢ ( ( ( 𝐴 ∖ { 𝑄 } ) ∪ { 𝑄 } ) = 𝐴 → ( ∃! 𝑔 ∈ ( ( 𝐴 ∖ { 𝑄 } ) ∪ { 𝑄 } ) ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ↔ ∃! 𝑔 ∈ 𝐴 ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ) )
54	41 51 52 53	4syl	⊢ ( 𝜑 → ( ∃! 𝑔 ∈ ( ( 𝐴 ∖ { 𝑄 } ) ∪ { 𝑄 } ) ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ↔ ∃! 𝑔 ∈ 𝐴 ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ) )
55	50 54	bitr3d	⊢ ( 𝜑 → ( ∃! 𝑔 ∈ ( 𝐴 ∖ { 𝑄 } ) ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ↔ ∃! 𝑔 ∈ 𝐴 ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ) )