hdmap14lem11

Description: Part of proof of part 14 in Baer p. 50 line 3. (Contributed by NM, 3-Jun-2015)

	Ref	Expression
Hypotheses	hdmap14lem8.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hdmap14lem8.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hdmap14lem8.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hdmap14lem8.q	⊢ + = ( +_g ‘ 𝑈 )
	hdmap14lem8.t	⊢ · = ( ·_𝑠 ‘ 𝑈 )
	hdmap14lem8.o	⊢ 0 = ( 0_g ‘ 𝑈 )
	hdmap14lem8.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	hdmap14lem8.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
	hdmap14lem8.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	hdmap14lem8.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	hdmap14lem8.d	⊢ ✚ = ( +_g ‘ 𝐶 )
	hdmap14lem8.e	⊢ ∙ = ( ·_𝑠 ‘ 𝐶 )
	hdmap14lem8.p	⊢ 𝑃 = ( Scalar ‘ 𝐶 )
	hdmap14lem8.a	⊢ 𝐴 = ( Base ‘ 𝑃 )
	hdmap14lem8.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
	hdmap14lem8.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hdmap14lem8.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	hdmap14lem8.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	hdmap14lem8.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
	hdmap14lem8.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐴 )
	hdmap14lem8.i	⊢ ( 𝜑 → 𝐼 ∈ 𝐴 )
	hdmap14lem8.xx	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) )
	hdmap14lem8.yy	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 𝑌 ) ) = ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) )
Assertion	hdmap14lem11	⊢ ( 𝜑 → 𝐺 = 𝐼 )

Proof

Step	Hyp	Ref	Expression
1		hdmap14lem8.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hdmap14lem8.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hdmap14lem8.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hdmap14lem8.q	⊢ + = ( +_g ‘ 𝑈 )
5		hdmap14lem8.t	⊢ · = ( ·_𝑠 ‘ 𝑈 )
6		hdmap14lem8.o	⊢ 0 = ( 0_g ‘ 𝑈 )
7		hdmap14lem8.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
8		hdmap14lem8.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
9		hdmap14lem8.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
10		hdmap14lem8.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
11		hdmap14lem8.d	⊢ ✚ = ( +_g ‘ 𝐶 )
12		hdmap14lem8.e	⊢ ∙ = ( ·_𝑠 ‘ 𝐶 )
13		hdmap14lem8.p	⊢ 𝑃 = ( Scalar ‘ 𝐶 )
14		hdmap14lem8.a	⊢ 𝐴 = ( Base ‘ 𝑃 )
15		hdmap14lem8.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
16		hdmap14lem8.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
17		hdmap14lem8.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
18		hdmap14lem8.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
19		hdmap14lem8.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
20		hdmap14lem8.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐴 )
21		hdmap14lem8.i	⊢ ( 𝜑 → 𝐼 ∈ 𝐴 )
22		hdmap14lem8.xx	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) )
23		hdmap14lem8.yy	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 𝑌 ) ) = ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) )
24	17	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
25	18	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
26	1 2 3 7 16 24 25	dvh3dim	⊢ ( 𝜑 → ∃ 𝑧 ∈ 𝑉 ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
27		eqid	⊢ ( LSpan ‘ 𝐶 ) = ( LSpan ‘ 𝐶 )
28	16	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
29		simp2	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑧 ∈ 𝑉 )
30	19	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝐹 ∈ 𝐵 )
31	1 2 3 5 8 9 10 12 27 13 14 15 28 29 30	hdmap14lem2a	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ∃ 𝑔 ∈ 𝐴 ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) )
32		simp11	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝜑 )
33	32 16	syl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
34		eqid	⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
35	1 2 16	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
36	32 35	syl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝑈 ∈ LMod )
37	3 34 7 35 24 25	lspprcl	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
38	32 37	syl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
39		simp12	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝑧 ∈ 𝑉 )
40		simp13	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
41	6 34 36 38 39 40	lssneln0	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝑧 ∈ ( 𝑉 ∖ { 0 } ) )
42	32 17	syl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
43	32 19	syl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝐹 ∈ 𝐵 )
44		simp2	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝑔 ∈ 𝐴 )
45	32 20	syl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝐺 ∈ 𝐴 )
46		simp3	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) )
47	32 22	syl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) )
48	1 2 16	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
49	32 48	syl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝑈 ∈ LVec )
50	32 24	syl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝑋 ∈ 𝑉 )
51	32 25	syl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝑌 ∈ 𝑉 )
52	3 7 49 39 50 51 40	lspindpi	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → ( ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ∧ ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) )
53	52	simpld	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
54	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 33 41 42 43 44 45 46 47 53	hdmap14lem10	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝑔 = 𝐺 )
55	32 18	syl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
56	32 21	syl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝐼 ∈ 𝐴 )
57	32 23	syl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → ( 𝑆 ‘ ( 𝐹 · 𝑌 ) ) = ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) )
58	52	simprd	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
59	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 33 41 55 43 44 56 46 57 58	hdmap14lem10	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝑔 = 𝐼 )
60	54 59	eqtr3d	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) ) → 𝐺 = 𝐼 )
61	60	rexlimdv3a	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( ∃ 𝑔 ∈ 𝐴 ( 𝑆 ‘ ( 𝐹 · 𝑧 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑧 ) ) → 𝐺 = 𝐼 ) )
62	31 61	mpd	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝐺 = 𝐼 )
63	62	rexlimdv3a	⊢ ( 𝜑 → ( ∃ 𝑧 ∈ 𝑉 ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) → 𝐺 = 𝐼 ) )
64	26 63	mpd	⊢ ( 𝜑 → 𝐺 = 𝐼 )

Metamath Proof Explorer

Theorem hdmap14lem11

Proof