Theorem hdmap14lem9

Description: Part of proof of part 14 in Baer p. 49 line 38. (Contributed by NM, 1-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 = ( 0g ‘ 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 ) ) )
	hdmap14lem8.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
	hdmap14lem8.j	⊢ ( 𝜑 → 𝐽 ∈ 𝐴 )
	hdmap14lem8.xy	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝐽 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) )
Assertion	hdmap14lem9	⊢ ( 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 = ( 0g ‘ 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		hdmap14lem8.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
25		hdmap14lem8.j	⊢ ( 𝜑 → 𝐽 ∈ 𝐴 )
26		hdmap14lem8.xy	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝐽 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) )
27		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
28		eqid	⊢ ( 0g ‘ 𝐶 ) = ( 0g ‘ 𝐶 )
29		eqid	⊢ ( LSpan ‘ 𝐶 ) = ( LSpan ‘ 𝐶 )
30	1 10 16	lcdlvec	⊢ ( 𝜑 → 𝐶 ∈ LVec )
31	1 2 3 6 10 28 27 15 16 17	hdmapnzcl	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑋 ) ∈ ( ( Base ‘ 𝐶 ) ∖ { ( 0g ‘ 𝐶 ) } ) )
32	1 2 3 6 10 28 27 15 16 18	hdmapnzcl	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑌 ) ∈ ( ( Base ‘ 𝐶 ) ∖ { ( 0g ‘ 𝐶 ) } ) )
33		eqid	⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
34		eqid	⊢ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
35	1 2 16	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
36	17	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
37	3 33 7	lspsncl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) )
38	35 36 37	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) )
39	18	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
40	3 33 7	lspsncl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
41	35 39 40	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
42	1 2 33 34 16 38 41	mapd11	⊢ ( 𝜑 → ( ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑁 ‘ { 𝑌 } ) ) ↔ ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) ) )
43	42	necon3bid	⊢ ( 𝜑 → ( ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑁 ‘ { 𝑋 } ) ) ≠ ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑁 ‘ { 𝑌 } ) ) ↔ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) )
44	24 43	mpbird	⊢ ( 𝜑 → ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑁 ‘ { 𝑋 } ) ) ≠ ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑁 ‘ { 𝑌 } ) ) )
45	1 2 3 7 10 29 34 15 16 36	hdmap10	⊢ ( 𝜑 → ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( ( LSpan ‘ 𝐶 ) ‘ { ( 𝑆 ‘ 𝑋 ) } ) )
46	1 2 3 7 10 29 34 15 16 39	hdmap10	⊢ ( 𝜑 → ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( ( LSpan ‘ 𝐶 ) ‘ { ( 𝑆 ‘ 𝑌 ) } ) )
47	44 45 46	3netr3d	⊢ ( 𝜑 → ( ( LSpan ‘ 𝐶 ) ‘ { ( 𝑆 ‘ 𝑋 ) } ) ≠ ( ( LSpan ‘ 𝐶 ) ‘ { ( 𝑆 ‘ 𝑌 ) } ) )
48	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26	hdmap14lem8	⊢ ( 𝜑 → ( ( 𝐽 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐽 ∙ ( 𝑆 ‘ 𝑌 ) ) ) = ( ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) ) )
49	27 11 13 14 12 28 29 30 31 32 25 25 20 21 47 48	lvecindp2	⊢ ( 𝜑 → ( 𝐽 = 𝐺 ∧ 𝐽 = 𝐼 ) )
50	49	simpld	⊢ ( 𝜑 → 𝐽 = 𝐺 )
51	49	simprd	⊢ ( 𝜑 → 𝐽 = 𝐼 )
52	50 51	eqtr3d	⊢ ( 𝜑 → 𝐺 = 𝐼 )