Theorem hdmap14lem10

Description: Part of proof of part 14 in Baer p. 49 line 38. (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 = ( 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 } ) )
Assertion	hdmap14lem10	⊢ ( 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		eqid	⊢ ( LSpan ‘ 𝐶 ) = ( LSpan ‘ 𝐶 )
26	1 2 16	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
27	17	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
28	18	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
29	3 4	lmodvacl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 + 𝑌 ) ∈ 𝑉 )
30	26 27 28 29	syl3anc	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝑉 )
31	1 2 3 5 8 9 10 12 25 13 14 15 16 30 19	hdmap14lem2a	⊢ ( 𝜑 → ∃ 𝑔 ∈ 𝐴 ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑔 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) )
32	16	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑔 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
33	17	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑔 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
34	18	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑔 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
35	19	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑔 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) ) → 𝐹 ∈ 𝐵 )
36	20	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑔 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) ) → 𝐺 ∈ 𝐴 )
37	21	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑔 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) ) → 𝐼 ∈ 𝐴 )
38	22	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑔 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) ) → ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) )
39	23	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑔 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) ) → ( 𝑆 ‘ ( 𝐹 · 𝑌 ) ) = ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) )
40	24	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑔 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
41		simp2	⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑔 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) ) → 𝑔 ∈ 𝐴 )
42		simp3	⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑔 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) ) → ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑔 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) )
43	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 32 33 34 35 36 37 38 39 40 41 42	hdmap14lem9	⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐴 ∧ ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑔 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) ) → 𝐺 = 𝐼 )
44	43	rexlimdv3a	⊢ ( 𝜑 → ( ∃ 𝑔 ∈ 𝐴 ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑔 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) → 𝐺 = 𝐼 ) )
45	31 44	mpd	⊢ ( 𝜑 → 𝐺 = 𝐼 )