Theorem mapdpglem21

Description: Lemma for mapdpg . (Contributed by NM, 20-Mar-2015)

	Ref	Expression
Hypotheses	mapdpglem.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdpglem.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdpglem.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdpglem.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	mapdpglem.s	⊢ − = ( -g ‘ 𝑈 )
	mapdpglem.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	mapdpglem.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	mapdpglem.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdpglem.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	mapdpglem.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
	mapdpglem1.p	⊢ ⊕ = ( LSSum ‘ 𝐶 )
	mapdpglem2.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
	mapdpglem3.f	⊢ 𝐹 = ( Base ‘ 𝐶 )
	mapdpglem3.te	⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
	mapdpglem3.a	⊢ 𝐴 = ( Scalar ‘ 𝑈 )
	mapdpglem3.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
	mapdpglem3.t	⊢ · = ( ·𝑠 ‘ 𝐶 )
	mapdpglem3.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
	mapdpglem3.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
	mapdpglem3.e	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
	mapdpglem4.q	⊢ 𝑄 = ( 0g ‘ 𝑈 )
	mapdpglem.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
	mapdpglem4.jt	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) )
	mapdpglem4.z	⊢ 0 = ( 0g ‘ 𝐴 )
	mapdpglem4.g4	⊢ ( 𝜑 → 𝑔 ∈ 𝐵 )
	mapdpglem4.z4	⊢ ( 𝜑 → 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) )
	mapdpglem4.t4	⊢ ( 𝜑 → 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )
	mapdpglem4.xn	⊢ ( 𝜑 → 𝑋 ≠ 𝑄 )
	mapdpglem12.yn	⊢ ( 𝜑 → 𝑌 ≠ 𝑄 )
	mapdpglem17.ep	⊢ 𝐸 = ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · 𝑧 )
Assertion	mapdpglem21	⊢ ( 𝜑 → ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · 𝑡 ) = ( 𝐺 𝑅 𝐸 ) )

Proof

Step	Hyp	Ref	Expression
1		mapdpglem.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdpglem.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
3		mapdpglem.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		mapdpglem.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		mapdpglem.s	⊢ − = ( -g ‘ 𝑈 )
6		mapdpglem.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
7		mapdpglem.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8		mapdpglem.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		mapdpglem.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
10		mapdpglem.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
11		mapdpglem1.p	⊢ ⊕ = ( LSSum ‘ 𝐶 )
12		mapdpglem2.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
13		mapdpglem3.f	⊢ 𝐹 = ( Base ‘ 𝐶 )
14		mapdpglem3.te	⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
15		mapdpglem3.a	⊢ 𝐴 = ( Scalar ‘ 𝑈 )
16		mapdpglem3.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
17		mapdpglem3.t	⊢ · = ( ·𝑠 ‘ 𝐶 )
18		mapdpglem3.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
19		mapdpglem3.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
20		mapdpglem3.e	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
21		mapdpglem4.q	⊢ 𝑄 = ( 0g ‘ 𝑈 )
22		mapdpglem.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
23		mapdpglem4.jt	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) )
24		mapdpglem4.z	⊢ 0 = ( 0g ‘ 𝐴 )
25		mapdpglem4.g4	⊢ ( 𝜑 → 𝑔 ∈ 𝐵 )
26		mapdpglem4.z4	⊢ ( 𝜑 → 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) )
27		mapdpglem4.t4	⊢ ( 𝜑 → 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )
28		mapdpglem4.xn	⊢ ( 𝜑 → 𝑋 ≠ 𝑄 )
29		mapdpglem12.yn	⊢ ( 𝜑 → 𝑌 ≠ 𝑄 )
30		mapdpglem17.ep	⊢ 𝐸 = ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · 𝑧 )
31	27	oveq2d	⊢ ( 𝜑 → ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · 𝑡 ) = ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) )
32		eqid	⊢ ( Scalar ‘ 𝐶 ) = ( Scalar ‘ 𝐶 )
33		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝐶 ) ) = ( Base ‘ ( Scalar ‘ 𝐶 ) )
34	1 7 8	lcdlmod	⊢ ( 𝜑 → 𝐶 ∈ LMod )
35	1 3 8	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
36	15	lvecdrng	⊢ ( 𝑈 ∈ LVec → 𝐴 ∈ DivRing )
37	35 36	syl	⊢ ( 𝜑 → 𝐴 ∈ DivRing )
38	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 27 28	mapdpglem11	⊢ ( 𝜑 → 𝑔 ≠ 0 )
39		eqid	⊢ ( invr ‘ 𝐴 ) = ( invr ‘ 𝐴 )
40	16 24 39	drnginvrcl	⊢ ( ( 𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) ∈ 𝐵 )
41	37 25 38 40	syl3anc	⊢ ( 𝜑 → ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) ∈ 𝐵 )
42	1 3 15 16 7 32 33 8	lcdsbase	⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐶 ) ) = 𝐵 )
43	41 42	eleqtrrd	⊢ ( 𝜑 → ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) )
44	1 3 15 16 7 13 17 8 25 19	lcdvscl	⊢ ( 𝜑 → ( 𝑔 · 𝐺 ) ∈ 𝐹 )
45		eqid	⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
46		eqid	⊢ ( LSubSp ‘ 𝐶 ) = ( LSubSp ‘ 𝐶 )
47	1 3 8	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
48	4 45 6	lspsncl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
49	47 10 48	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
50	1 2 3 45 7 46 8 49	mapdcl2	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( LSubSp ‘ 𝐶 ) )
51	13 46	lssss	⊢ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( LSubSp ‘ 𝐶 ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ⊆ 𝐹 )
52	50 51	syl	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ⊆ 𝐹 )
53	52 26	sseldd	⊢ ( 𝜑 → 𝑧 ∈ 𝐹 )
54	13 17 32 33 18 34 43 44 53	lmodsubdi	⊢ ( 𝜑 → ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) = ( ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · ( 𝑔 · 𝐺 ) ) 𝑅 ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · 𝑧 ) ) )
55		eqid	⊢ ( .r ‘ 𝐴 ) = ( .r ‘ 𝐴 )
56		eqid	⊢ ( 1r ‘ 𝐴 ) = ( 1r ‘ 𝐴 )
57	16 24 55 56 39	drnginvrr	⊢ ( ( 𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ( 𝑔 ( .r ‘ 𝐴 ) ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) ) = ( 1r ‘ 𝐴 ) )
58	37 25 38 57	syl3anc	⊢ ( 𝜑 → ( 𝑔 ( .r ‘ 𝐴 ) ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) ) = ( 1r ‘ 𝐴 ) )
59		eqid	⊢ ( 1r ‘ ( Scalar ‘ 𝐶 ) ) = ( 1r ‘ ( Scalar ‘ 𝐶 ) )
60	1 3 15 56 7 32 59 8	lcd1	⊢ ( 𝜑 → ( 1r ‘ ( Scalar ‘ 𝐶 ) ) = ( 1r ‘ 𝐴 ) )
61	58 60	eqtr4d	⊢ ( 𝜑 → ( 𝑔 ( .r ‘ 𝐴 ) ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) ) = ( 1r ‘ ( Scalar ‘ 𝐶 ) ) )
62	61	oveq1d	⊢ ( 𝜑 → ( ( 𝑔 ( .r ‘ 𝐴 ) ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) ) · 𝐺 ) = ( ( 1r ‘ ( Scalar ‘ 𝐶 ) ) · 𝐺 ) )
63	1 3 15 16 55 7 13 17 8 41 25 19	lcdvsass	⊢ ( 𝜑 → ( ( 𝑔 ( .r ‘ 𝐴 ) ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) ) · 𝐺 ) = ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · ( 𝑔 · 𝐺 ) ) )
64	13 32 17 59	lmodvs1	⊢ ( ( 𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → ( ( 1r ‘ ( Scalar ‘ 𝐶 ) ) · 𝐺 ) = 𝐺 )
65	34 19 64	syl2anc	⊢ ( 𝜑 → ( ( 1r ‘ ( Scalar ‘ 𝐶 ) ) · 𝐺 ) = 𝐺 )
66	62 63 65	3eqtr3d	⊢ ( 𝜑 → ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · ( 𝑔 · 𝐺 ) ) = 𝐺 )
67	66	oveq1d	⊢ ( 𝜑 → ( ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · ( 𝑔 · 𝐺 ) ) 𝑅 ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · 𝑧 ) ) = ( 𝐺 𝑅 ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · 𝑧 ) ) )
68	30	oveq2i	⊢ ( 𝐺 𝑅 𝐸 ) = ( 𝐺 𝑅 ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · 𝑧 ) )
69	67 68	eqtr4di	⊢ ( 𝜑 → ( ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · ( 𝑔 · 𝐺 ) ) 𝑅 ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · 𝑧 ) ) = ( 𝐺 𝑅 𝐸 ) )
70	31 54 69	3eqtrd	⊢ ( 𝜑 → ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · 𝑡 ) = ( 𝐺 𝑅 𝐸 ) )