Metamath Proof Explorer

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 ) )