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 |
|
mapdpglem12.g0 |
⊢ ( 𝜑 → 𝑧 = ( 0g ‘ 𝐶 ) ) |
31 |
1 3 8
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
32 |
|
eqid |
⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 ) |
33 |
4 32 5
|
lmodvnpcan |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝑌 − 𝑋 ) ( +g ‘ 𝑈 ) 𝑋 ) = 𝑌 ) |
34 |
31 10 9 33
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑌 − 𝑋 ) ( +g ‘ 𝑈 ) 𝑋 ) = 𝑌 ) |
35 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
36 |
4 35 6
|
lspsncl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
37 |
31 9 36
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
38 |
|
lmodgrp |
⊢ ( 𝑈 ∈ LMod → 𝑈 ∈ Grp ) |
39 |
31 38
|
syl |
⊢ ( 𝜑 → 𝑈 ∈ Grp ) |
40 |
|
eqid |
⊢ ( invg ‘ 𝑈 ) = ( invg ‘ 𝑈 ) |
41 |
4 5 40
|
grpinvsub |
⊢ ( ( 𝑈 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( ( invg ‘ 𝑈 ) ‘ ( 𝑋 − 𝑌 ) ) = ( 𝑌 − 𝑋 ) ) |
42 |
39 9 10 41
|
syl3anc |
⊢ ( 𝜑 → ( ( invg ‘ 𝑈 ) ‘ ( 𝑋 − 𝑌 ) ) = ( 𝑌 − 𝑋 ) ) |
43 |
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 29 30
|
mapdpglem13 |
⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ⊆ ( 𝑁 ‘ { 𝑋 } ) ) |
44 |
4 5
|
lmodvsubcl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 − 𝑌 ) ∈ 𝑉 ) |
45 |
31 9 10 44
|
syl3anc |
⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) ∈ 𝑉 ) |
46 |
4 6
|
lspsnid |
⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑋 − 𝑌 ) ∈ 𝑉 ) → ( 𝑋 − 𝑌 ) ∈ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) |
47 |
31 45 46
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) ∈ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) |
48 |
43 47
|
sseldd |
⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
49 |
35 40
|
lssvnegcl |
⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) ∧ ( 𝑋 − 𝑌 ) ∈ ( 𝑁 ‘ { 𝑋 } ) ) → ( ( invg ‘ 𝑈 ) ‘ ( 𝑋 − 𝑌 ) ) ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
50 |
31 37 48 49
|
syl3anc |
⊢ ( 𝜑 → ( ( invg ‘ 𝑈 ) ‘ ( 𝑋 − 𝑌 ) ) ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
51 |
42 50
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝑌 − 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
52 |
4 6
|
lspsnid |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
53 |
31 9 52
|
syl2anc |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
54 |
32 35
|
lssvacl |
⊢ ( ( ( 𝑈 ∈ LMod ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) ) ∧ ( ( 𝑌 − 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) ) → ( ( 𝑌 − 𝑋 ) ( +g ‘ 𝑈 ) 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
55 |
31 37 51 53 54
|
syl22anc |
⊢ ( 𝜑 → ( ( 𝑌 − 𝑋 ) ( +g ‘ 𝑈 ) 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
56 |
34 55
|
eqeltrrd |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑁 ‘ { 𝑋 } ) ) |