Step |
Hyp |
Ref |
Expression |
1 |
|
hdmapglem7.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hdmapglem7.e |
⊢ 𝐸 = 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 |
3 |
|
hdmapglem7.o |
⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
hdmapglem7.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
hdmapglem7.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
6 |
|
hdmapglem7.p |
⊢ + = ( +g ‘ 𝑈 ) |
7 |
|
hdmapglem7.q |
⊢ · = ( ·𝑠 ‘ 𝑈 ) |
8 |
|
hdmapglem7.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
9 |
|
hdmapglem7.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
10 |
|
hdmapglem7.a |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
11 |
|
hdmapglem7.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
12 |
|
hdmapglem7.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
13 |
|
hdmapglem7.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
14 |
|
hdmapglem7.t |
⊢ × = ( .r ‘ 𝑅 ) |
15 |
|
hdmapglem7.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
16 |
|
hdmapglem7.c |
⊢ ✚ = ( +g ‘ 𝑅 ) |
17 |
|
hdmapglem7.s |
⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) |
18 |
|
hdmapglem7.g |
⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) |
19 |
|
hdmapglem7b.u |
⊢ ( 𝜑 → 𝑥 ∈ ( 𝑂 ‘ { 𝐸 } ) ) |
20 |
|
hdmapglem7b.v |
⊢ ( 𝜑 → 𝑦 ∈ ( 𝑂 ‘ { 𝐸 } ) ) |
21 |
|
hdmapglem7b.k |
⊢ ( 𝜑 → 𝑚 ∈ 𝐵 ) |
22 |
|
hdmapglem7b.l |
⊢ ( 𝜑 → 𝑛 ∈ 𝐵 ) |
23 |
1 4 12
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
24 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
25 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
26 |
|
eqid |
⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 ) |
27 |
1 24 25 4 5 26 2 12
|
dvheveccl |
⊢ ( 𝜑 → 𝐸 ∈ ( 𝑉 ∖ { ( 0g ‘ 𝑈 ) } ) ) |
28 |
27
|
eldifad |
⊢ ( 𝜑 → 𝐸 ∈ 𝑉 ) |
29 |
5 8 7 9
|
lmodvscl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑛 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉 ) → ( 𝑛 · 𝐸 ) ∈ 𝑉 ) |
30 |
23 22 28 29
|
syl3anc |
⊢ ( 𝜑 → ( 𝑛 · 𝐸 ) ∈ 𝑉 ) |
31 |
28
|
snssd |
⊢ ( 𝜑 → { 𝐸 } ⊆ 𝑉 ) |
32 |
1 4 5 3
|
dochssv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ { 𝐸 } ⊆ 𝑉 ) → ( 𝑂 ‘ { 𝐸 } ) ⊆ 𝑉 ) |
33 |
12 31 32
|
syl2anc |
⊢ ( 𝜑 → ( 𝑂 ‘ { 𝐸 } ) ⊆ 𝑉 ) |
34 |
33 20
|
sseldd |
⊢ ( 𝜑 → 𝑦 ∈ 𝑉 ) |
35 |
5 6
|
lmodvacl |
⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑛 · 𝐸 ) ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → ( ( 𝑛 · 𝐸 ) + 𝑦 ) ∈ 𝑉 ) |
36 |
23 30 34 35
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑛 · 𝐸 ) + 𝑦 ) ∈ 𝑉 ) |
37 |
33 19
|
sseldd |
⊢ ( 𝜑 → 𝑥 ∈ 𝑉 ) |
38 |
1 4 5 6 7 8 9 16 14 17 18 12 36 28 37 21
|
hdmapgln2 |
⊢ ( 𝜑 → ( ( 𝑆 ‘ ( ( 𝑚 · 𝐸 ) + 𝑥 ) ) ‘ ( ( 𝑛 · 𝐸 ) + 𝑦 ) ) = ( ( ( ( 𝑆 ‘ 𝐸 ) ‘ ( ( 𝑛 · 𝐸 ) + 𝑦 ) ) × ( 𝐺 ‘ 𝑚 ) ) ✚ ( ( 𝑆 ‘ 𝑥 ) ‘ ( ( 𝑛 · 𝐸 ) + 𝑦 ) ) ) ) |
39 |
1 4 5 6 7 8 9 16 14 17 12 28 34 28 22
|
hdmapln1 |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐸 ) ‘ ( ( 𝑛 · 𝐸 ) + 𝑦 ) ) = ( ( 𝑛 × ( ( 𝑆 ‘ 𝐸 ) ‘ 𝐸 ) ) ✚ ( ( 𝑆 ‘ 𝐸 ) ‘ 𝑦 ) ) ) |
40 |
|
eqid |
⊢ ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) |
41 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
42 |
1 2 40 17 12 4 8 41
|
hdmapevec2 |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐸 ) ‘ 𝐸 ) = ( 1r ‘ 𝑅 ) ) |
43 |
42
|
oveq2d |
⊢ ( 𝜑 → ( 𝑛 × ( ( 𝑆 ‘ 𝐸 ) ‘ 𝐸 ) ) = ( 𝑛 × ( 1r ‘ 𝑅 ) ) ) |
44 |
8
|
lmodring |
⊢ ( 𝑈 ∈ LMod → 𝑅 ∈ Ring ) |
45 |
23 44
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
46 |
9 14 41
|
ringridm |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑛 ∈ 𝐵 ) → ( 𝑛 × ( 1r ‘ 𝑅 ) ) = 𝑛 ) |
47 |
45 22 46
|
syl2anc |
⊢ ( 𝜑 → ( 𝑛 × ( 1r ‘ 𝑅 ) ) = 𝑛 ) |
48 |
43 47
|
eqtrd |
⊢ ( 𝜑 → ( 𝑛 × ( ( 𝑆 ‘ 𝐸 ) ‘ 𝐸 ) ) = 𝑛 ) |
49 |
1 2 3 4 5 8 9 14 15 17 12 20
|
hdmapinvlem1 |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐸 ) ‘ 𝑦 ) = 0 ) |
50 |
48 49
|
oveq12d |
⊢ ( 𝜑 → ( ( 𝑛 × ( ( 𝑆 ‘ 𝐸 ) ‘ 𝐸 ) ) ✚ ( ( 𝑆 ‘ 𝐸 ) ‘ 𝑦 ) ) = ( 𝑛 ✚ 0 ) ) |
51 |
|
ringgrp |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp ) |
52 |
45 51
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
53 |
9 16 15
|
grprid |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝑛 ∈ 𝐵 ) → ( 𝑛 ✚ 0 ) = 𝑛 ) |
54 |
52 22 53
|
syl2anc |
⊢ ( 𝜑 → ( 𝑛 ✚ 0 ) = 𝑛 ) |
55 |
39 50 54
|
3eqtrd |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐸 ) ‘ ( ( 𝑛 · 𝐸 ) + 𝑦 ) ) = 𝑛 ) |
56 |
55
|
oveq1d |
⊢ ( 𝜑 → ( ( ( 𝑆 ‘ 𝐸 ) ‘ ( ( 𝑛 · 𝐸 ) + 𝑦 ) ) × ( 𝐺 ‘ 𝑚 ) ) = ( 𝑛 × ( 𝐺 ‘ 𝑚 ) ) ) |
57 |
1 4 5 6 7 8 9 16 14 17 12 28 34 37 22
|
hdmapln1 |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑥 ) ‘ ( ( 𝑛 · 𝐸 ) + 𝑦 ) ) = ( ( 𝑛 × ( ( 𝑆 ‘ 𝑥 ) ‘ 𝐸 ) ) ✚ ( ( 𝑆 ‘ 𝑥 ) ‘ 𝑦 ) ) ) |
58 |
1 2 3 4 5 8 9 14 15 17 12 19
|
hdmapinvlem2 |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑥 ) ‘ 𝐸 ) = 0 ) |
59 |
58
|
oveq2d |
⊢ ( 𝜑 → ( 𝑛 × ( ( 𝑆 ‘ 𝑥 ) ‘ 𝐸 ) ) = ( 𝑛 × 0 ) ) |
60 |
9 14 15
|
ringrz |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑛 ∈ 𝐵 ) → ( 𝑛 × 0 ) = 0 ) |
61 |
45 22 60
|
syl2anc |
⊢ ( 𝜑 → ( 𝑛 × 0 ) = 0 ) |
62 |
59 61
|
eqtrd |
⊢ ( 𝜑 → ( 𝑛 × ( ( 𝑆 ‘ 𝑥 ) ‘ 𝐸 ) ) = 0 ) |
63 |
62
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝑛 × ( ( 𝑆 ‘ 𝑥 ) ‘ 𝐸 ) ) ✚ ( ( 𝑆 ‘ 𝑥 ) ‘ 𝑦 ) ) = ( 0 ✚ ( ( 𝑆 ‘ 𝑥 ) ‘ 𝑦 ) ) ) |
64 |
1 4 5 8 9 17 12 34 37
|
hdmapipcl |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑥 ) ‘ 𝑦 ) ∈ 𝐵 ) |
65 |
9 16 15
|
grplid |
⊢ ( ( 𝑅 ∈ Grp ∧ ( ( 𝑆 ‘ 𝑥 ) ‘ 𝑦 ) ∈ 𝐵 ) → ( 0 ✚ ( ( 𝑆 ‘ 𝑥 ) ‘ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) ‘ 𝑦 ) ) |
66 |
52 64 65
|
syl2anc |
⊢ ( 𝜑 → ( 0 ✚ ( ( 𝑆 ‘ 𝑥 ) ‘ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) ‘ 𝑦 ) ) |
67 |
57 63 66
|
3eqtrd |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑥 ) ‘ ( ( 𝑛 · 𝐸 ) + 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) ‘ 𝑦 ) ) |
68 |
56 67
|
oveq12d |
⊢ ( 𝜑 → ( ( ( ( 𝑆 ‘ 𝐸 ) ‘ ( ( 𝑛 · 𝐸 ) + 𝑦 ) ) × ( 𝐺 ‘ 𝑚 ) ) ✚ ( ( 𝑆 ‘ 𝑥 ) ‘ ( ( 𝑛 · 𝐸 ) + 𝑦 ) ) ) = ( ( 𝑛 × ( 𝐺 ‘ 𝑚 ) ) ✚ ( ( 𝑆 ‘ 𝑥 ) ‘ 𝑦 ) ) ) |
69 |
38 68
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑆 ‘ ( ( 𝑚 · 𝐸 ) + 𝑥 ) ) ‘ ( ( 𝑛 · 𝐸 ) + 𝑦 ) ) = ( ( 𝑛 × ( 𝐺 ‘ 𝑚 ) ) ✚ ( ( 𝑆 ‘ 𝑥 ) ‘ 𝑦 ) ) ) |