Step |
Hyp |
Ref |
Expression |
1 |
|
hdmaprnlem1.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hdmaprnlem1.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hdmaprnlem1.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
hdmaprnlem1.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
5 |
|
hdmaprnlem1.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
hdmaprnlem1.l |
⊢ 𝐿 = ( LSpan ‘ 𝐶 ) |
7 |
|
hdmaprnlem1.m |
⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
hdmaprnlem1.s |
⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
hdmaprnlem1.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
10 |
|
hdmaprnlem1.se |
⊢ ( 𝜑 → 𝑠 ∈ ( 𝐷 ∖ { 𝑄 } ) ) |
11 |
|
hdmaprnlem1.ve |
⊢ ( 𝜑 → 𝑣 ∈ 𝑉 ) |
12 |
|
hdmaprnlem1.e |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑣 } ) ) = ( 𝐿 ‘ { 𝑠 } ) ) |
13 |
|
hdmaprnlem1.ue |
⊢ ( 𝜑 → 𝑢 ∈ 𝑉 ) |
14 |
|
hdmaprnlem1.un |
⊢ ( 𝜑 → ¬ 𝑢 ∈ ( 𝑁 ‘ { 𝑣 } ) ) |
15 |
|
hdmaprnlem1.d |
⊢ 𝐷 = ( Base ‘ 𝐶 ) |
16 |
|
hdmaprnlem1.q |
⊢ 𝑄 = ( 0g ‘ 𝐶 ) |
17 |
|
hdmaprnlem1.o |
⊢ 0 = ( 0g ‘ 𝑈 ) |
18 |
|
hdmaprnlem1.a |
⊢ ✚ = ( +g ‘ 𝐶 ) |
19 |
|
hdmaprnlem1.t2 |
⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) ) |
20 |
|
hdmaprnlem1.p |
⊢ + = ( +g ‘ 𝑈 ) |
21 |
|
hdmaprnlem1.pt |
⊢ ( 𝜑 → ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) ) |
22 |
1 5 9
|
lcdlmod |
⊢ ( 𝜑 → 𝐶 ∈ LMod ) |
23 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
24 |
|
eqid |
⊢ ( LSubSp ‘ 𝐶 ) = ( LSubSp ‘ 𝐶 ) |
25 |
1 2 9
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
26 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
|
hdmaprnlem4tN |
⊢ ( 𝜑 → 𝑡 ∈ 𝑉 ) |
27 |
3 23 4
|
lspsncl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑡 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑡 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
28 |
25 26 27
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑡 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
29 |
1 7 2 23 5 24 9 28
|
mapdcl2 |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) ∈ ( LSubSp ‘ 𝐶 ) ) |
30 |
10
|
eldifad |
⊢ ( 𝜑 → 𝑠 ∈ 𝐷 ) |
31 |
15 6
|
lspsnid |
⊢ ( ( 𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷 ) → 𝑠 ∈ ( 𝐿 ‘ { 𝑠 } ) ) |
32 |
22 30 31
|
syl2anc |
⊢ ( 𝜑 → 𝑠 ∈ ( 𝐿 ‘ { 𝑠 } ) ) |
33 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
|
hdmaprnlem4N |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) = ( 𝐿 ‘ { 𝑠 } ) ) |
34 |
32 33
|
eleqtrrd |
⊢ ( 𝜑 → 𝑠 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) ) |
35 |
1 2 3 5 15 8 9 26
|
hdmapcl |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑡 ) ∈ 𝐷 ) |
36 |
15 6
|
lspsnid |
⊢ ( ( 𝐶 ∈ LMod ∧ ( 𝑆 ‘ 𝑡 ) ∈ 𝐷 ) → ( 𝑆 ‘ 𝑡 ) ∈ ( 𝐿 ‘ { ( 𝑆 ‘ 𝑡 ) } ) ) |
37 |
22 35 36
|
syl2anc |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑡 ) ∈ ( 𝐿 ‘ { ( 𝑆 ‘ 𝑡 ) } ) ) |
38 |
1 2 3 4 5 6 7 8 9 26
|
hdmap10 |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) = ( 𝐿 ‘ { ( 𝑆 ‘ 𝑡 ) } ) ) |
39 |
37 38
|
eleqtrrd |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑡 ) ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) ) |
40 |
|
eqid |
⊢ ( -g ‘ 𝐶 ) = ( -g ‘ 𝐶 ) |
41 |
40 24
|
lssvsubcl |
⊢ ( ( ( 𝐶 ∈ LMod ∧ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) ∈ ( LSubSp ‘ 𝐶 ) ) ∧ ( 𝑠 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) ∧ ( 𝑆 ‘ 𝑡 ) ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) ) ) → ( 𝑠 ( -g ‘ 𝐶 ) ( 𝑆 ‘ 𝑡 ) ) ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) ) |
42 |
22 29 34 39 41
|
syl22anc |
⊢ ( 𝜑 → ( 𝑠 ( -g ‘ 𝐶 ) ( 𝑆 ‘ 𝑡 ) ) ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) ) |