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 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
20 |
1 2 9
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
21 |
3 19 4
|
lspsncl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑢 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑢 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
22 |
20 13 21
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑢 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
23 |
1 7 2 19 9 22
|
mapdcnvid1N |
⊢ ( 𝜑 → ( ◡ 𝑀 ‘ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑢 } ) ) ) = ( 𝑁 ‘ { 𝑢 } ) ) |
24 |
1 2 3 4 5 6 7 8 9 13
|
hdmap10 |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑢 } ) ) = ( 𝐿 ‘ { ( 𝑆 ‘ 𝑢 ) } ) ) |
25 |
1 5 9
|
lcdlvec |
⊢ ( 𝜑 → 𝐶 ∈ LVec ) |
26 |
1 2 3 5 15 8 9 13
|
hdmapcl |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑢 ) ∈ 𝐷 ) |
27 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14
|
hdmaprnlem1N |
⊢ ( 𝜑 → ( 𝐿 ‘ { ( 𝑆 ‘ 𝑢 ) } ) ≠ ( 𝐿 ‘ { 𝑠 } ) ) |
28 |
15 18 16 6 25 26 10 27
|
lspindp3 |
⊢ ( 𝜑 → ( 𝐿 ‘ { ( 𝑆 ‘ 𝑢 ) } ) ≠ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) |
29 |
24 28
|
eqnetrd |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑢 } ) ) ≠ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) |
30 |
1 7 2 19 9 22
|
mapdcl |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑢 } ) ) ∈ ran 𝑀 ) |
31 |
1 5 9
|
lcdlmod |
⊢ ( 𝜑 → 𝐶 ∈ LMod ) |
32 |
10
|
eldifad |
⊢ ( 𝜑 → 𝑠 ∈ 𝐷 ) |
33 |
15 18
|
lmodvacl |
⊢ ( ( 𝐶 ∈ LMod ∧ ( 𝑆 ‘ 𝑢 ) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷 ) → ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ∈ 𝐷 ) |
34 |
31 26 32 33
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ∈ 𝐷 ) |
35 |
|
eqid |
⊢ ( LSubSp ‘ 𝐶 ) = ( LSubSp ‘ 𝐶 ) |
36 |
15 35 6
|
lspsncl |
⊢ ( ( 𝐶 ∈ LMod ∧ ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ∈ 𝐷 ) → ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ∈ ( LSubSp ‘ 𝐶 ) ) |
37 |
31 34 36
|
syl2anc |
⊢ ( 𝜑 → ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ∈ ( LSubSp ‘ 𝐶 ) ) |
38 |
1 7 5 35 9
|
mapdrn2 |
⊢ ( 𝜑 → ran 𝑀 = ( LSubSp ‘ 𝐶 ) ) |
39 |
37 38
|
eleqtrrd |
⊢ ( 𝜑 → ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ∈ ran 𝑀 ) |
40 |
1 7 9 30 39
|
mapdcnv11N |
⊢ ( 𝜑 → ( ( ◡ 𝑀 ‘ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑢 } ) ) ) = ( ◡ 𝑀 ‘ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) ↔ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑢 } ) ) = ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) ) |
41 |
40
|
necon3bid |
⊢ ( 𝜑 → ( ( ◡ 𝑀 ‘ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑢 } ) ) ) ≠ ( ◡ 𝑀 ‘ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) ↔ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑢 } ) ) ≠ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) ) |
42 |
29 41
|
mpbird |
⊢ ( 𝜑 → ( ◡ 𝑀 ‘ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑢 } ) ) ) ≠ ( ◡ 𝑀 ‘ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) ) |
43 |
23 42
|
eqnetrrd |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑢 } ) ≠ ( ◡ 𝑀 ‘ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) ) |