Step |
Hyp |
Ref |
Expression |
1 |
|
hgmaprnlem1.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hgmaprnlem1.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hgmaprnlem1.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
hgmaprnlem1.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
5 |
|
hgmaprnlem1.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
6 |
|
hgmaprnlem1.t |
⊢ · = ( ·𝑠 ‘ 𝑈 ) |
7 |
|
hgmaprnlem1.o |
⊢ 0 = ( 0g ‘ 𝑈 ) |
8 |
|
hgmaprnlem1.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
hgmaprnlem1.d |
⊢ 𝐷 = ( Base ‘ 𝐶 ) |
10 |
|
hgmaprnlem1.p |
⊢ 𝑃 = ( Scalar ‘ 𝐶 ) |
11 |
|
hgmaprnlem1.a |
⊢ 𝐴 = ( Base ‘ 𝑃 ) |
12 |
|
hgmaprnlem1.e |
⊢ ∙ = ( ·𝑠 ‘ 𝐶 ) |
13 |
|
hgmaprnlem1.q |
⊢ 𝑄 = ( 0g ‘ 𝐶 ) |
14 |
|
hgmaprnlem1.s |
⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) |
15 |
|
hgmaprnlem1.g |
⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) |
16 |
|
hgmaprnlem1.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
17 |
|
hgmaprnlem1.z |
⊢ ( 𝜑 → 𝑧 ∈ 𝐴 ) |
18 |
|
hgmaprnlem1.t2 |
⊢ ( 𝜑 → 𝑡 ∈ ( 𝑉 ∖ { 0 } ) ) |
19 |
|
hgmaprnlem1.s2 |
⊢ ( 𝜑 → 𝑠 ∈ 𝑉 ) |
20 |
|
hgmaprnlem1.sz |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) ) |
21 |
|
hgmaprnlem1.m |
⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) |
22 |
|
hgmaprnlem1.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
23 |
|
hgmaprnlem1.l |
⊢ 𝐿 = ( LSpan ‘ 𝐶 ) |
24 |
1 8 16
|
lcdlmod |
⊢ ( 𝜑 → 𝐶 ∈ LMod ) |
25 |
18
|
eldifad |
⊢ ( 𝜑 → 𝑡 ∈ 𝑉 ) |
26 |
1 2 3 8 9 14 16 25
|
hdmapcl |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑡 ) ∈ 𝐷 ) |
27 |
10 11 9 12 23
|
lspsnvsi |
⊢ ( ( 𝐶 ∈ LMod ∧ 𝑧 ∈ 𝐴 ∧ ( 𝑆 ‘ 𝑡 ) ∈ 𝐷 ) → ( 𝐿 ‘ { ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) } ) ⊆ ( 𝐿 ‘ { ( 𝑆 ‘ 𝑡 ) } ) ) |
28 |
24 17 26 27
|
syl3anc |
⊢ ( 𝜑 → ( 𝐿 ‘ { ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) } ) ⊆ ( 𝐿 ‘ { ( 𝑆 ‘ 𝑡 ) } ) ) |
29 |
1 2 3 22 8 23 21 14 16 19
|
hdmap10 |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑠 } ) ) = ( 𝐿 ‘ { ( 𝑆 ‘ 𝑠 ) } ) ) |
30 |
20
|
sneqd |
⊢ ( 𝜑 → { ( 𝑆 ‘ 𝑠 ) } = { ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) } ) |
31 |
30
|
fveq2d |
⊢ ( 𝜑 → ( 𝐿 ‘ { ( 𝑆 ‘ 𝑠 ) } ) = ( 𝐿 ‘ { ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) } ) ) |
32 |
29 31
|
eqtrd |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑠 } ) ) = ( 𝐿 ‘ { ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) } ) ) |
33 |
1 2 3 22 8 23 21 14 16 25
|
hdmap10 |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) = ( 𝐿 ‘ { ( 𝑆 ‘ 𝑡 ) } ) ) |
34 |
28 32 33
|
3sstr4d |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑠 } ) ) ⊆ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) ) |
35 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
36 |
1 2 16
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
37 |
3 35 22
|
lspsncl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑠 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑠 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
38 |
36 19 37
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑠 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
39 |
3 35 22
|
lspsncl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑡 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑡 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
40 |
36 25 39
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑡 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
41 |
1 2 35 21 16 38 40
|
mapdord |
⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑠 } ) ) ⊆ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑡 } ) ) ↔ ( 𝑁 ‘ { 𝑠 } ) ⊆ ( 𝑁 ‘ { 𝑡 } ) ) ) |
42 |
34 41
|
mpbid |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑠 } ) ⊆ ( 𝑁 ‘ { 𝑡 } ) ) |