Step |
Hyp |
Ref |
Expression |
1 |
|
lspsnneg.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lspsnneg.m |
⊢ 𝑀 = ( invg ‘ 𝑊 ) |
3 |
|
lspsnneg.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
4 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
5 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 ) |
6 |
|
eqid |
⊢ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) = ( 1r ‘ ( Scalar ‘ 𝑊 ) ) |
7 |
|
eqid |
⊢ ( invg ‘ ( Scalar ‘ 𝑊 ) ) = ( invg ‘ ( Scalar ‘ 𝑊 ) ) |
8 |
1 2 4 5 6 7
|
lmodvneg1 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) = ( 𝑀 ‘ 𝑋 ) ) |
9 |
8
|
sneqd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → { ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) } = { ( 𝑀 ‘ 𝑋 ) } ) |
10 |
9
|
fveq2d |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) } ) = ( 𝑁 ‘ { ( 𝑀 ‘ 𝑋 ) } ) ) |
11 |
|
simpl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑊 ∈ LMod ) |
12 |
4
|
lmodfgrp |
⊢ ( 𝑊 ∈ LMod → ( Scalar ‘ 𝑊 ) ∈ Grp ) |
13 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) |
14 |
4 13 6
|
lmod1cl |
⊢ ( 𝑊 ∈ LMod → ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
15 |
13 7
|
grpinvcl |
⊢ ( ( ( Scalar ‘ 𝑊 ) ∈ Grp ∧ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) → ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
16 |
12 14 15
|
syl2anc |
⊢ ( 𝑊 ∈ LMod → ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
17 |
16
|
adantr |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
18 |
|
simpr |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ 𝑉 ) |
19 |
4 13 1 5 3
|
lspsnvsi |
⊢ ( ( 𝑊 ∈ LMod ∧ ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) } ) ⊆ ( 𝑁 ‘ { 𝑋 } ) ) |
20 |
11 17 18 19
|
syl3anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) } ) ⊆ ( 𝑁 ‘ { 𝑋 } ) ) |
21 |
10 20
|
eqsstrrd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { ( 𝑀 ‘ 𝑋 ) } ) ⊆ ( 𝑁 ‘ { 𝑋 } ) ) |
22 |
1 2
|
lmodvnegcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑀 ‘ 𝑋 ) ∈ 𝑉 ) |
23 |
1 2 4 5 6 7
|
lmodvneg1 |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑀 ‘ 𝑋 ) ∈ 𝑉 ) → ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) ( 𝑀 ‘ 𝑋 ) ) = ( 𝑀 ‘ ( 𝑀 ‘ 𝑋 ) ) ) |
24 |
22 23
|
syldan |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) ( 𝑀 ‘ 𝑋 ) ) = ( 𝑀 ‘ ( 𝑀 ‘ 𝑋 ) ) ) |
25 |
|
lmodgrp |
⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Grp ) |
26 |
1 2
|
grpinvinv |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ) → ( 𝑀 ‘ ( 𝑀 ‘ 𝑋 ) ) = 𝑋 ) |
27 |
25 26
|
sylan |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑀 ‘ ( 𝑀 ‘ 𝑋 ) ) = 𝑋 ) |
28 |
24 27
|
eqtrd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) ( 𝑀 ‘ 𝑋 ) ) = 𝑋 ) |
29 |
28
|
sneqd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → { ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) ( 𝑀 ‘ 𝑋 ) ) } = { 𝑋 } ) |
30 |
29
|
fveq2d |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) ( 𝑀 ‘ 𝑋 ) ) } ) = ( 𝑁 ‘ { 𝑋 } ) ) |
31 |
4 13 1 5 3
|
lspsnvsi |
⊢ ( ( 𝑊 ∈ LMod ∧ ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑀 ‘ 𝑋 ) ∈ 𝑉 ) → ( 𝑁 ‘ { ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) ( 𝑀 ‘ 𝑋 ) ) } ) ⊆ ( 𝑁 ‘ { ( 𝑀 ‘ 𝑋 ) } ) ) |
32 |
11 17 22 31
|
syl3anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { ( ( ( invg ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) ( ·𝑠 ‘ 𝑊 ) ( 𝑀 ‘ 𝑋 ) ) } ) ⊆ ( 𝑁 ‘ { ( 𝑀 ‘ 𝑋 ) } ) ) |
33 |
30 32
|
eqsstrrd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { ( 𝑀 ‘ 𝑋 ) } ) ) |
34 |
21 33
|
eqssd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { ( 𝑀 ‘ 𝑋 ) } ) = ( 𝑁 ‘ { 𝑋 } ) ) |