Step |
Hyp |
Ref |
Expression |
1 |
|
lspsn.f |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
2 |
|
lspsn.k |
⊢ 𝐾 = ( Base ‘ 𝐹 ) |
3 |
|
lspsn.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
4 |
|
lspsn.t |
⊢ · = ( ·𝑠 ‘ 𝑊 ) |
5 |
|
lspsn.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
6 |
|
eqid |
⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 ) |
7 |
|
simpl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑊 ∈ LMod ) |
8 |
3 1 4 2 6
|
lss1d |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } ∈ ( LSubSp ‘ 𝑊 ) ) |
9 |
|
eqid |
⊢ ( 1r ‘ 𝐹 ) = ( 1r ‘ 𝐹 ) |
10 |
1 2 9
|
lmod1cl |
⊢ ( 𝑊 ∈ LMod → ( 1r ‘ 𝐹 ) ∈ 𝐾 ) |
11 |
3 1 4 9
|
lmodvs1 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( 1r ‘ 𝐹 ) · 𝑋 ) = 𝑋 ) |
12 |
11
|
eqcomd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 = ( ( 1r ‘ 𝐹 ) · 𝑋 ) ) |
13 |
|
oveq1 |
⊢ ( 𝑘 = ( 1r ‘ 𝐹 ) → ( 𝑘 · 𝑋 ) = ( ( 1r ‘ 𝐹 ) · 𝑋 ) ) |
14 |
13
|
rspceeqv |
⊢ ( ( ( 1r ‘ 𝐹 ) ∈ 𝐾 ∧ 𝑋 = ( ( 1r ‘ 𝐹 ) · 𝑋 ) ) → ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑋 ) ) |
15 |
10 12 14
|
syl2an2r |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑋 ) ) |
16 |
|
eqeq1 |
⊢ ( 𝑣 = 𝑋 → ( 𝑣 = ( 𝑘 · 𝑋 ) ↔ 𝑋 = ( 𝑘 · 𝑋 ) ) ) |
17 |
16
|
rexbidv |
⊢ ( 𝑣 = 𝑋 → ( ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) ↔ ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑋 ) ) ) |
18 |
17
|
elabg |
⊢ ( 𝑋 ∈ 𝑉 → ( 𝑋 ∈ { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } ↔ ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑋 ) ) ) |
19 |
18
|
adantl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑋 ∈ { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } ↔ ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑋 ) ) ) |
20 |
15 19
|
mpbird |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } ) |
21 |
6 5 7 8 20
|
lspsnel5a |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ⊆ { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } ) |
22 |
7
|
adantr |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → 𝑊 ∈ LMod ) |
23 |
3 6 5
|
lspsncl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
24 |
23
|
adantr |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
25 |
|
simpr |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → 𝑘 ∈ 𝐾 ) |
26 |
3 5
|
lspsnid |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
27 |
26
|
adantr |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
28 |
1 4 2 6
|
lssvscl |
⊢ ( ( ( 𝑊 ∈ LMod ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) ∧ ( 𝑘 ∈ 𝐾 ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) ) → ( 𝑘 · 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
29 |
22 24 25 27 28
|
syl22anc |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → ( 𝑘 · 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
30 |
|
eleq1a |
⊢ ( ( 𝑘 · 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) → ( 𝑣 = ( 𝑘 · 𝑋 ) → 𝑣 ∈ ( 𝑁 ‘ { 𝑋 } ) ) ) |
31 |
29 30
|
syl |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → ( 𝑣 = ( 𝑘 · 𝑋 ) → 𝑣 ∈ ( 𝑁 ‘ { 𝑋 } ) ) ) |
32 |
31
|
rexlimdva |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) → 𝑣 ∈ ( 𝑁 ‘ { 𝑋 } ) ) ) |
33 |
32
|
abssdv |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } ⊆ ( 𝑁 ‘ { 𝑋 } ) ) |
34 |
21 33
|
eqssd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) = { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } ) |