Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
2 |
1
|
linds1 |
⊢ ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) → 𝐹 ⊆ ( Base ‘ 𝑊 ) ) |
3 |
|
eldifi |
⊢ ( 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) → 𝑋 ∈ ( Base ‘ 𝑊 ) ) |
4 |
3
|
snssd |
⊢ ( 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) → { 𝑋 } ⊆ ( Base ‘ 𝑊 ) ) |
5 |
|
unss |
⊢ ( ( 𝐹 ⊆ ( Base ‘ 𝑊 ) ∧ { 𝑋 } ⊆ ( Base ‘ 𝑊 ) ) ↔ ( 𝐹 ∪ { 𝑋 } ) ⊆ ( Base ‘ 𝑊 ) ) |
6 |
5
|
biimpi |
⊢ ( ( 𝐹 ⊆ ( Base ‘ 𝑊 ) ∧ { 𝑋 } ⊆ ( Base ‘ 𝑊 ) ) → ( 𝐹 ∪ { 𝑋 } ) ⊆ ( Base ‘ 𝑊 ) ) |
7 |
2 4 6
|
syl2an |
⊢ ( ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ( 𝐹 ∪ { 𝑋 } ) ⊆ ( Base ‘ 𝑊 ) ) |
8 |
7
|
3adant1 |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ( 𝐹 ∪ { 𝑋 } ) ⊆ ( Base ‘ 𝑊 ) ) |
9 |
|
eldifn |
⊢ ( 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) → ¬ 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) |
10 |
9
|
3ad2ant3 |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ¬ 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) |
11 |
10
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ¬ 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) |
12 |
|
simpll1 |
⊢ ( ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) ∧ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) ) ) → 𝑊 ∈ LVec ) |
13 |
2
|
ssdifssd |
⊢ ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) → ( 𝐹 ∖ { 𝑥 } ) ⊆ ( Base ‘ 𝑊 ) ) |
14 |
13
|
3ad2ant2 |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ( 𝐹 ∖ { 𝑥 } ) ⊆ ( Base ‘ 𝑊 ) ) |
15 |
14
|
ad2antrr |
⊢ ( ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) ∧ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) ) ) → ( 𝐹 ∖ { 𝑥 } ) ⊆ ( Base ‘ 𝑊 ) ) |
16 |
3
|
3ad2ant3 |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → 𝑋 ∈ ( Base ‘ 𝑊 ) ) |
17 |
16
|
ad2antrr |
⊢ ( ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) ∧ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) ) ) → 𝑋 ∈ ( Base ‘ 𝑊 ) ) |
18 |
|
simpr |
⊢ ( ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) ∧ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) ) ) → 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) ) ) |
19 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
20 |
19
|
ad2antrr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → 𝑊 ∈ LMod ) |
21 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
22 |
21
|
lmodring |
⊢ ( 𝑊 ∈ LMod → ( Scalar ‘ 𝑊 ) ∈ Ring ) |
23 |
19 22
|
syl |
⊢ ( 𝑊 ∈ LVec → ( Scalar ‘ 𝑊 ) ∈ Ring ) |
24 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) |
25 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) |
26 |
24 25
|
ringelnzr |
⊢ ( ( ( Scalar ‘ 𝑊 ) ∈ Ring ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → ( Scalar ‘ 𝑊 ) ∈ NzRing ) |
27 |
23 26
|
sylan |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → ( Scalar ‘ 𝑊 ) ∈ NzRing ) |
28 |
27
|
ad2ant2rl |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ( Scalar ‘ 𝑊 ) ∈ NzRing ) |
29 |
|
simplr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) |
30 |
|
simprl |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → 𝑥 ∈ 𝐹 ) |
31 |
|
eqid |
⊢ ( LSpan ‘ 𝑊 ) = ( LSpan ‘ 𝑊 ) |
32 |
31 21
|
lindsind2 |
⊢ ( ( ( 𝑊 ∈ LMod ∧ ( Scalar ‘ 𝑊 ) ∈ NzRing ) ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑥 ∈ 𝐹 ) → ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑥 } ) ) ) |
33 |
20 28 29 30 32
|
syl211anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑥 } ) ) ) |
34 |
33
|
3adantl3 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑥 } ) ) ) |
35 |
34
|
adantr |
⊢ ( ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) ∧ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) ) ) → ¬ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑥 } ) ) ) |
36 |
18 35
|
eldifd |
⊢ ( ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) ∧ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) ) ) → 𝑥 ∈ ( ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑥 } ) ) ) ) |
37 |
|
eqid |
⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 ) |
38 |
1 37 31
|
lspsolv |
⊢ ( ( 𝑊 ∈ LVec ∧ ( ( 𝐹 ∖ { 𝑥 } ) ⊆ ( Base ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ∧ 𝑥 ∈ ( ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑥 } ) ) ) ) ) → 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑥 } ) ) ) |
39 |
12 15 17 36 38
|
syl13anc |
⊢ ( ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) ∧ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) ) ) → 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑥 } ) ) ) |
40 |
39
|
ex |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ( 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) ) → 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑥 } ) ) ) ) |
41 |
|
eldif |
⊢ ( 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ↔ ( 𝑋 ∈ ( Base ‘ 𝑊 ) ∧ ¬ 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
42 |
|
snssi |
⊢ ( 𝑋 ∈ 𝐹 → { 𝑋 } ⊆ 𝐹 ) |
43 |
1 31
|
lspss |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐹 ⊆ ( Base ‘ 𝑊 ) ∧ { 𝑋 } ⊆ 𝐹 ) → ( ( LSpan ‘ 𝑊 ) ‘ { 𝑋 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) |
44 |
19 2 42 43
|
syl3an |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ 𝐹 ) → ( ( LSpan ‘ 𝑊 ) ‘ { 𝑋 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) |
45 |
44
|
ad4ant124 |
⊢ ( ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑋 ∈ 𝐹 ) → ( ( LSpan ‘ 𝑊 ) ‘ { 𝑋 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) |
46 |
1 31
|
lspsnid |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑋 } ) ) |
47 |
19 46
|
sylan |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑋 } ) ) |
48 |
47
|
ad4ant13 |
⊢ ( ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑋 ∈ 𝐹 ) → 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑋 } ) ) |
49 |
45 48
|
sseldd |
⊢ ( ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑋 ∈ 𝐹 ) → 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) |
50 |
49
|
ex |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑋 ∈ 𝐹 → 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
51 |
50
|
con3d |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → ( ¬ 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) → ¬ 𝑋 ∈ 𝐹 ) ) |
52 |
51
|
expimpd |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) → ( ( 𝑋 ∈ ( Base ‘ 𝑊 ) ∧ ¬ 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) → ¬ 𝑋 ∈ 𝐹 ) ) |
53 |
52
|
3impia |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ ( 𝑋 ∈ ( Base ‘ 𝑊 ) ∧ ¬ 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ¬ 𝑋 ∈ 𝐹 ) |
54 |
41 53
|
syl3an3b |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ¬ 𝑋 ∈ 𝐹 ) |
55 |
|
eleq1 |
⊢ ( 𝑋 = 𝑥 → ( 𝑋 ∈ 𝐹 ↔ 𝑥 ∈ 𝐹 ) ) |
56 |
55
|
notbid |
⊢ ( 𝑋 = 𝑥 → ( ¬ 𝑋 ∈ 𝐹 ↔ ¬ 𝑥 ∈ 𝐹 ) ) |
57 |
54 56
|
syl5ibcom |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ( 𝑋 = 𝑥 → ¬ 𝑥 ∈ 𝐹 ) ) |
58 |
57
|
necon2ad |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ( 𝑥 ∈ 𝐹 → 𝑋 ≠ 𝑥 ) ) |
59 |
58
|
imp |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ 𝑥 ∈ 𝐹 ) → 𝑋 ≠ 𝑥 ) |
60 |
|
disjsn2 |
⊢ ( 𝑋 ≠ 𝑥 → ( { 𝑋 } ∩ { 𝑥 } ) = ∅ ) |
61 |
59 60
|
syl |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ 𝑥 ∈ 𝐹 ) → ( { 𝑋 } ∩ { 𝑥 } ) = ∅ ) |
62 |
|
disj3 |
⊢ ( ( { 𝑋 } ∩ { 𝑥 } ) = ∅ ↔ { 𝑋 } = ( { 𝑋 } ∖ { 𝑥 } ) ) |
63 |
61 62
|
sylib |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ 𝑥 ∈ 𝐹 ) → { 𝑋 } = ( { 𝑋 } ∖ { 𝑥 } ) ) |
64 |
63
|
uneq2d |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ 𝑥 ∈ 𝐹 ) → ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) = ( ( 𝐹 ∖ { 𝑥 } ) ∪ ( { 𝑋 } ∖ { 𝑥 } ) ) ) |
65 |
|
difundir |
⊢ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) = ( ( 𝐹 ∖ { 𝑥 } ) ∪ ( { 𝑋 } ∖ { 𝑥 } ) ) |
66 |
64 65
|
eqtr4di |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ 𝑥 ∈ 𝐹 ) → ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) = ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) |
67 |
66
|
fveq2d |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ 𝑥 ∈ 𝐹 ) → ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) ) = ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) |
68 |
67
|
eleq2d |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ 𝑥 ∈ 𝐹 ) → ( 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) ) ↔ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) ) |
69 |
68
|
adantrr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ( 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) ) ↔ 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) ) |
70 |
|
simpl |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → 𝑊 ∈ LVec ) |
71 |
|
eldifsn |
⊢ ( 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ↔ ( 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑘 ≠ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) |
72 |
71
|
biimpi |
⊢ ( 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) → ( 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑘 ≠ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) |
73 |
72
|
adantl |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → ( 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑘 ≠ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) |
74 |
2
|
sselda |
⊢ ( ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑥 ∈ 𝐹 ) → 𝑥 ∈ ( Base ‘ 𝑊 ) ) |
75 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 ) |
76 |
1 21 75 25 24 31
|
lspsnvs |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑘 ≠ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( ( LSpan ‘ 𝑊 ) ‘ { ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) } ) = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ) |
77 |
70 73 74 76
|
syl2an3an |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ∧ ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑥 ∈ 𝐹 ) ) → ( ( LSpan ‘ 𝑊 ) ‘ { ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) } ) = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ) |
78 |
77
|
an42s |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ( ( LSpan ‘ 𝑊 ) ‘ { ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) } ) = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ) |
79 |
78
|
sseq1d |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ( ( ( LSpan ‘ 𝑊 ) ‘ { ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ↔ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) ) |
80 |
79
|
3adantl3 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ( ( ( LSpan ‘ 𝑊 ) ‘ { ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ↔ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) ) |
81 |
|
eldifi |
⊢ ( 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) → 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
82 |
19
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → 𝑊 ∈ LMod ) |
83 |
82
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ) → 𝑊 ∈ LMod ) |
84 |
|
snssi |
⊢ ( 𝑋 ∈ ( Base ‘ 𝑊 ) → { 𝑋 } ⊆ ( Base ‘ 𝑊 ) ) |
85 |
2 84 6
|
syl2an |
⊢ ( ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → ( 𝐹 ∪ { 𝑋 } ) ⊆ ( Base ‘ 𝑊 ) ) |
86 |
85
|
ssdifssd |
⊢ ( ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ⊆ ( Base ‘ 𝑊 ) ) |
87 |
1 37 31
|
lspcl |
⊢ ( ( 𝑊 ∈ LMod ∧ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ⊆ ( Base ‘ 𝑊 ) ) → ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ∈ ( LSubSp ‘ 𝑊 ) ) |
88 |
19 86 87
|
syl2an |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ) → ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ∈ ( LSubSp ‘ 𝑊 ) ) |
89 |
88
|
3impb |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ∈ ( LSubSp ‘ 𝑊 ) ) |
90 |
89
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ) → ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ∈ ( LSubSp ‘ 𝑊 ) ) |
91 |
19
|
anim1i |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) → ( 𝑊 ∈ LMod ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ) |
92 |
1 21 75 25
|
lmodvscl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( Base ‘ 𝑊 ) ) |
93 |
92
|
3expa |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( Base ‘ 𝑊 ) ) |
94 |
91 74 93
|
syl2an |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑥 ∈ 𝐹 ) ) → ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( Base ‘ 𝑊 ) ) |
95 |
94
|
an42s |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ) → ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( Base ‘ 𝑊 ) ) |
96 |
95
|
3adantl3 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ) → ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( Base ‘ 𝑊 ) ) |
97 |
1 37 31 83 90 96
|
lspsnel5 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ) → ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ↔ ( ( LSpan ‘ 𝑊 ) ‘ { ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) ) |
98 |
81 97
|
sylanr2 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ↔ ( ( LSpan ‘ 𝑊 ) ‘ { ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) ) |
99 |
82
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑥 ∈ 𝐹 ) → 𝑊 ∈ LMod ) |
100 |
89
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑥 ∈ 𝐹 ) → ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ∈ ( LSubSp ‘ 𝑊 ) ) |
101 |
74
|
3ad2antl2 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑥 ∈ 𝐹 ) → 𝑥 ∈ ( Base ‘ 𝑊 ) ) |
102 |
1 37 31 99 100 101
|
lspsnel5 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑥 ∈ 𝐹 ) → ( 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ↔ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) ) |
103 |
102
|
adantrr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ( 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ↔ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) ) |
104 |
80 98 103
|
3bitr4rd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ( 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ↔ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) ) |
105 |
3 104
|
syl3anl3 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ( 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ↔ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) ) |
106 |
69 105
|
bitrd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ( 𝑥 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑋 } ) ) ↔ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) ) |
107 |
|
difsnid |
⊢ ( 𝑥 ∈ 𝐹 → ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑥 } ) = 𝐹 ) |
108 |
107
|
fveq2d |
⊢ ( 𝑥 ∈ 𝐹 → ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑥 } ) ) = ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) |
109 |
108
|
eleq2d |
⊢ ( 𝑥 ∈ 𝐹 → ( 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑥 } ) ) ↔ 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
110 |
109
|
ad2antrl |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ( 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∖ { 𝑥 } ) ∪ { 𝑥 } ) ) ↔ 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
111 |
40 106 110
|
3imtr3d |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) → 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
112 |
11 111
|
mtod |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) ) → ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) |
113 |
112
|
ralrimivva |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ∀ 𝑥 ∈ 𝐹 ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) |
114 |
10
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → ¬ 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) |
115 |
|
difsn |
⊢ ( ¬ 𝑋 ∈ 𝐹 → ( 𝐹 ∖ { 𝑋 } ) = 𝐹 ) |
116 |
54 115
|
syl |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ( 𝐹 ∖ { 𝑋 } ) = 𝐹 ) |
117 |
116
|
fveq2d |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑋 } ) ) = ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) |
118 |
117
|
eleq2d |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑋 } ) ) ↔ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
119 |
118
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑋 } ) ) ↔ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
120 |
1 21 75 25 24 31
|
lspsnvs |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑘 ≠ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → ( ( LSpan ‘ 𝑊 ) ‘ { ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) } ) = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑋 } ) ) |
121 |
120
|
3expa |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑘 ≠ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → ( ( LSpan ‘ 𝑊 ) ‘ { ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) } ) = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑋 } ) ) |
122 |
121
|
an32s |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ ( 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑘 ≠ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) → ( ( LSpan ‘ 𝑊 ) ‘ { ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) } ) = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑋 } ) ) |
123 |
71 122
|
sylan2b |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → ( ( LSpan ‘ 𝑊 ) ‘ { ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) } ) = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑋 } ) ) |
124 |
123
|
sseq1d |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → ( ( ( LSpan ‘ 𝑊 ) ‘ { ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ↔ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑋 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
125 |
124
|
3adantl2 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → ( ( ( LSpan ‘ 𝑊 ) ‘ { ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ↔ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑋 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
126 |
82
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) → 𝑊 ∈ LMod ) |
127 |
1 37 31
|
lspcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐹 ⊆ ( Base ‘ 𝑊 ) ) → ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ∈ ( LSubSp ‘ 𝑊 ) ) |
128 |
19 2 127
|
syl2an |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) → ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ∈ ( LSubSp ‘ 𝑊 ) ) |
129 |
128
|
3adant3 |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ∈ ( LSubSp ‘ 𝑊 ) ) |
130 |
129
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) → ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ∈ ( LSubSp ‘ 𝑊 ) ) |
131 |
1 21 75 25
|
lmodvscl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( Base ‘ 𝑊 ) ) |
132 |
131
|
3expa |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( Base ‘ 𝑊 ) ) |
133 |
132
|
an32s |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) → ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( Base ‘ 𝑊 ) ) |
134 |
19 133
|
sylanl1 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) → ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( Base ‘ 𝑊 ) ) |
135 |
134
|
3adantl2 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) → ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( Base ‘ 𝑊 ) ) |
136 |
1 37 31 126 130 135
|
lspsnel5 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) → ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ↔ ( ( LSpan ‘ 𝑊 ) ‘ { ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
137 |
81 136
|
sylan2 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ↔ ( ( LSpan ‘ 𝑊 ) ‘ { ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
138 |
|
simp3 |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → 𝑋 ∈ ( Base ‘ 𝑊 ) ) |
139 |
1 37 31 82 129 138
|
lspsnel5 |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ↔ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑋 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
140 |
139
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → ( 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ↔ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑋 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
141 |
125 137 140
|
3bitr4d |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ↔ 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
142 |
3 141
|
syl3anl3 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ↔ 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
143 |
119 142
|
bitrd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑋 } ) ) ↔ 𝑋 ∈ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) |
144 |
114 143
|
mtbird |
⊢ ( ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) ∧ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) → ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑋 } ) ) ) |
145 |
144
|
ralrimiva |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑋 } ) ) ) |
146 |
|
oveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) = ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) |
147 |
|
sneq |
⊢ ( 𝑥 = 𝑋 → { 𝑥 } = { 𝑋 } ) |
148 |
147
|
difeq2d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) = ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑋 } ) ) |
149 |
|
difun2 |
⊢ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑋 } ) = ( 𝐹 ∖ { 𝑋 } ) |
150 |
148 149
|
eqtrdi |
⊢ ( 𝑥 = 𝑋 → ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) = ( 𝐹 ∖ { 𝑋 } ) ) |
151 |
150
|
fveq2d |
⊢ ( 𝑥 = 𝑋 → ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) = ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑋 } ) ) ) |
152 |
146 151
|
eleq12d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ↔ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑋 } ) ) ) ) |
153 |
152
|
notbid |
⊢ ( 𝑥 = 𝑋 → ( ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ↔ ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑋 } ) ) ) ) |
154 |
153
|
ralbidv |
⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ↔ ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑋 } ) ) ) ) |
155 |
154
|
ralsng |
⊢ ( 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) → ( ∀ 𝑥 ∈ { 𝑋 } ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ↔ ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑋 } ) ) ) ) |
156 |
155
|
3ad2ant3 |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ( ∀ 𝑥 ∈ { 𝑋 } ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ↔ ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( 𝐹 ∖ { 𝑋 } ) ) ) ) |
157 |
145 156
|
mpbird |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ∀ 𝑥 ∈ { 𝑋 } ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) |
158 |
|
ralunb |
⊢ ( ∀ 𝑥 ∈ ( 𝐹 ∪ { 𝑋 } ) ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ↔ ( ∀ 𝑥 ∈ 𝐹 ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ∧ ∀ 𝑥 ∈ { 𝑋 } ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) ) |
159 |
113 157 158
|
sylanbrc |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ∀ 𝑥 ∈ ( 𝐹 ∪ { 𝑋 } ) ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) |
160 |
1 75 31 21 25 24
|
islinds2 |
⊢ ( 𝑊 ∈ LVec → ( ( 𝐹 ∪ { 𝑋 } ) ∈ ( LIndS ‘ 𝑊 ) ↔ ( ( 𝐹 ∪ { 𝑋 } ) ⊆ ( Base ‘ 𝑊 ) ∧ ∀ 𝑥 ∈ ( 𝐹 ∪ { 𝑋 } ) ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) ) ) |
161 |
160
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ( ( 𝐹 ∪ { 𝑋 } ) ∈ ( LIndS ‘ 𝑊 ) ↔ ( ( 𝐹 ∪ { 𝑋 } ) ⊆ ( Base ‘ 𝑊 ) ∧ ∀ 𝑥 ∈ ( 𝐹 ∪ { 𝑋 } ) ∀ 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ¬ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ∈ ( ( LSpan ‘ 𝑊 ) ‘ ( ( 𝐹 ∪ { 𝑋 } ) ∖ { 𝑥 } ) ) ) ) ) |
162 |
8 159 161
|
mpbir2and |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ∧ 𝑋 ∈ ( ( Base ‘ 𝑊 ) ∖ ( ( LSpan ‘ 𝑊 ) ‘ 𝐹 ) ) ) → ( 𝐹 ∪ { 𝑋 } ) ∈ ( LIndS ‘ 𝑊 ) ) |