Step |
Hyp |
Ref |
Expression |
1 |
|
lspsolv.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lspsolv.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
3 |
|
lspsolv.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
4 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
5 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) |
6 |
|
eqid |
⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 ) |
7 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 ) |
8 |
|
eqid |
⊢ { 𝑧 ∈ 𝑉 ∣ ∃ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ( 𝑧 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) } = { 𝑧 ∈ 𝑉 ∣ ∃ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ( 𝑧 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) } |
9 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
10 |
9
|
adantr |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) → 𝑊 ∈ LMod ) |
11 |
|
simpr1 |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) → 𝐴 ⊆ 𝑉 ) |
12 |
|
simpr2 |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) → 𝑌 ∈ 𝑉 ) |
13 |
|
simpr3 |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) → 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) |
14 |
13
|
eldifad |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) → 𝑋 ∈ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ) |
15 |
1 2 3 4 5 6 7 8 10 11 12 14
|
lspsolvlem |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) → ∃ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) |
16 |
4
|
lvecdrng |
⊢ ( 𝑊 ∈ LVec → ( Scalar ‘ 𝑊 ) ∈ DivRing ) |
17 |
16
|
ad2antrr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( Scalar ‘ 𝑊 ) ∈ DivRing ) |
18 |
|
simprl |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
19 |
10
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → 𝑊 ∈ LMod ) |
20 |
12
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → 𝑌 ∈ 𝑉 ) |
21 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) |
22 |
|
eqid |
⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 ) |
23 |
1 4 7 21 22
|
lmod0vs |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) = ( 0g ‘ 𝑊 ) ) |
24 |
19 20 23
|
syl2anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) = ( 0g ‘ 𝑊 ) ) |
25 |
24
|
oveq2d |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( 𝑋 ( +g ‘ 𝑊 ) ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) = ( 𝑋 ( +g ‘ 𝑊 ) ( 0g ‘ 𝑊 ) ) ) |
26 |
11
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → 𝐴 ⊆ 𝑉 ) |
27 |
20
|
snssd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → { 𝑌 } ⊆ 𝑉 ) |
28 |
26 27
|
unssd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( 𝐴 ∪ { 𝑌 } ) ⊆ 𝑉 ) |
29 |
1 3
|
lspssv |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐴 ∪ { 𝑌 } ) ⊆ 𝑉 ) → ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ⊆ 𝑉 ) |
30 |
19 28 29
|
syl2anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ⊆ 𝑉 ) |
31 |
30
|
ssdifssd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ⊆ 𝑉 ) |
32 |
13
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) |
33 |
31 32
|
sseldd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → 𝑋 ∈ 𝑉 ) |
34 |
1 6 22
|
lmod0vrid |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑋 ( +g ‘ 𝑊 ) ( 0g ‘ 𝑊 ) ) = 𝑋 ) |
35 |
19 33 34
|
syl2anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( 𝑋 ( +g ‘ 𝑊 ) ( 0g ‘ 𝑊 ) ) = 𝑋 ) |
36 |
25 35
|
eqtrd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( 𝑋 ( +g ‘ 𝑊 ) ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) = 𝑋 ) |
37 |
36 32
|
eqeltrd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( 𝑋 ( +g ‘ 𝑊 ) ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) |
38 |
37
|
eldifbd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ¬ ( 𝑋 ( +g ‘ 𝑊 ) ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) |
39 |
|
simprr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) |
40 |
|
oveq1 |
⊢ ( 𝑟 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) → ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) = ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) |
41 |
40
|
oveq2d |
⊢ ( 𝑟 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) → ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) = ( 𝑋 ( +g ‘ 𝑊 ) ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ) |
42 |
41
|
eleq1d |
⊢ ( 𝑟 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) → ( ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ↔ ( 𝑋 ( +g ‘ 𝑊 ) ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) |
43 |
39 42
|
syl5ibcom |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( 𝑟 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) → ( 𝑋 ( +g ‘ 𝑊 ) ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) |
44 |
43
|
necon3bd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( ¬ ( 𝑋 ( +g ‘ 𝑊 ) ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) → 𝑟 ≠ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) |
45 |
38 44
|
mpd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → 𝑟 ≠ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
46 |
|
eqid |
⊢ ( .r ‘ ( Scalar ‘ 𝑊 ) ) = ( .r ‘ ( Scalar ‘ 𝑊 ) ) |
47 |
|
eqid |
⊢ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) = ( 1r ‘ ( Scalar ‘ 𝑊 ) ) |
48 |
|
eqid |
⊢ ( invr ‘ ( Scalar ‘ 𝑊 ) ) = ( invr ‘ ( Scalar ‘ 𝑊 ) ) |
49 |
5 21 46 47 48
|
drnginvrl |
⊢ ( ( ( Scalar ‘ 𝑊 ) ∈ DivRing ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑟 ≠ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) → ( ( ( invr ‘ ( Scalar ‘ 𝑊 ) ) ‘ 𝑟 ) ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) = ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) |
50 |
17 18 45 49
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( ( ( invr ‘ ( Scalar ‘ 𝑊 ) ) ‘ 𝑟 ) ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) = ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ) |
51 |
50
|
oveq1d |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( ( ( ( invr ‘ ( Scalar ‘ 𝑊 ) ) ‘ 𝑟 ) ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) = ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) |
52 |
5 21 48
|
drnginvrcl |
⊢ ( ( ( Scalar ‘ 𝑊 ) ∈ DivRing ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑟 ≠ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) → ( ( invr ‘ ( Scalar ‘ 𝑊 ) ) ‘ 𝑟 ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
53 |
17 18 45 52
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( ( invr ‘ ( Scalar ‘ 𝑊 ) ) ‘ 𝑟 ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
54 |
1 4 7 5 46
|
lmodvsass |
⊢ ( ( 𝑊 ∈ LMod ∧ ( ( ( invr ‘ ( Scalar ‘ 𝑊 ) ) ‘ 𝑟 ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑌 ∈ 𝑉 ) ) → ( ( ( ( invr ‘ ( Scalar ‘ 𝑊 ) ) ‘ 𝑟 ) ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) = ( ( ( invr ‘ ( Scalar ‘ 𝑊 ) ) ‘ 𝑟 ) ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ) |
55 |
19 53 18 20 54
|
syl13anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( ( ( ( invr ‘ ( Scalar ‘ 𝑊 ) ) ‘ 𝑟 ) ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) = ( ( ( invr ‘ ( Scalar ‘ 𝑊 ) ) ‘ 𝑟 ) ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ) |
56 |
1 4 7 47
|
lmodvs1 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) = 𝑌 ) |
57 |
19 20 56
|
syl2anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) = 𝑌 ) |
58 |
51 55 57
|
3eqtr3d |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( ( ( invr ‘ ( Scalar ‘ 𝑊 ) ) ‘ 𝑟 ) ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) = 𝑌 ) |
59 |
33
|
snssd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → { 𝑋 } ⊆ 𝑉 ) |
60 |
26 59
|
unssd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( 𝐴 ∪ { 𝑋 } ) ⊆ 𝑉 ) |
61 |
1 2 3
|
lspcl |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐴 ∪ { 𝑋 } ) ⊆ 𝑉 ) → ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ∈ 𝑆 ) |
62 |
19 60 61
|
syl2anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ∈ 𝑆 ) |
63 |
1 4 7 5
|
lmodvscl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑌 ∈ 𝑉 ) → ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ∈ 𝑉 ) |
64 |
19 18 20 63
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ∈ 𝑉 ) |
65 |
|
eqid |
⊢ ( -g ‘ 𝑊 ) = ( -g ‘ 𝑊 ) |
66 |
1 6 65
|
lmodvpncan |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) → ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ( +g ‘ 𝑊 ) 𝑋 ) ( -g ‘ 𝑊 ) 𝑋 ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) |
67 |
19 64 33 66
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ( +g ‘ 𝑊 ) 𝑋 ) ( -g ‘ 𝑊 ) 𝑋 ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) |
68 |
1 6
|
lmodcom |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ( +g ‘ 𝑊 ) 𝑋 ) = ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ) |
69 |
19 64 33 68
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ( +g ‘ 𝑊 ) 𝑋 ) = ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ) |
70 |
|
ssun1 |
⊢ 𝐴 ⊆ ( 𝐴 ∪ { 𝑋 } ) |
71 |
70
|
a1i |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → 𝐴 ⊆ ( 𝐴 ∪ { 𝑋 } ) ) |
72 |
1 3
|
lspss |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐴 ∪ { 𝑋 } ) ⊆ 𝑉 ∧ 𝐴 ⊆ ( 𝐴 ∪ { 𝑋 } ) ) → ( 𝑁 ‘ 𝐴 ) ⊆ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) |
73 |
19 60 71 72
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( 𝑁 ‘ 𝐴 ) ⊆ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) |
74 |
73 39
|
sseldd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) |
75 |
69 74
|
eqeltrd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ( +g ‘ 𝑊 ) 𝑋 ) ∈ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) |
76 |
1 3
|
lspssid |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐴 ∪ { 𝑋 } ) ⊆ 𝑉 ) → ( 𝐴 ∪ { 𝑋 } ) ⊆ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) |
77 |
19 60 76
|
syl2anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( 𝐴 ∪ { 𝑋 } ) ⊆ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) |
78 |
|
snidg |
⊢ ( 𝑋 ∈ 𝑉 → 𝑋 ∈ { 𝑋 } ) |
79 |
|
elun2 |
⊢ ( 𝑋 ∈ { 𝑋 } → 𝑋 ∈ ( 𝐴 ∪ { 𝑋 } ) ) |
80 |
33 78 79
|
3syl |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → 𝑋 ∈ ( 𝐴 ∪ { 𝑋 } ) ) |
81 |
77 80
|
sseldd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → 𝑋 ∈ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) |
82 |
65 2
|
lssvsubcl |
⊢ ( ( ( 𝑊 ∈ LMod ∧ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ∈ 𝑆 ) ∧ ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ( +g ‘ 𝑊 ) 𝑋 ) ∈ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ∧ 𝑋 ∈ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) ) → ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ( +g ‘ 𝑊 ) 𝑋 ) ( -g ‘ 𝑊 ) 𝑋 ) ∈ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) |
83 |
19 62 75 81 82
|
syl22anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ( +g ‘ 𝑊 ) 𝑋 ) ( -g ‘ 𝑊 ) 𝑋 ) ∈ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) |
84 |
67 83
|
eqeltrrd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ∈ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) |
85 |
4 7 5 2
|
lssvscl |
⊢ ( ( ( 𝑊 ∈ LMod ∧ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ∈ 𝑆 ) ∧ ( ( ( invr ‘ ( Scalar ‘ 𝑊 ) ) ‘ 𝑟 ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ∈ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) ) → ( ( ( invr ‘ ( Scalar ‘ 𝑊 ) ) ‘ 𝑟 ) ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) |
86 |
19 62 53 84 85
|
syl22anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → ( ( ( invr ‘ ( Scalar ‘ 𝑊 ) ) ‘ 𝑟 ) ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) |
87 |
58 86
|
eqeltrrd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑋 ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ∈ ( 𝑁 ‘ 𝐴 ) ) ) → 𝑌 ∈ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) |
88 |
15 87
|
rexlimddv |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ( ( 𝑁 ‘ ( 𝐴 ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ 𝐴 ) ) ) ) → 𝑌 ∈ ( 𝑁 ‘ ( 𝐴 ∪ { 𝑋 } ) ) ) |