Step |
Hyp |
Ref |
Expression |
1 |
|
lspprat.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lspprat.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
3 |
|
lspprat.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
4 |
|
lspprat.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
5 |
|
lspprat.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑆 ) |
6 |
|
lspprat.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
7 |
|
lspprat.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
8 |
|
lspprat.p |
⊢ ( 𝜑 → 𝑈 ⊊ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) |
9 |
|
lsppratlem1.o |
⊢ 0 = ( 0g ‘ 𝑊 ) |
10 |
|
lsppratlem1.x2 |
⊢ ( 𝜑 → 𝑥 ∈ ( 𝑈 ∖ { 0 } ) ) |
11 |
|
lsppratlem1.y2 |
⊢ ( 𝜑 → 𝑦 ∈ ( 𝑈 ∖ ( 𝑁 ‘ { 𝑥 } ) ) ) |
12 |
|
lsppratlem4.x3 |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑁 ‘ { 𝑥 , 𝑌 } ) ) |
13 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
14 |
4 13
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
15 |
1 2
|
lssss |
⊢ ( 𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉 ) |
16 |
5 15
|
syl |
⊢ ( 𝜑 → 𝑈 ⊆ 𝑉 ) |
17 |
16
|
ssdifssd |
⊢ ( 𝜑 → ( 𝑈 ∖ { 0 } ) ⊆ 𝑉 ) |
18 |
17 10
|
sseldd |
⊢ ( 𝜑 → 𝑥 ∈ 𝑉 ) |
19 |
16
|
ssdifssd |
⊢ ( 𝜑 → ( 𝑈 ∖ ( 𝑁 ‘ { 𝑥 } ) ) ⊆ 𝑉 ) |
20 |
19 11
|
sseldd |
⊢ ( 𝜑 → 𝑦 ∈ 𝑉 ) |
21 |
1 2 3 14 18 20
|
lspprcl |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑥 , 𝑦 } ) ∈ 𝑆 ) |
22 |
|
df-pr |
⊢ { 𝑥 , 𝑌 } = ( { 𝑥 } ∪ { 𝑌 } ) |
23 |
|
snsspr1 |
⊢ { 𝑥 } ⊆ { 𝑥 , 𝑦 } |
24 |
18 20
|
prssd |
⊢ ( 𝜑 → { 𝑥 , 𝑦 } ⊆ 𝑉 ) |
25 |
1 3
|
lspssid |
⊢ ( ( 𝑊 ∈ LMod ∧ { 𝑥 , 𝑦 } ⊆ 𝑉 ) → { 𝑥 , 𝑦 } ⊆ ( 𝑁 ‘ { 𝑥 , 𝑦 } ) ) |
26 |
14 24 25
|
syl2anc |
⊢ ( 𝜑 → { 𝑥 , 𝑦 } ⊆ ( 𝑁 ‘ { 𝑥 , 𝑦 } ) ) |
27 |
23 26
|
sstrid |
⊢ ( 𝜑 → { 𝑥 } ⊆ ( 𝑁 ‘ { 𝑥 , 𝑦 } ) ) |
28 |
18
|
snssd |
⊢ ( 𝜑 → { 𝑥 } ⊆ 𝑉 ) |
29 |
8
|
pssssd |
⊢ ( 𝜑 → 𝑈 ⊆ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) |
30 |
1 2 3 14 18 7
|
lspprcl |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑥 , 𝑌 } ) ∈ 𝑆 ) |
31 |
|
df-pr |
⊢ { 𝑋 , 𝑌 } = ( { 𝑋 } ∪ { 𝑌 } ) |
32 |
12
|
snssd |
⊢ ( 𝜑 → { 𝑋 } ⊆ ( 𝑁 ‘ { 𝑥 , 𝑌 } ) ) |
33 |
|
snsspr2 |
⊢ { 𝑌 } ⊆ { 𝑥 , 𝑌 } |
34 |
18 7
|
prssd |
⊢ ( 𝜑 → { 𝑥 , 𝑌 } ⊆ 𝑉 ) |
35 |
1 3
|
lspssid |
⊢ ( ( 𝑊 ∈ LMod ∧ { 𝑥 , 𝑌 } ⊆ 𝑉 ) → { 𝑥 , 𝑌 } ⊆ ( 𝑁 ‘ { 𝑥 , 𝑌 } ) ) |
36 |
14 34 35
|
syl2anc |
⊢ ( 𝜑 → { 𝑥 , 𝑌 } ⊆ ( 𝑁 ‘ { 𝑥 , 𝑌 } ) ) |
37 |
33 36
|
sstrid |
⊢ ( 𝜑 → { 𝑌 } ⊆ ( 𝑁 ‘ { 𝑥 , 𝑌 } ) ) |
38 |
32 37
|
unssd |
⊢ ( 𝜑 → ( { 𝑋 } ∪ { 𝑌 } ) ⊆ ( 𝑁 ‘ { 𝑥 , 𝑌 } ) ) |
39 |
31 38
|
eqsstrid |
⊢ ( 𝜑 → { 𝑋 , 𝑌 } ⊆ ( 𝑁 ‘ { 𝑥 , 𝑌 } ) ) |
40 |
2 3
|
lspssp |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑁 ‘ { 𝑥 , 𝑌 } ) ∈ 𝑆 ∧ { 𝑋 , 𝑌 } ⊆ ( 𝑁 ‘ { 𝑥 , 𝑌 } ) ) → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ⊆ ( 𝑁 ‘ { 𝑥 , 𝑌 } ) ) |
41 |
14 30 39 40
|
syl3anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ⊆ ( 𝑁 ‘ { 𝑥 , 𝑌 } ) ) |
42 |
29 41
|
sstrd |
⊢ ( 𝜑 → 𝑈 ⊆ ( 𝑁 ‘ { 𝑥 , 𝑌 } ) ) |
43 |
22
|
fveq2i |
⊢ ( 𝑁 ‘ { 𝑥 , 𝑌 } ) = ( 𝑁 ‘ ( { 𝑥 } ∪ { 𝑌 } ) ) |
44 |
42 43
|
sseqtrdi |
⊢ ( 𝜑 → 𝑈 ⊆ ( 𝑁 ‘ ( { 𝑥 } ∪ { 𝑌 } ) ) ) |
45 |
44
|
ssdifd |
⊢ ( 𝜑 → ( 𝑈 ∖ ( 𝑁 ‘ { 𝑥 } ) ) ⊆ ( ( 𝑁 ‘ ( { 𝑥 } ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ { 𝑥 } ) ) ) |
46 |
45 11
|
sseldd |
⊢ ( 𝜑 → 𝑦 ∈ ( ( 𝑁 ‘ ( { 𝑥 } ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ { 𝑥 } ) ) ) |
47 |
1 2 3
|
lspsolv |
⊢ ( ( 𝑊 ∈ LVec ∧ ( { 𝑥 } ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑦 ∈ ( ( 𝑁 ‘ ( { 𝑥 } ∪ { 𝑌 } ) ) ∖ ( 𝑁 ‘ { 𝑥 } ) ) ) ) → 𝑌 ∈ ( 𝑁 ‘ ( { 𝑥 } ∪ { 𝑦 } ) ) ) |
48 |
4 28 7 46 47
|
syl13anc |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑁 ‘ ( { 𝑥 } ∪ { 𝑦 } ) ) ) |
49 |
|
df-pr |
⊢ { 𝑥 , 𝑦 } = ( { 𝑥 } ∪ { 𝑦 } ) |
50 |
49
|
fveq2i |
⊢ ( 𝑁 ‘ { 𝑥 , 𝑦 } ) = ( 𝑁 ‘ ( { 𝑥 } ∪ { 𝑦 } ) ) |
51 |
48 50
|
eleqtrrdi |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑁 ‘ { 𝑥 , 𝑦 } ) ) |
52 |
51
|
snssd |
⊢ ( 𝜑 → { 𝑌 } ⊆ ( 𝑁 ‘ { 𝑥 , 𝑦 } ) ) |
53 |
27 52
|
unssd |
⊢ ( 𝜑 → ( { 𝑥 } ∪ { 𝑌 } ) ⊆ ( 𝑁 ‘ { 𝑥 , 𝑦 } ) ) |
54 |
22 53
|
eqsstrid |
⊢ ( 𝜑 → { 𝑥 , 𝑌 } ⊆ ( 𝑁 ‘ { 𝑥 , 𝑦 } ) ) |
55 |
2 3
|
lspssp |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑁 ‘ { 𝑥 , 𝑦 } ) ∈ 𝑆 ∧ { 𝑥 , 𝑌 } ⊆ ( 𝑁 ‘ { 𝑥 , 𝑦 } ) ) → ( 𝑁 ‘ { 𝑥 , 𝑌 } ) ⊆ ( 𝑁 ‘ { 𝑥 , 𝑦 } ) ) |
56 |
14 21 54 55
|
syl3anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑥 , 𝑌 } ) ⊆ ( 𝑁 ‘ { 𝑥 , 𝑦 } ) ) |
57 |
56 12
|
sseldd |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑁 ‘ { 𝑥 , 𝑦 } ) ) |
58 |
57 51
|
jca |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑥 , 𝑦 } ) ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑥 , 𝑦 } ) ) ) |