Step |
Hyp |
Ref |
Expression |
1 |
|
lshpnelb.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lshpnelb.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
3 |
|
lshpnelb.p |
⊢ ⊕ = ( LSSum ‘ 𝑊 ) |
4 |
|
lshpnelb.h |
⊢ 𝐻 = ( LSHyp ‘ 𝑊 ) |
5 |
|
lshpnelb.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
6 |
|
lshpnelb.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝐻 ) |
7 |
|
lshpnelb.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
8 |
|
eqid |
⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 ) |
9 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
10 |
5 9
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
11 |
1 2 8 3 4 10
|
islshpsm |
⊢ ( 𝜑 → ( 𝑈 ∈ 𝐻 ↔ ( 𝑈 ∈ ( LSubSp ‘ 𝑊 ) ∧ 𝑈 ≠ 𝑉 ∧ ∃ 𝑣 ∈ 𝑉 ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) ) ) |
12 |
6 11
|
mpbid |
⊢ ( 𝜑 → ( 𝑈 ∈ ( LSubSp ‘ 𝑊 ) ∧ 𝑈 ≠ 𝑉 ∧ ∃ 𝑣 ∈ 𝑉 ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) ) |
13 |
12
|
simp3d |
⊢ ( 𝜑 → ∃ 𝑣 ∈ 𝑉 ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) |
14 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ 𝑈 ) → ∃ 𝑣 ∈ 𝑉 ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) |
15 |
|
simp1l |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑋 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑉 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) → 𝜑 ) |
16 |
|
simp2 |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑋 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑉 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) → 𝑣 ∈ 𝑉 ) |
17 |
8
|
lsssssubg |
⊢ ( 𝑊 ∈ LMod → ( LSubSp ‘ 𝑊 ) ⊆ ( SubGrp ‘ 𝑊 ) ) |
18 |
10 17
|
syl |
⊢ ( 𝜑 → ( LSubSp ‘ 𝑊 ) ⊆ ( SubGrp ‘ 𝑊 ) ) |
19 |
8 4 10 6
|
lshplss |
⊢ ( 𝜑 → 𝑈 ∈ ( LSubSp ‘ 𝑊 ) ) |
20 |
18 19
|
sseldd |
⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝑊 ) ) |
21 |
1 8 2
|
lspsncl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
22 |
10 7 21
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
23 |
18 22
|
sseldd |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝑊 ) ) |
24 |
3
|
lsmub1 |
⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝑊 ) ) → 𝑈 ⊆ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ) |
25 |
20 23 24
|
syl2anc |
⊢ ( 𝜑 → 𝑈 ⊆ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ) |
26 |
25
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ 𝑈 ) → 𝑈 ⊆ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ) |
27 |
3
|
lsmub2 |
⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝑊 ) ) → ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ) |
28 |
20 23 27
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ) |
29 |
1 2
|
lspsnid |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
30 |
10 7 29
|
syl2anc |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
31 |
28 30
|
sseldd |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ) |
32 |
|
nelne1 |
⊢ ( ( 𝑋 ∈ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ∧ ¬ 𝑋 ∈ 𝑈 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ≠ 𝑈 ) |
33 |
31 32
|
sylan |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ 𝑈 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ≠ 𝑈 ) |
34 |
33
|
necomd |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ 𝑈 ) → 𝑈 ≠ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ) |
35 |
|
df-pss |
⊢ ( 𝑈 ⊊ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ↔ ( 𝑈 ⊆ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑈 ≠ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ) ) |
36 |
26 34 35
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ 𝑈 ) → 𝑈 ⊊ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ) |
37 |
36
|
3ad2ant1 |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑋 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑉 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) → 𝑈 ⊊ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ) |
38 |
8 3
|
lsmcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ ( LSubSp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( LSubSp ‘ 𝑊 ) ) |
39 |
10 19 22 38
|
syl3anc |
⊢ ( 𝜑 → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( LSubSp ‘ 𝑊 ) ) |
40 |
1 8
|
lssss |
⊢ ( ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( LSubSp ‘ 𝑊 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ⊆ 𝑉 ) |
41 |
39 40
|
syl |
⊢ ( 𝜑 → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ⊆ 𝑉 ) |
42 |
41
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ⊆ 𝑉 ) |
43 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) |
44 |
42 43
|
sseqtrrd |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ⊆ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) ) |
45 |
44
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑋 ∈ 𝑈 ) ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ⊆ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) ) |
46 |
45
|
3adant2 |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑋 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑉 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ⊆ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) ) |
47 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → 𝑊 ∈ LVec ) |
48 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → 𝑈 ∈ ( LSubSp ‘ 𝑊 ) ) |
49 |
39
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( LSubSp ‘ 𝑊 ) ) |
50 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) → 𝑣 ∈ 𝑉 ) |
51 |
1 8 2 3 47 48 49 50
|
lsmcv |
⊢ ( ( ( 𝜑 ∧ 𝑣 ∈ 𝑉 ) ∧ 𝑈 ⊊ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ⊆ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) ) |
52 |
15 16 37 46 51
|
syl211anc |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑋 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑉 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) ) |
53 |
|
simp3 |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑋 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑉 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) |
54 |
52 53
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑋 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑉 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) |
55 |
54
|
rexlimdv3a |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ 𝑈 ) → ( ∃ 𝑣 ∈ 𝑉 ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑣 } ) ) = 𝑉 → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) ) |
56 |
14 55
|
mpd |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ 𝑈 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) |
57 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) → 𝑊 ∈ LMod ) |
58 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) → 𝑈 ∈ 𝐻 ) |
59 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) → 𝑋 ∈ 𝑉 ) |
60 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) |
61 |
1 2 3 4 57 58 59 60
|
lshpnel |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) → ¬ 𝑋 ∈ 𝑈 ) |
62 |
56 61
|
impbida |
⊢ ( 𝜑 → ( ¬ 𝑋 ∈ 𝑈 ↔ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) ) |