Step |
Hyp |
Ref |
Expression |
1 |
|
lshpnel.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lshpnel.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
3 |
|
lshpnel.p |
⊢ ⊕ = ( LSSum ‘ 𝑊 ) |
4 |
|
lshpnel.h |
⊢ 𝐻 = ( LSHyp ‘ 𝑊 ) |
5 |
|
lshpnel.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
6 |
|
lshpnel.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝐻 ) |
7 |
|
lshpnel.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
8 |
|
lshpnel.e |
⊢ ( 𝜑 → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) |
9 |
1 4 5 6
|
lshpne |
⊢ ( 𝜑 → 𝑈 ≠ 𝑉 ) |
10 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → 𝑊 ∈ LMod ) |
11 |
|
eqid |
⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 ) |
12 |
11
|
lsssssubg |
⊢ ( 𝑊 ∈ LMod → ( LSubSp ‘ 𝑊 ) ⊆ ( SubGrp ‘ 𝑊 ) ) |
13 |
10 12
|
syl |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → ( LSubSp ‘ 𝑊 ) ⊆ ( SubGrp ‘ 𝑊 ) ) |
14 |
11 4 5 6
|
lshplss |
⊢ ( 𝜑 → 𝑈 ∈ ( LSubSp ‘ 𝑊 ) ) |
15 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → 𝑈 ∈ ( LSubSp ‘ 𝑊 ) ) |
16 |
13 15
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → 𝑈 ∈ ( SubGrp ‘ 𝑊 ) ) |
17 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → 𝑋 ∈ 𝑉 ) |
18 |
1 11 2
|
lspsncl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
19 |
10 17 18
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
20 |
13 19
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝑊 ) ) |
21 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → 𝑋 ∈ 𝑈 ) |
22 |
11 2 10 15 21
|
lspsnel5a |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → ( 𝑁 ‘ { 𝑋 } ) ⊆ 𝑈 ) |
23 |
3
|
lsmss2 |
⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { 𝑋 } ) ⊆ 𝑈 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑈 ) |
24 |
16 20 22 23
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑈 ) |
25 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) |
26 |
24 25
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑈 ) → 𝑈 = 𝑉 ) |
27 |
26
|
ex |
⊢ ( 𝜑 → ( 𝑋 ∈ 𝑈 → 𝑈 = 𝑉 ) ) |
28 |
27
|
necon3ad |
⊢ ( 𝜑 → ( 𝑈 ≠ 𝑉 → ¬ 𝑋 ∈ 𝑈 ) ) |
29 |
9 28
|
mpd |
⊢ ( 𝜑 → ¬ 𝑋 ∈ 𝑈 ) |