Step |
Hyp |
Ref |
Expression |
1 |
|
lshpnel2.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lshpnel2.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
3 |
|
lshpnel2.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
4 |
|
lshpnel2.p |
⊢ ⊕ = ( LSSum ‘ 𝑊 ) |
5 |
|
lshpnel2.h |
⊢ 𝐻 = ( LSHyp ‘ 𝑊 ) |
6 |
|
lshpnel2.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
7 |
|
lshpnel2.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑆 ) |
8 |
|
lshpnel2.t |
⊢ ( 𝜑 → 𝑈 ≠ 𝑉 ) |
9 |
|
lshpnel2.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
10 |
|
lshpnel2.e |
⊢ ( 𝜑 → ¬ 𝑋 ∈ 𝑈 ) |
11 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑈 ∈ 𝐻 ) → ¬ 𝑋 ∈ 𝑈 ) |
12 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑈 ∈ 𝐻 ) → 𝑊 ∈ LVec ) |
13 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑈 ∈ 𝐻 ) → 𝑈 ∈ 𝐻 ) |
14 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑈 ∈ 𝐻 ) → 𝑋 ∈ 𝑉 ) |
15 |
1 3 4 5 12 13 14
|
lshpnelb |
⊢ ( ( 𝜑 ∧ 𝑈 ∈ 𝐻 ) → ( ¬ 𝑋 ∈ 𝑈 ↔ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) ) |
16 |
11 15
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑈 ∈ 𝐻 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) |
17 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) → 𝑈 ∈ 𝑆 ) |
18 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) → 𝑈 ≠ 𝑉 ) |
19 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) → 𝑋 ∈ 𝑉 ) |
20 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
21 |
6 20
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
22 |
2 3
|
lspid |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ) → ( 𝑁 ‘ 𝑈 ) = 𝑈 ) |
23 |
21 7 22
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝑈 ) = 𝑈 ) |
24 |
23
|
uneq1d |
⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑈 ) ∪ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝑈 ∪ ( 𝑁 ‘ { 𝑋 } ) ) ) |
25 |
24
|
fveq2d |
⊢ ( 𝜑 → ( 𝑁 ‘ ( ( 𝑁 ‘ 𝑈 ) ∪ ( 𝑁 ‘ { 𝑋 } ) ) ) = ( 𝑁 ‘ ( 𝑈 ∪ ( 𝑁 ‘ { 𝑋 } ) ) ) ) |
26 |
1 2
|
lssss |
⊢ ( 𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉 ) |
27 |
7 26
|
syl |
⊢ ( 𝜑 → 𝑈 ⊆ 𝑉 ) |
28 |
9
|
snssd |
⊢ ( 𝜑 → { 𝑋 } ⊆ 𝑉 ) |
29 |
1 3
|
lspun |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ { 𝑋 } ⊆ 𝑉 ) → ( 𝑁 ‘ ( 𝑈 ∪ { 𝑋 } ) ) = ( 𝑁 ‘ ( ( 𝑁 ‘ 𝑈 ) ∪ ( 𝑁 ‘ { 𝑋 } ) ) ) ) |
30 |
21 27 28 29
|
syl3anc |
⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑈 ∪ { 𝑋 } ) ) = ( 𝑁 ‘ ( ( 𝑁 ‘ 𝑈 ) ∪ ( 𝑁 ‘ { 𝑋 } ) ) ) ) |
31 |
1 2 3
|
lspsncl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ 𝑆 ) |
32 |
21 9 31
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ 𝑆 ) |
33 |
2 3 4
|
lsmsp |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ 𝑆 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝑁 ‘ ( 𝑈 ∪ ( 𝑁 ‘ { 𝑋 } ) ) ) ) |
34 |
21 7 32 33
|
syl3anc |
⊢ ( 𝜑 → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝑁 ‘ ( 𝑈 ∪ ( 𝑁 ‘ { 𝑋 } ) ) ) ) |
35 |
25 30 34
|
3eqtr4rd |
⊢ ( 𝜑 → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝑁 ‘ ( 𝑈 ∪ { 𝑋 } ) ) ) |
36 |
35
|
eqeq1d |
⊢ ( 𝜑 → ( ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ↔ ( 𝑁 ‘ ( 𝑈 ∪ { 𝑋 } ) ) = 𝑉 ) ) |
37 |
36
|
biimpa |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) → ( 𝑁 ‘ ( 𝑈 ∪ { 𝑋 } ) ) = 𝑉 ) |
38 |
|
sneq |
⊢ ( 𝑣 = 𝑋 → { 𝑣 } = { 𝑋 } ) |
39 |
38
|
uneq2d |
⊢ ( 𝑣 = 𝑋 → ( 𝑈 ∪ { 𝑣 } ) = ( 𝑈 ∪ { 𝑋 } ) ) |
40 |
39
|
fveqeq2d |
⊢ ( 𝑣 = 𝑋 → ( ( 𝑁 ‘ ( 𝑈 ∪ { 𝑣 } ) ) = 𝑉 ↔ ( 𝑁 ‘ ( 𝑈 ∪ { 𝑋 } ) ) = 𝑉 ) ) |
41 |
40
|
rspcev |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ ( 𝑁 ‘ ( 𝑈 ∪ { 𝑋 } ) ) = 𝑉 ) → ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑈 ∪ { 𝑣 } ) ) = 𝑉 ) |
42 |
19 37 41
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) → ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑈 ∪ { 𝑣 } ) ) = 𝑉 ) |
43 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) → 𝑊 ∈ LVec ) |
44 |
1 3 2 5
|
islshp |
⊢ ( 𝑊 ∈ LVec → ( 𝑈 ∈ 𝐻 ↔ ( 𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑈 ∪ { 𝑣 } ) ) = 𝑉 ) ) ) |
45 |
43 44
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) → ( 𝑈 ∈ 𝐻 ↔ ( 𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃ 𝑣 ∈ 𝑉 ( 𝑁 ‘ ( 𝑈 ∪ { 𝑣 } ) ) = 𝑉 ) ) ) |
46 |
17 18 42 45
|
mpbir3and |
⊢ ( ( 𝜑 ∧ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) → 𝑈 ∈ 𝐻 ) |
47 |
16 46
|
impbida |
⊢ ( 𝜑 → ( 𝑈 ∈ 𝐻 ↔ ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) ) |