Step |
Hyp |
Ref |
Expression |
1 |
|
lkrpss.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
2 |
|
lkrpss.k |
⊢ 𝐾 = ( LKer ‘ 𝑊 ) |
3 |
|
lkrpss.d |
⊢ 𝐷 = ( LDual ‘ 𝑊 ) |
4 |
|
lkrpss.o |
⊢ 0 = ( 0g ‘ 𝐷 ) |
5 |
|
lkrpss.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
6 |
|
lkrpss.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
7 |
|
lkrpss.h |
⊢ ( 𝜑 → 𝐻 ∈ 𝐹 ) |
8 |
|
df-pss |
⊢ ( ( 𝐾 ‘ 𝐺 ) ⊊ ( 𝐾 ‘ 𝐻 ) ↔ ( ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ∧ ( 𝐾 ‘ 𝐺 ) ≠ ( 𝐾 ‘ 𝐻 ) ) ) |
9 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊊ ( 𝐾 ‘ 𝐻 ) ) → ( 𝐾 ‘ 𝐺 ) ⊊ ( 𝐾 ‘ 𝐻 ) ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
11 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
12 |
5 11
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
13 |
10 1 2 12 7
|
lkrssv |
⊢ ( 𝜑 → ( 𝐾 ‘ 𝐻 ) ⊆ ( Base ‘ 𝑊 ) ) |
14 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊊ ( 𝐾 ‘ 𝐻 ) ) → ( 𝐾 ‘ 𝐻 ) ⊆ ( Base ‘ 𝑊 ) ) |
15 |
9 14
|
psssstrd |
⊢ ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊊ ( 𝐾 ‘ 𝐻 ) ) → ( 𝐾 ‘ 𝐺 ) ⊊ ( Base ‘ 𝑊 ) ) |
16 |
15
|
pssned |
⊢ ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊊ ( 𝐾 ‘ 𝐻 ) ) → ( 𝐾 ‘ 𝐺 ) ≠ ( Base ‘ 𝑊 ) ) |
17 |
8 16
|
sylan2br |
⊢ ( ( 𝜑 ∧ ( ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ∧ ( 𝐾 ‘ 𝐺 ) ≠ ( 𝐾 ‘ 𝐻 ) ) ) → ( 𝐾 ‘ 𝐺 ) ≠ ( Base ‘ 𝑊 ) ) |
18 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) → ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) |
19 |
|
eqid |
⊢ ( LSHyp ‘ 𝑊 ) = ( LSHyp ‘ 𝑊 ) |
20 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) → 𝑊 ∈ LVec ) |
21 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾 ‘ 𝐺 ) ∈ ( LSHyp ‘ 𝑊 ) ) → ( 𝐾 ‘ 𝐺 ) ∈ ( LSHyp ‘ 𝑊 ) ) |
22 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾 ‘ 𝐺 ) = ( Base ‘ 𝑊 ) ) → ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) |
23 |
13
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾 ‘ 𝐺 ) = ( Base ‘ 𝑊 ) ) → ( 𝐾 ‘ 𝐻 ) ⊆ ( Base ‘ 𝑊 ) ) |
24 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾 ‘ 𝐺 ) = ( Base ‘ 𝑊 ) ) → ( 𝐾 ‘ 𝐺 ) = ( Base ‘ 𝑊 ) ) |
25 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾 ‘ 𝐺 ) = ( Base ‘ 𝑊 ) ) → ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) |
26 |
24 25
|
eqsstrrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾 ‘ 𝐺 ) = ( Base ‘ 𝑊 ) ) → ( Base ‘ 𝑊 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) |
27 |
23 26
|
eqssd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾 ‘ 𝐺 ) = ( Base ‘ 𝑊 ) ) → ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) |
28 |
10 19 1 2 5 7
|
lkrshp4 |
⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐻 ) ≠ ( Base ‘ 𝑊 ) ↔ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ) |
29 |
28
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾 ‘ 𝐺 ) = ( Base ‘ 𝑊 ) ) → ( ( 𝐾 ‘ 𝐻 ) ≠ ( Base ‘ 𝑊 ) ↔ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ) |
30 |
29
|
necon1bbid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾 ‘ 𝐺 ) = ( Base ‘ 𝑊 ) ) → ( ¬ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ↔ ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) ) |
31 |
27 30
|
mpbird |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾 ‘ 𝐺 ) = ( Base ‘ 𝑊 ) ) → ¬ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) |
32 |
22 31
|
pm2.21dd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾 ‘ 𝐺 ) = ( Base ‘ 𝑊 ) ) → ( 𝐾 ‘ 𝐺 ) ∈ ( LSHyp ‘ 𝑊 ) ) |
33 |
10 19 1 2 5 6
|
lkrshpor |
⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) ∈ ( LSHyp ‘ 𝑊 ) ∨ ( 𝐾 ‘ 𝐺 ) = ( Base ‘ 𝑊 ) ) ) |
34 |
33
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) → ( ( 𝐾 ‘ 𝐺 ) ∈ ( LSHyp ‘ 𝑊 ) ∨ ( 𝐾 ‘ 𝐺 ) = ( Base ‘ 𝑊 ) ) ) |
35 |
21 32 34
|
mpjaodan |
⊢ ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) → ( 𝐾 ‘ 𝐺 ) ∈ ( LSHyp ‘ 𝑊 ) ) |
36 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) → ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) |
37 |
19 20 35 36
|
lshpcmp |
⊢ ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) → ( ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ↔ ( 𝐾 ‘ 𝐺 ) = ( 𝐾 ‘ 𝐻 ) ) ) |
38 |
18 37
|
mpbid |
⊢ ( ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) ∧ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) → ( 𝐾 ‘ 𝐺 ) = ( 𝐾 ‘ 𝐻 ) ) |
39 |
38
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) → ( ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) → ( 𝐾 ‘ 𝐺 ) = ( 𝐾 ‘ 𝐻 ) ) ) |
40 |
39
|
necon3ad |
⊢ ( ( 𝜑 ∧ ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) → ( ( 𝐾 ‘ 𝐺 ) ≠ ( 𝐾 ‘ 𝐻 ) → ¬ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ) |
41 |
40
|
impr |
⊢ ( ( 𝜑 ∧ ( ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ∧ ( 𝐾 ‘ 𝐺 ) ≠ ( 𝐾 ‘ 𝐻 ) ) ) → ¬ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) |
42 |
28
|
necon1bbid |
⊢ ( 𝜑 → ( ¬ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ↔ ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) ) |
43 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ∧ ( 𝐾 ‘ 𝐺 ) ≠ ( 𝐾 ‘ 𝐻 ) ) ) → ( ¬ ( 𝐾 ‘ 𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ↔ ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) ) |
44 |
41 43
|
mpbid |
⊢ ( ( 𝜑 ∧ ( ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ∧ ( 𝐾 ‘ 𝐺 ) ≠ ( 𝐾 ‘ 𝐻 ) ) ) → ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) |
45 |
17 44
|
jca |
⊢ ( ( 𝜑 ∧ ( ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ∧ ( 𝐾 ‘ 𝐺 ) ≠ ( 𝐾 ‘ 𝐻 ) ) ) → ( ( 𝐾 ‘ 𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) ) |
46 |
10 1 2 12 6
|
lkrssv |
⊢ ( 𝜑 → ( 𝐾 ‘ 𝐺 ) ⊆ ( Base ‘ 𝑊 ) ) |
47 |
46
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝐾 ‘ 𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) ) → ( 𝐾 ‘ 𝐺 ) ⊆ ( Base ‘ 𝑊 ) ) |
48 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( ( 𝐾 ‘ 𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) ) → ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) |
49 |
48
|
eqcomd |
⊢ ( ( 𝜑 ∧ ( ( 𝐾 ‘ 𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) ) → ( Base ‘ 𝑊 ) = ( 𝐾 ‘ 𝐻 ) ) |
50 |
47 49
|
sseqtrd |
⊢ ( ( 𝜑 ∧ ( ( 𝐾 ‘ 𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) ) → ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ) |
51 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( ( 𝐾 ‘ 𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) ) → ( 𝐾 ‘ 𝐺 ) ≠ ( Base ‘ 𝑊 ) ) |
52 |
51 49
|
neeqtrd |
⊢ ( ( 𝜑 ∧ ( ( 𝐾 ‘ 𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) ) → ( 𝐾 ‘ 𝐺 ) ≠ ( 𝐾 ‘ 𝐻 ) ) |
53 |
50 52
|
jca |
⊢ ( ( 𝜑 ∧ ( ( 𝐾 ‘ 𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) ) → ( ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ∧ ( 𝐾 ‘ 𝐺 ) ≠ ( 𝐾 ‘ 𝐻 ) ) ) |
54 |
45 53
|
impbida |
⊢ ( 𝜑 → ( ( ( 𝐾 ‘ 𝐺 ) ⊆ ( 𝐾 ‘ 𝐻 ) ∧ ( 𝐾 ‘ 𝐺 ) ≠ ( 𝐾 ‘ 𝐻 ) ) ↔ ( ( 𝐾 ‘ 𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) ) ) |
55 |
8 54
|
syl5bb |
⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) ⊊ ( 𝐾 ‘ 𝐻 ) ↔ ( ( 𝐾 ‘ 𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) ) ) |
56 |
10 1 2 3 4 12 6
|
lkr0f2 |
⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) = ( Base ‘ 𝑊 ) ↔ 𝐺 = 0 ) ) |
57 |
56
|
necon3bid |
⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) ≠ ( Base ‘ 𝑊 ) ↔ 𝐺 ≠ 0 ) ) |
58 |
10 1 2 3 4 12 7
|
lkr0f2 |
⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ↔ 𝐻 = 0 ) ) |
59 |
57 58
|
anbi12d |
⊢ ( 𝜑 → ( ( ( 𝐾 ‘ 𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾 ‘ 𝐻 ) = ( Base ‘ 𝑊 ) ) ↔ ( 𝐺 ≠ 0 ∧ 𝐻 = 0 ) ) ) |
60 |
55 59
|
bitrd |
⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) ⊊ ( 𝐾 ‘ 𝐻 ) ↔ ( 𝐺 ≠ 0 ∧ 𝐻 = 0 ) ) ) |