Step |
Hyp |
Ref |
Expression |
1 |
|
dochkrshp.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dochkrshp.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dochkrshp.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dochkrshp.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
dochkrshp.y |
⊢ 𝑌 = ( LSHyp ‘ 𝑈 ) |
6 |
|
dochkrshp.f |
⊢ 𝐹 = ( LFnl ‘ 𝑈 ) |
7 |
|
dochkrshp.l |
⊢ 𝐿 = ( LKer ‘ 𝑈 ) |
8 |
|
dochkrshp.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
9 |
|
dochkrshp.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
10 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ ( 𝐿 ‘ 𝐺 ) ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ ( 𝐿 ‘ 𝐺 ) ) |
11 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ ( 𝐿 ‘ 𝐺 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
12 |
|
2fveq3 |
⊢ ( ( 𝐿 ‘ 𝐺 ) = 𝑉 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( ⊥ ‘ ( ⊥ ‘ 𝑉 ) ) ) |
13 |
1 3 2 4 8
|
dochoc1 |
⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ 𝑉 ) ) = 𝑉 ) |
14 |
12 13
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐺 ) = 𝑉 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = 𝑉 ) |
15 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐺 ) = 𝑉 ) → ( 𝐿 ‘ 𝐺 ) = 𝑉 ) |
16 |
14 15
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐺 ) = 𝑉 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ) |
17 |
16
|
ex |
⊢ ( 𝜑 → ( ( 𝐿 ‘ 𝐺 ) = 𝑉 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ) ) |
18 |
17
|
necon3d |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ ( 𝐿 ‘ 𝐺 ) → ( 𝐿 ‘ 𝐺 ) ≠ 𝑉 ) ) |
19 |
|
df-ne |
⊢ ( ( 𝐿 ‘ 𝐺 ) ≠ 𝑉 ↔ ¬ ( 𝐿 ‘ 𝐺 ) = 𝑉 ) |
20 |
1 3 8
|
dvhlvec |
⊢ ( 𝜑 → 𝑈 ∈ LVec ) |
21 |
4 5 6 7 20 9
|
lkrshpor |
⊢ ( 𝜑 → ( ( 𝐿 ‘ 𝐺 ) ∈ 𝑌 ∨ ( 𝐿 ‘ 𝐺 ) = 𝑉 ) ) |
22 |
21
|
orcomd |
⊢ ( 𝜑 → ( ( 𝐿 ‘ 𝐺 ) = 𝑉 ∨ ( 𝐿 ‘ 𝐺 ) ∈ 𝑌 ) ) |
23 |
22
|
ord |
⊢ ( 𝜑 → ( ¬ ( 𝐿 ‘ 𝐺 ) = 𝑉 → ( 𝐿 ‘ 𝐺 ) ∈ 𝑌 ) ) |
24 |
19 23
|
syl5bi |
⊢ ( 𝜑 → ( ( 𝐿 ‘ 𝐺 ) ≠ 𝑉 → ( 𝐿 ‘ 𝐺 ) ∈ 𝑌 ) ) |
25 |
18 24
|
syld |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ ( 𝐿 ‘ 𝐺 ) → ( 𝐿 ‘ 𝐺 ) ∈ 𝑌 ) ) |
26 |
25
|
imp |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ ( 𝐿 ‘ 𝐺 ) ) → ( 𝐿 ‘ 𝐺 ) ∈ 𝑌 ) |
27 |
1 2 3 4 5 11 26
|
dochshpncl |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ ( 𝐿 ‘ 𝐺 ) ) → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ ( 𝐿 ‘ 𝐺 ) ↔ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = 𝑉 ) ) |
28 |
10 27
|
mpbid |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ ( 𝐿 ‘ 𝐺 ) ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = 𝑉 ) |
29 |
28
|
ex |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ ( 𝐿 ‘ 𝐺 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = 𝑉 ) ) |
30 |
29
|
necon1d |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ) ) |
31 |
14
|
ex |
⊢ ( 𝜑 → ( ( 𝐿 ‘ 𝐺 ) = 𝑉 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = 𝑉 ) ) |
32 |
31
|
necon3ad |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 → ¬ ( 𝐿 ‘ 𝐺 ) = 𝑉 ) ) |
33 |
32 23
|
syld |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 → ( 𝐿 ‘ 𝐺 ) ∈ 𝑌 ) ) |
34 |
30 33
|
jcad |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( 𝐿 ‘ 𝐺 ) ∈ 𝑌 ) ) ) |
35 |
1 2 3 6 5 7 8 9
|
dochlkr |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∈ 𝑌 ↔ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( 𝐿 ‘ 𝐺 ) ∈ 𝑌 ) ) ) |
36 |
34 35
|
sylibrd |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∈ 𝑌 ) ) |
37 |
1 3 8
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
38 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∈ 𝑌 ) → 𝑈 ∈ LMod ) |
39 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∈ 𝑌 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∈ 𝑌 ) |
40 |
4 5 38 39
|
lshpne |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∈ 𝑌 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 ) |
41 |
40
|
ex |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∈ 𝑌 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 ) ) |
42 |
36 41
|
impbid |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 ↔ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∈ 𝑌 ) ) |