Step |
Hyp |
Ref |
Expression |
1 |
|
lhpocnle.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
lhpocnle.o |
⊢ ⊥ = ( oc ‘ 𝐾 ) |
3 |
|
lhpocnle.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
hlatl |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat ) |
5 |
4
|
adantr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐾 ∈ AtLat ) |
6 |
|
simpr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ∈ 𝐻 ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
8 |
7 3
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
9 |
|
eqid |
⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 ) |
10 |
7 2 9 3
|
lhpoc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑊 ∈ 𝐻 ↔ ( ⊥ ‘ 𝑊 ) ∈ ( Atoms ‘ 𝐾 ) ) ) |
11 |
8 10
|
sylan2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑊 ∈ 𝐻 ↔ ( ⊥ ‘ 𝑊 ) ∈ ( Atoms ‘ 𝐾 ) ) ) |
12 |
6 11
|
mpbid |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ 𝑊 ) ∈ ( Atoms ‘ 𝐾 ) ) |
13 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
14 |
13 9
|
atn0 |
⊢ ( ( 𝐾 ∈ AtLat ∧ ( ⊥ ‘ 𝑊 ) ∈ ( Atoms ‘ 𝐾 ) ) → ( ⊥ ‘ 𝑊 ) ≠ ( 0. ‘ 𝐾 ) ) |
15 |
5 12 14
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ 𝑊 ) ≠ ( 0. ‘ 𝐾 ) ) |
16 |
15
|
neneqd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ¬ ( ⊥ ‘ 𝑊 ) = ( 0. ‘ 𝐾 ) ) |
17 |
|
simpr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ) → ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ) |
18 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
19 |
18
|
ad2antrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ) → 𝐾 ∈ Lat ) |
20 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
21 |
20
|
ad2antrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ) → 𝐾 ∈ OP ) |
22 |
8
|
ad2antlr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
23 |
7 2
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ⊥ ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
24 |
21 22 23
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ) → ( ⊥ ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
25 |
7 1
|
latref |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ⊥ ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → ( ⊥ ‘ 𝑊 ) ≤ ( ⊥ ‘ 𝑊 ) ) |
26 |
19 24 25
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ) → ( ⊥ ‘ 𝑊 ) ≤ ( ⊥ ‘ 𝑊 ) ) |
27 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
28 |
7 1 27
|
latlem12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ⊥ ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ∧ ( ⊥ ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ∧ ( ⊥ ‘ 𝑊 ) ≤ ( ⊥ ‘ 𝑊 ) ) ↔ ( ⊥ ‘ 𝑊 ) ≤ ( 𝑊 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑊 ) ) ) ) |
29 |
19 24 22 24 28
|
syl13anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ) → ( ( ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ∧ ( ⊥ ‘ 𝑊 ) ≤ ( ⊥ ‘ 𝑊 ) ) ↔ ( ⊥ ‘ 𝑊 ) ≤ ( 𝑊 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑊 ) ) ) ) |
30 |
17 26 29
|
mpbi2and |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ) → ( ⊥ ‘ 𝑊 ) ≤ ( 𝑊 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑊 ) ) ) |
31 |
7 2 27 13
|
opnoncon |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑊 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑊 ) ) = ( 0. ‘ 𝐾 ) ) |
32 |
21 22 31
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ) → ( 𝑊 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑊 ) ) = ( 0. ‘ 𝐾 ) ) |
33 |
30 32
|
breqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ) → ( ⊥ ‘ 𝑊 ) ≤ ( 0. ‘ 𝐾 ) ) |
34 |
7 1 13
|
ople0 |
⊢ ( ( 𝐾 ∈ OP ∧ ( ⊥ ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( ⊥ ‘ 𝑊 ) ≤ ( 0. ‘ 𝐾 ) ↔ ( ⊥ ‘ 𝑊 ) = ( 0. ‘ 𝐾 ) ) ) |
35 |
21 24 34
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ) → ( ( ⊥ ‘ 𝑊 ) ≤ ( 0. ‘ 𝐾 ) ↔ ( ⊥ ‘ 𝑊 ) = ( 0. ‘ 𝐾 ) ) ) |
36 |
33 35
|
mpbid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ) → ( ⊥ ‘ 𝑊 ) = ( 0. ‘ 𝐾 ) ) |
37 |
16 36
|
mtand |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ¬ ( ⊥ ‘ 𝑊 ) ≤ 𝑊 ) |