Proof of Theorem lhpocnle
| Step | Hyp | Ref
| Expression |
| 1 | | hlatl 39361 |
. . . . 5
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
| 2 | 1 | adantr 480 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ AtLat) |
| 3 | | simpr 484 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ 𝐻) |
| 4 | | eqid 2737 |
. . . . . . 7
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 5 | | lhpocnle.h |
. . . . . . 7
⊢ 𝐻 = (LHyp‘𝐾) |
| 6 | 4, 5 | lhpbase 40000 |
. . . . . 6
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
| 7 | | lhpocnle.o |
. . . . . . 7
⊢ ⊥ =
(oc‘𝐾) |
| 8 | | eqid 2737 |
. . . . . . 7
⊢
(Atoms‘𝐾) =
(Atoms‘𝐾) |
| 9 | 4, 7, 8, 5 | lhpoc 40016 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑊 ∈ 𝐻 ↔ ( ⊥ ‘𝑊) ∈ (Atoms‘𝐾))) |
| 10 | 6, 9 | sylan2 593 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑊 ∈ 𝐻 ↔ ( ⊥ ‘𝑊) ∈ (Atoms‘𝐾))) |
| 11 | 3, 10 | mpbid 232 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑊) ∈ (Atoms‘𝐾)) |
| 12 | | eqid 2737 |
. . . . 5
⊢
(0.‘𝐾) =
(0.‘𝐾) |
| 13 | 12, 8 | atn0 39309 |
. . . 4
⊢ ((𝐾 ∈ AtLat ∧ ( ⊥
‘𝑊) ∈
(Atoms‘𝐾)) → (
⊥
‘𝑊) ≠
(0.‘𝐾)) |
| 14 | 2, 11, 13 | syl2anc 584 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑊) ≠ (0.‘𝐾)) |
| 15 | 14 | neneqd 2945 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ¬ ( ⊥ ‘𝑊) = (0.‘𝐾)) |
| 16 | | simpr 484 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑊) ≤ 𝑊) → ( ⊥ ‘𝑊) ≤ 𝑊) |
| 17 | | hllat 39364 |
. . . . . . 7
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
| 18 | 17 | ad2antrr 726 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑊) ≤ 𝑊) → 𝐾 ∈ Lat) |
| 19 | | hlop 39363 |
. . . . . . . 8
⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) |
| 20 | 19 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑊) ≤ 𝑊) → 𝐾 ∈ OP) |
| 21 | 6 | ad2antlr 727 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑊) ≤ 𝑊) → 𝑊 ∈ (Base‘𝐾)) |
| 22 | 4, 7 | opoccl 39195 |
. . . . . . 7
⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ( ⊥ ‘𝑊) ∈ (Base‘𝐾)) |
| 23 | 20, 21, 22 | syl2anc 584 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑊) ≤ 𝑊) → ( ⊥ ‘𝑊) ∈ (Base‘𝐾)) |
| 24 | | lhpocnle.l |
. . . . . . 7
⊢ ≤ =
(le‘𝐾) |
| 25 | 4, 24 | latref 18486 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ ( ⊥
‘𝑊) ∈
(Base‘𝐾)) → (
⊥
‘𝑊) ≤ ( ⊥
‘𝑊)) |
| 26 | 18, 23, 25 | syl2anc 584 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑊) ≤ 𝑊) → ( ⊥ ‘𝑊) ≤ ( ⊥ ‘𝑊)) |
| 27 | | eqid 2737 |
. . . . . . 7
⊢
(meet‘𝐾) =
(meet‘𝐾) |
| 28 | 4, 24, 27 | latlem12 18511 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ (( ⊥
‘𝑊) ∈
(Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾) ∧ ( ⊥ ‘𝑊) ∈ (Base‘𝐾))) → ((( ⊥
‘𝑊) ≤ 𝑊 ∧ ( ⊥ ‘𝑊) ≤ ( ⊥ ‘𝑊)) ↔ ( ⊥ ‘𝑊) ≤ (𝑊(meet‘𝐾)( ⊥ ‘𝑊)))) |
| 29 | 18, 23, 21, 23, 28 | syl13anc 1374 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑊) ≤ 𝑊) → ((( ⊥ ‘𝑊) ≤ 𝑊 ∧ ( ⊥ ‘𝑊) ≤ ( ⊥ ‘𝑊)) ↔ ( ⊥ ‘𝑊) ≤ (𝑊(meet‘𝐾)( ⊥ ‘𝑊)))) |
| 30 | 16, 26, 29 | mpbi2and 712 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑊) ≤ 𝑊) → ( ⊥ ‘𝑊) ≤ (𝑊(meet‘𝐾)( ⊥ ‘𝑊))) |
| 31 | 4, 7, 27, 12 | opnoncon 39209 |
. . . . 5
⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑊(meet‘𝐾)( ⊥ ‘𝑊)) = (0.‘𝐾)) |
| 32 | 20, 21, 31 | syl2anc 584 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑊) ≤ 𝑊) → (𝑊(meet‘𝐾)( ⊥ ‘𝑊)) = (0.‘𝐾)) |
| 33 | 30, 32 | breqtrd 5169 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑊) ≤ 𝑊) → ( ⊥ ‘𝑊) ≤ (0.‘𝐾)) |
| 34 | 4, 24, 12 | ople0 39188 |
. . . 4
⊢ ((𝐾 ∈ OP ∧ ( ⊥
‘𝑊) ∈
(Base‘𝐾)) → ((
⊥
‘𝑊) ≤
(0.‘𝐾) ↔ ( ⊥
‘𝑊) = (0.‘𝐾))) |
| 35 | 20, 23, 34 | syl2anc 584 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑊) ≤ 𝑊) → (( ⊥ ‘𝑊) ≤ (0.‘𝐾) ↔ ( ⊥ ‘𝑊) = (0.‘𝐾))) |
| 36 | 33, 35 | mpbid 232 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑊) ≤ 𝑊) → ( ⊥ ‘𝑊) = (0.‘𝐾)) |
| 37 | 15, 36 | mtand 816 |
1
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ¬ ( ⊥ ‘𝑊) ≤ 𝑊) |