Step | Hyp | Ref
| Expression |
1 | | simp2 1136 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑋 ∈ 𝑉) |
2 | | simp1 1135 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝐾 ∈ HL) |
3 | | eqid 2738 |
. . . . . . 7
⊢
(Base‘𝐾) =
(Base‘𝐾) |
4 | | lvolcmp.v |
. . . . . . 7
⊢ 𝑉 = (LVols‘𝐾) |
5 | 3, 4 | lvolbase 37578 |
. . . . . 6
⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Base‘𝐾)) |
6 | 5 | 3ad2ant2 1133 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑋 ∈ (Base‘𝐾)) |
7 | | eqid 2738 |
. . . . . 6
⊢ ( ⋖
‘𝐾) = ( ⋖
‘𝐾) |
8 | | eqid 2738 |
. . . . . 6
⊢
(LPlanes‘𝐾) =
(LPlanes‘𝐾) |
9 | 3, 7, 8, 4 | islvol4 37574 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋 ∈ 𝑉 ↔ ∃𝑧 ∈ (LPlanes‘𝐾)𝑧( ⋖ ‘𝐾)𝑋)) |
10 | 2, 6, 9 | syl2anc 584 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 ∈ 𝑉 ↔ ∃𝑧 ∈ (LPlanes‘𝐾)𝑧( ⋖ ‘𝐾)𝑋)) |
11 | 1, 10 | mpbid 231 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∃𝑧 ∈ (LPlanes‘𝐾)𝑧( ⋖ ‘𝐾)𝑋) |
12 | | simpr3 1195 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ≤ 𝑌) |
13 | | hlpos 37366 |
. . . . . . . . 9
⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) |
14 | 13 | 3ad2ant1 1132 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝐾 ∈ Poset) |
15 | 14 | adantr 481 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ Poset) |
16 | 6 | adantr 481 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ∈ (Base‘𝐾)) |
17 | | simpl3 1192 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑌 ∈ 𝑉) |
18 | 3, 4 | lvolbase 37578 |
. . . . . . . 8
⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ (Base‘𝐾)) |
19 | 17, 18 | syl 17 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑌 ∈ (Base‘𝐾)) |
20 | | simpr1 1193 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ∈ (LPlanes‘𝐾)) |
21 | 3, 8 | lplnbase 37534 |
. . . . . . . 8
⊢ (𝑧 ∈ (LPlanes‘𝐾) → 𝑧 ∈ (Base‘𝐾)) |
22 | 20, 21 | syl 17 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ∈ (Base‘𝐾)) |
23 | | simpr2 1194 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧( ⋖ ‘𝐾)𝑋) |
24 | | simpl1 1190 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ HL) |
25 | | lvolcmp.l |
. . . . . . . . . . 11
⊢ ≤ =
(le‘𝐾) |
26 | 3, 25, 7 | cvrle 37278 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑧( ⋖ ‘𝐾)𝑋) → 𝑧 ≤ 𝑋) |
27 | 24, 22, 16, 23, 26 | syl31anc 1372 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ≤ 𝑋) |
28 | 3, 25 | postr 18026 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ Poset ∧ (𝑧 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑧 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) → 𝑧 ≤ 𝑌)) |
29 | 15, 22, 16, 19, 28 | syl13anc 1371 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → ((𝑧 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) → 𝑧 ≤ 𝑌)) |
30 | 27, 12, 29 | mp2and 696 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ≤ 𝑌) |
31 | 25, 7, 8, 4 | lplncvrlvol2 37615 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑌 ∈ 𝑉) ∧ 𝑧 ≤ 𝑌) → 𝑧( ⋖ ‘𝐾)𝑌) |
32 | 24, 20, 17, 30, 31 | syl31anc 1372 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧( ⋖ ‘𝐾)𝑌) |
33 | 3, 25, 7 | cvrcmp 37283 |
. . . . . . 7
⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) ∧ (𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑧( ⋖ ‘𝐾)𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌)) |
34 | 15, 16, 19, 22, 23, 32, 33 | syl132anc 1387 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌)) |
35 | 12, 34 | mpbid 231 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 = 𝑌) |
36 | 35 | 3exp2 1353 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑧 ∈ (LPlanes‘𝐾) → (𝑧( ⋖ ‘𝐾)𝑋 → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌)))) |
37 | 36 | rexlimdv 3210 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (∃𝑧 ∈ (LPlanes‘𝐾)𝑧( ⋖ ‘𝐾)𝑋 → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))) |
38 | 11, 37 | mpd 15 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌)) |
39 | 3, 25 | posref 18024 |
. . . 4
⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋 ≤ 𝑋) |
40 | 14, 6, 39 | syl2anc 584 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑋 ≤ 𝑋) |
41 | | breq2 5078 |
. . 3
⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌)) |
42 | 40, 41 | syl5ibcom 244 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌)) |
43 | 38, 42 | impbid 211 |
1
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌)) |