Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trlval3 Structured version   Visualization version   GIF version

Theorem trlval3 36875
Description: The value of the trace of a lattice translation in terms of 2 atoms. TODO: Try to shorten proof. (Contributed by NM, 3-May-2013.)
Hypotheses
Ref Expression
trlval3.l = (le‘𝐾)
trlval3.j = (join‘𝐾)
trlval3.m = (meet‘𝐾)
trlval3.a 𝐴 = (Atoms‘𝐾)
trlval3.h 𝐻 = (LHyp‘𝐾)
trlval3.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
trlval3.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
trlval3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))

Proof of Theorem trlval3
StepHypRef Expression
1 simpl1 1184 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl31 1247 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 simpl2 1185 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝐹𝑇)
4 simpr 485 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝐹𝑃) = 𝑃)
5 trlval3.l . . . . 5 = (le‘𝐾)
6 eqid 2797 . . . . 5 (0.‘𝐾) = (0.‘𝐾)
7 trlval3.a . . . . 5 𝐴 = (Atoms‘𝐾)
8 trlval3.h . . . . 5 𝐻 = (LHyp‘𝐾)
9 trlval3.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
10 trlval3.r . . . . 5 𝑅 = ((trL‘𝐾)‘𝑊)
115, 6, 7, 8, 9, 10trl0 36858 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑅𝐹) = (0.‘𝐾))
121, 2, 3, 4, 11syl112anc 1367 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑅𝐹) = (0.‘𝐾))
13 simpl33 1249 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))
14 simpl1l 1217 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝐾 ∈ HL)
15 hlatl 36048 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
1614, 15syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝐾 ∈ AtLat)
174oveq2d 7039 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) = (𝑃 𝑃))
18 simp31l 1289 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → 𝑃𝐴)
1918adantr 481 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝑃𝐴)
20 trlval3.j . . . . . . . . 9 = (join‘𝐾)
2120, 7hlatjidm 36057 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑃 𝑃) = 𝑃)
2214, 19, 21syl2anc 584 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 𝑃) = 𝑃)
2317, 22eqtrd 2833 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) = 𝑃)
2423, 19eqeltrd 2885 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) ∈ 𝐴)
25 simp1 1129 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
26 simp2 1130 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → 𝐹𝑇)
27 simp31 1202 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
28 simp32 1203 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
295, 7, 8, 9ltrn2ateq 36868 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝐹𝑃) = 𝑃 ↔ (𝐹𝑄) = 𝑄))
3025, 26, 27, 28, 29syl13anc 1365 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → ((𝐹𝑃) = 𝑃 ↔ (𝐹𝑄) = 𝑄))
3130biimpa 477 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝐹𝑄) = 𝑄)
3231oveq2d 7039 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝐹𝑄)) = (𝑄 𝑄))
33 simp32l 1291 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → 𝑄𝐴)
3433adantr 481 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝑄𝐴)
3520, 7hlatjidm 36057 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
3614, 34, 35syl2anc 584 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 𝑄) = 𝑄)
3732, 36eqtrd 2833 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝐹𝑄)) = 𝑄)
3837, 34eqeltrd 2885 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝐹𝑄)) ∈ 𝐴)
39 trlval3.m . . . . . 6 = (meet‘𝐾)
4039, 6, 7atnem0 36006 . . . . 5 ((𝐾 ∈ AtLat ∧ (𝑃 (𝐹𝑃)) ∈ 𝐴 ∧ (𝑄 (𝐹𝑄)) ∈ 𝐴) → ((𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)) ↔ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾)))
4116, 24, 38, 40syl3anc 1364 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → ((𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)) ↔ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾)))
4213, 41mpbid 233 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾))
4312, 42eqtr4d 2836 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))
44 simpl1 1184 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐾 ∈ HL ∧ 𝑊𝐻))
45 simpl2 1185 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐹𝑇)
46 simpl31 1247 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
475, 20, 39, 7, 8, 9, 10trlval2 36851 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
4844, 45, 46, 47syl3anc 1364 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
49 simpl1l 1217 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐾 ∈ HL)
5049hllatd 36052 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐾 ∈ Lat)
5118adantr 481 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑃𝐴)
525, 7, 8, 9ltrnat 36828 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃𝐴) → (𝐹𝑃) ∈ 𝐴)
5344, 45, 51, 52syl3anc 1364 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑃) ∈ 𝐴)
54 eqid 2797 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
5554, 20, 7hlatjcl 36055 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
5649, 51, 53, 55syl3anc 1364 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
57 simpl1r 1218 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑊𝐻)
5854, 8lhpbase 36686 . . . . . . 7 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
5957, 58syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑊 ∈ (Base‘𝐾))
6054, 5, 39latmle1 17519 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) 𝑊) (𝑃 (𝐹𝑃)))
6150, 56, 59, 60syl3anc 1364 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) 𝑊) (𝑃 (𝐹𝑃)))
6248, 61eqbrtrd 4990 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) (𝑃 (𝐹𝑃)))
63 simpl32 1248 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
645, 20, 39, 7, 8, 9, 10trlval2 36851 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑅𝐹) = ((𝑄 (𝐹𝑄)) 𝑊))
6544, 45, 63, 64syl3anc 1364 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) = ((𝑄 (𝐹𝑄)) 𝑊))
6633adantr 481 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑄𝐴)
675, 7, 8, 9ltrnat 36828 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑄𝐴) → (𝐹𝑄) ∈ 𝐴)
6844, 45, 66, 67syl3anc 1364 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑄) ∈ 𝐴)
6954, 20, 7hlatjcl 36055 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑄𝐴 ∧ (𝐹𝑄) ∈ 𝐴) → (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾))
7049, 66, 68, 69syl3anc 1364 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾))
7154, 5, 39latmle1 17519 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑄 (𝐹𝑄)) 𝑊) (𝑄 (𝐹𝑄)))
7250, 70, 59, 71syl3anc 1364 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑄 (𝐹𝑄)) 𝑊) (𝑄 (𝐹𝑄)))
7365, 72eqbrtrd 4990 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) (𝑄 (𝐹𝑄)))
7454, 8, 9, 10trlcl 36852 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) ∈ (Base‘𝐾))
7544, 45, 74syl2anc 584 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) ∈ (Base‘𝐾))
7654, 5, 39latlem12 17521 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑅𝐹) ∈ (Base‘𝐾) ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾))) → (((𝑅𝐹) (𝑃 (𝐹𝑃)) ∧ (𝑅𝐹) (𝑄 (𝐹𝑄))) ↔ (𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄)))))
7750, 75, 56, 70, 76syl13anc 1365 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (((𝑅𝐹) (𝑃 (𝐹𝑃)) ∧ (𝑅𝐹) (𝑄 (𝐹𝑄))) ↔ (𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄)))))
7862, 73, 77mpbi2and 708 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))
7949, 15syl 17 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐾 ∈ AtLat)
80 simpr 485 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑃) ≠ 𝑃)
815, 7, 8, 9, 10trlat 36857 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
8244, 46, 45, 80, 81syl112anc 1367 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) ∈ 𝐴)
8354, 39latmcl 17495 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ (Base‘𝐾))
8450, 56, 70, 83syl3anc 1364 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ (Base‘𝐾))
8554, 5, 6, 7atlen0 35998 . . . . . . 7 (((𝐾 ∈ AtLat ∧ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ (Base‘𝐾) ∧ (𝑅𝐹) ∈ 𝐴) ∧ (𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄)))) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ≠ (0.‘𝐾))
8679, 84, 82, 78, 85syl31anc 1366 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ≠ (0.‘𝐾))
8786neneqd 2991 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ¬ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾))
88 simpl33 1249 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))
8920, 39, 6, 72atmat0 36214 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) ∧ (𝑄𝐴 ∧ (𝐹𝑄) ∈ 𝐴 ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴 ∨ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾)))
9049, 51, 53, 66, 68, 88, 89syl33anc 1378 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴 ∨ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾)))
9190ord 859 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (¬ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴 → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾)))
9287, 91mt3d 150 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴)
935, 7atcmp 35999 . . . 4 ((𝐾 ∈ AtLat ∧ (𝑅𝐹) ∈ 𝐴 ∧ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴) → ((𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ↔ (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄)))))
9479, 82, 92, 93syl3anc 1364 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ↔ (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄)))))
9578, 94mpbid 233 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))
9643, 95pm2.61dane 3074 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 842  w3a 1080   = wceq 1525  wcel 2083  wne 2986   class class class wbr 4968  cfv 6232  (class class class)co 7023  Basecbs 16316  lecple 16405  joincjn 17387  meetcmee 17388  0.cp0 17480  Latclat 17488  Atomscatm 35951  AtLatcal 35952  HLchlt 36038  LHypclh 36672  LTrncltrn 36789  trLctrl 36846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-map 8265  df-proset 17371  df-poset 17389  df-plt 17401  df-lub 17417  df-glb 17418  df-join 17419  df-meet 17420  df-p0 17482  df-p1 17483  df-lat 17489  df-clat 17551  df-oposet 35864  df-ol 35866  df-oml 35867  df-covers 35954  df-ats 35955  df-atl 35986  df-cvlat 36010  df-hlat 36039  df-llines 36186  df-lhyp 36676  df-laut 36677  df-ldil 36792  df-ltrn 36793  df-trl 36847
This theorem is referenced by:  trlval4  36876
  Copyright terms: Public domain W3C validator