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Theorem trlval3 39514
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 1188 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl31 1251 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 simpl2 1189 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝐹𝑇)
4 simpr 484 . . . 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 2724 . . . . 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 39497 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑅𝐹) = (0.‘𝐾))
121, 2, 3, 4, 11syl112anc 1371 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑅𝐹) = (0.‘𝐾))
13 simpl33 1253 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))
14 simpl1l 1221 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝐾 ∈ HL)
15 hlatl 38686 . . . . . 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 7417 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) = (𝑃 𝑃))
18 simp31l 1293 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → 𝑃𝐴)
1918adantr 480 . . . . . . . 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 38695 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑃 𝑃) = 𝑃)
2214, 19, 21syl2anc 583 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 𝑃) = 𝑃)
2317, 22eqtrd 2764 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) = 𝑃)
2423, 19eqeltrd 2825 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) ∈ 𝐴)
25 simp1 1133 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
26 simp2 1134 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → 𝐹𝑇)
27 simp31 1206 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
28 simp32 1207 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
295, 7, 8, 9ltrn2ateq 39507 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝐹𝑃) = 𝑃 ↔ (𝐹𝑄) = 𝑄))
3025, 26, 27, 28, 29syl13anc 1369 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → ((𝐹𝑃) = 𝑃 ↔ (𝐹𝑄) = 𝑄))
3130biimpa 476 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝐹𝑄) = 𝑄)
3231oveq2d 7417 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝐹𝑄)) = (𝑄 𝑄))
33 simp32l 1295 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → 𝑄𝐴)
3433adantr 480 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝑄𝐴)
3520, 7hlatjidm 38695 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
3614, 34, 35syl2anc 583 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 𝑄) = 𝑄)
3732, 36eqtrd 2764 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝐹𝑄)) = 𝑄)
3837, 34eqeltrd 2825 . . . . 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 38644 . . . . 5 ((𝐾 ∈ AtLat ∧ (𝑃 (𝐹𝑃)) ∈ 𝐴 ∧ (𝑄 (𝐹𝑄)) ∈ 𝐴) → ((𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)) ↔ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾)))
4116, 24, 38, 40syl3anc 1368 . . . 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 231 . . 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 2767 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))
44 simpl1 1188 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐾 ∈ HL ∧ 𝑊𝐻))
45 simpl2 1189 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐹𝑇)
46 simpl31 1251 . . . . . 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 39490 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
4844, 45, 46, 47syl3anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
49 simpl1l 1221 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐾 ∈ HL)
5049hllatd 38690 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐾 ∈ Lat)
5118adantr 480 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑃𝐴)
525, 7, 8, 9ltrnat 39467 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃𝐴) → (𝐹𝑃) ∈ 𝐴)
5344, 45, 51, 52syl3anc 1368 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑃) ∈ 𝐴)
54 eqid 2724 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
5554, 20, 7hlatjcl 38693 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
5649, 51, 53, 55syl3anc 1368 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
57 simpl1r 1222 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑊𝐻)
5854, 8lhpbase 39325 . . . . . . 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 18416 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) 𝑊) (𝑃 (𝐹𝑃)))
6150, 56, 59, 60syl3anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) 𝑊) (𝑃 (𝐹𝑃)))
6248, 61eqbrtrd 5160 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) (𝑃 (𝐹𝑃)))
63 simpl32 1252 . . . . . 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 39490 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑅𝐹) = ((𝑄 (𝐹𝑄)) 𝑊))
6544, 45, 63, 64syl3anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) = ((𝑄 (𝐹𝑄)) 𝑊))
6633adantr 480 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑄𝐴)
675, 7, 8, 9ltrnat 39467 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑄𝐴) → (𝐹𝑄) ∈ 𝐴)
6844, 45, 66, 67syl3anc 1368 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑄) ∈ 𝐴)
6954, 20, 7hlatjcl 38693 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑄𝐴 ∧ (𝐹𝑄) ∈ 𝐴) → (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾))
7049, 66, 68, 69syl3anc 1368 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾))
7154, 5, 39latmle1 18416 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑄 (𝐹𝑄)) 𝑊) (𝑄 (𝐹𝑄)))
7250, 70, 59, 71syl3anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑄 (𝐹𝑄)) 𝑊) (𝑄 (𝐹𝑄)))
7365, 72eqbrtrd 5160 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) (𝑄 (𝐹𝑄)))
7454, 8, 9, 10trlcl 39491 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) ∈ (Base‘𝐾))
7544, 45, 74syl2anc 583 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) ∈ (Base‘𝐾))
7654, 5, 39latlem12 18418 . . . . 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 1369 . . . 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 709 . . 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 484 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑃) ≠ 𝑃)
815, 7, 8, 9, 10trlat 39496 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
8244, 46, 45, 80, 81syl112anc 1371 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) ∈ 𝐴)
8354, 39latmcl 18392 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ (Base‘𝐾))
8450, 56, 70, 83syl3anc 1368 . . . . . . 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 38636 . . . . . . 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 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ≠ (0.‘𝐾))
8786neneqd 2937 . . . . 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 1253 . . . . . . 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 38853 . . . . . . 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 1382 . . . . . 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 861 . . . . 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 148 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴)
935, 7atcmp 38637 . . . 4 ((𝐾 ∈ AtLat ∧ (𝑅𝐹) ∈ 𝐴 ∧ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴) → ((𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ↔ (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄)))))
9479, 82, 92, 93syl3anc 1368 . . 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 231 . 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 3021 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 205  wa 395  wo 844  w3a 1084   = wceq 1533  wcel 2098  wne 2932   class class class wbr 5138  cfv 6533  (class class class)co 7401  Basecbs 17140  lecple 17200  joincjn 18263  meetcmee 18264  0.cp0 18375  Latclat 18383  Atomscatm 38589  AtLatcal 38590  HLchlt 38676  LHypclh 39311  LTrncltrn 39428  trLctrl 39485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-map 8817  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-p1 18378  df-lat 18384  df-clat 18451  df-oposet 38502  df-ol 38504  df-oml 38505  df-covers 38592  df-ats 38593  df-atl 38624  df-cvlat 38648  df-hlat 38677  df-llines 38825  df-lhyp 39315  df-laut 39316  df-ldil 39431  df-ltrn 39432  df-trl 39486
This theorem is referenced by:  trlval4  39515
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