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Theorem trlval3 35792
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 1084 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl31 1162 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 simpl2 1085 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝐹𝑇)
4 simpr 476 . . . 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 2651 . . . . 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 35775 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑅𝐹) = (0.‘𝐾))
121, 2, 3, 4, 11syl112anc 1370 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑅𝐹) = (0.‘𝐾))
13 simpl33 1164 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))
14 simpl1l 1132 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝐾 ∈ HL)
15 hlatl 34965 . . . . . 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 6706 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) = (𝑃 𝑃))
18 simp31l 1204 . . . . . . . . 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 34973 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑃 𝑃) = 𝑃)
2214, 19, 21syl2anc 694 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 𝑃) = 𝑃)
2317, 22eqtrd 2685 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) = 𝑃)
2423, 19eqeltrd 2730 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) ∈ 𝐴)
25 simp1 1081 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
26 simp2 1082 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → 𝐹𝑇)
27 simp31 1117 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
28 simp32 1118 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
295, 7, 8, 9ltrn2ateq 35785 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝐹𝑃) = 𝑃 ↔ (𝐹𝑄) = 𝑄))
3025, 26, 27, 28, 29syl13anc 1368 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → ((𝐹𝑃) = 𝑃 ↔ (𝐹𝑄) = 𝑄))
3130biimpa 500 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝐹𝑄) = 𝑄)
3231oveq2d 6706 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝐹𝑄)) = (𝑄 𝑄))
33 simp32l 1206 . . . . . . . . 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 34973 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
3614, 34, 35syl2anc 694 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 𝑄) = 𝑄)
3732, 36eqtrd 2685 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝐹𝑄)) = 𝑄)
3837, 34eqeltrd 2730 . . . . 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 34923 . . . . 5 ((𝐾 ∈ AtLat ∧ (𝑃 (𝐹𝑃)) ∈ 𝐴 ∧ (𝑄 (𝐹𝑄)) ∈ 𝐴) → ((𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)) ↔ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾)))
4116, 24, 38, 40syl3anc 1366 . . . 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 222 . . 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 2688 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))
44 simpl1 1084 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐾 ∈ HL ∧ 𝑊𝐻))
45 simpl2 1085 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐹𝑇)
46 simpl31 1162 . . . . . 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 35768 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
4844, 45, 46, 47syl3anc 1366 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
49 simpl1l 1132 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐾 ∈ HL)
50 hllat 34968 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
5149, 50syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐾 ∈ Lat)
5218adantr 480 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑃𝐴)
535, 7, 8, 9ltrnat 35744 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃𝐴) → (𝐹𝑃) ∈ 𝐴)
5444, 45, 52, 53syl3anc 1366 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑃) ∈ 𝐴)
55 eqid 2651 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
5655, 20, 7hlatjcl 34971 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
5749, 52, 54, 56syl3anc 1366 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
58 simpl1r 1133 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑊𝐻)
5955, 8lhpbase 35602 . . . . . . 7 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
6058, 59syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑊 ∈ (Base‘𝐾))
6155, 5, 39latmle1 17123 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) 𝑊) (𝑃 (𝐹𝑃)))
6251, 57, 60, 61syl3anc 1366 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) 𝑊) (𝑃 (𝐹𝑃)))
6348, 62eqbrtrd 4707 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) (𝑃 (𝐹𝑃)))
64 simpl32 1163 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
655, 20, 39, 7, 8, 9, 10trlval2 35768 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑅𝐹) = ((𝑄 (𝐹𝑄)) 𝑊))
6644, 45, 64, 65syl3anc 1366 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) = ((𝑄 (𝐹𝑄)) 𝑊))
6733adantr 480 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑄𝐴)
685, 7, 8, 9ltrnat 35744 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑄𝐴) → (𝐹𝑄) ∈ 𝐴)
6944, 45, 67, 68syl3anc 1366 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑄) ∈ 𝐴)
7055, 20, 7hlatjcl 34971 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑄𝐴 ∧ (𝐹𝑄) ∈ 𝐴) → (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾))
7149, 67, 69, 70syl3anc 1366 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾))
7255, 5, 39latmle1 17123 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑄 (𝐹𝑄)) 𝑊) (𝑄 (𝐹𝑄)))
7351, 71, 60, 72syl3anc 1366 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑄 (𝐹𝑄)) 𝑊) (𝑄 (𝐹𝑄)))
7466, 73eqbrtrd 4707 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) (𝑄 (𝐹𝑄)))
7555, 8, 9, 10trlcl 35769 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) ∈ (Base‘𝐾))
7644, 45, 75syl2anc 694 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) ∈ (Base‘𝐾))
7755, 5, 39latlem12 17125 . . . . 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)))))
7851, 76, 57, 71, 77syl13anc 1368 . . . 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)))))
7963, 74, 78mpbi2and 976 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))
8049, 15syl 17 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐾 ∈ AtLat)
81 simpr 476 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑃) ≠ 𝑃)
825, 7, 8, 9, 10trlat 35774 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
8344, 46, 45, 81, 82syl112anc 1370 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) ∈ 𝐴)
8455, 39latmcl 17099 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ (Base‘𝐾))
8551, 57, 71, 84syl3anc 1366 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ (Base‘𝐾))
8655, 5, 6, 7atlen0 34915 . . . . . . 7 (((𝐾 ∈ AtLat ∧ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ (Base‘𝐾) ∧ (𝑅𝐹) ∈ 𝐴) ∧ (𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄)))) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ≠ (0.‘𝐾))
8780, 85, 83, 79, 86syl31anc 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ≠ (0.‘𝐾))
8887neneqd 2828 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ¬ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾))
89 simpl33 1164 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))
9020, 39, 6, 72atmat0 35130 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) ∧ (𝑄𝐴 ∧ (𝐹𝑄) ∈ 𝐴 ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴 ∨ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾)))
9149, 52, 54, 67, 69, 89, 90syl33anc 1381 . . . . . 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)))
9291ord 391 . . . . 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)))
9388, 92mt3d 140 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴)
945, 7atcmp 34916 . . . 4 ((𝐾 ∈ AtLat ∧ (𝑅𝐹) ∈ 𝐴 ∧ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴) → ((𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ↔ (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄)))))
9580, 83, 93, 94syl3anc 1366 . . 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)))))
9679, 95mpbid 222 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))
9743, 96pm2.61dane 2910 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 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  0.cp0 17084  Latclat 17092  Atomscatm 34868  AtLatcal 34869  HLchlt 34955  LHypclh 35588  LTrncltrn 35705  trLctrl 35763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lhyp 35592  df-laut 35593  df-ldil 35708  df-ltrn 35709  df-trl 35764
This theorem is referenced by:  trlval4  35793
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