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Theorem trlval3 35968
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 1235 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl31 1334 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 simpl2 1237 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝐹𝑇)
4 simpr 473 . . . 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 2806 . . . . 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 35951 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑅𝐹) = (0.‘𝐾))
121, 2, 3, 4, 11syl112anc 1486 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑅𝐹) = (0.‘𝐾))
13 simpl33 1338 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))
14 simpl1l 1286 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝐾 ∈ HL)
15 hlatl 35140 . . . . . 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 6890 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 (𝐹𝑃)) = (𝑃 𝑃))
18 simp31l 1388 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → 𝑃𝐴)
1918adantr 468 . . . . . . . 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 35149 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑃 𝑃) = 𝑃)
2214, 19, 21syl2anc 575 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑃 𝑃) = 𝑃)
2317, 22eqtrd 2840 . . . . . 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 1159 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
26 simp2 1160 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → 𝐹𝑇)
27 simp31 1259 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
28 simp32 1260 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
295, 7, 8, 9ltrn2ateq 35961 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝐹𝑃) = 𝑃 ↔ (𝐹𝑄) = 𝑄))
3025, 26, 27, 28, 29syl13anc 1484 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → ((𝐹𝑃) = 𝑃 ↔ (𝐹𝑄) = 𝑄))
3130biimpa 464 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝐹𝑄) = 𝑄)
3231oveq2d 6890 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 (𝐹𝑄)) = (𝑄 𝑄))
33 simp32l 1390 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → 𝑄𝐴)
3433adantr 468 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → 𝑄𝐴)
3520, 7hlatjidm 35149 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
3614, 34, 35syl2anc 575 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑄 𝑄) = 𝑄)
3732, 36eqtrd 2840 . . . . . 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 35098 . . . . 5 ((𝐾 ∈ AtLat ∧ (𝑃 (𝐹𝑃)) ∈ 𝐴 ∧ (𝑄 (𝐹𝑄)) ∈ 𝐴) → ((𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)) ↔ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) = (0.‘𝐾)))
4116, 24, 38, 40syl3anc 1483 . . . 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 223 . . 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 2843 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) = 𝑃) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))
44 simpl1 1235 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐾 ∈ HL ∧ 𝑊𝐻))
45 simpl2 1237 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐹𝑇)
46 simpl31 1334 . . . . . 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 35944 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
4844, 45, 46, 47syl3anc 1483 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
49 simpl1l 1286 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐾 ∈ HL)
5049hllatd 35144 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝐾 ∈ Lat)
5118adantr 468 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑃𝐴)
525, 7, 8, 9ltrnat 35920 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃𝐴) → (𝐹𝑃) ∈ 𝐴)
5344, 45, 51, 52syl3anc 1483 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑃) ∈ 𝐴)
54 eqid 2806 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
5554, 20, 7hlatjcl 35147 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
5649, 51, 53, 55syl3anc 1483 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
57 simpl1r 1288 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑊𝐻)
5854, 8lhpbase 35778 . . . . . . 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 17281 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) 𝑊) (𝑃 (𝐹𝑃)))
6150, 56, 59, 60syl3anc 1483 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) 𝑊) (𝑃 (𝐹𝑃)))
6248, 61eqbrtrd 4866 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) (𝑃 (𝐹𝑃)))
63 simpl32 1336 . . . . . 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 35944 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑅𝐹) = ((𝑄 (𝐹𝑄)) 𝑊))
6544, 45, 63, 64syl3anc 1483 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) = ((𝑄 (𝐹𝑄)) 𝑊))
6633adantr 468 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → 𝑄𝐴)
675, 7, 8, 9ltrnat 35920 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑄𝐴) → (𝐹𝑄) ∈ 𝐴)
6844, 45, 66, 67syl3anc 1483 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑄) ∈ 𝐴)
6954, 20, 7hlatjcl 35147 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑄𝐴 ∧ (𝐹𝑄) ∈ 𝐴) → (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾))
7049, 66, 68, 69syl3anc 1483 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾))
7154, 5, 39latmle1 17281 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑄 (𝐹𝑄)) 𝑊) (𝑄 (𝐹𝑄)))
7250, 70, 59, 71syl3anc 1483 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑄 (𝐹𝑄)) 𝑊) (𝑄 (𝐹𝑄)))
7365, 72eqbrtrd 4866 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) (𝑄 (𝐹𝑄)))
7454, 8, 9, 10trlcl 35945 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) ∈ (Base‘𝐾))
7544, 45, 74syl2anc 575 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) ∈ (Base‘𝐾))
7654, 5, 39latlem12 17283 . . . . 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 1484 . . . 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 694 . . 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 473 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑃) ≠ 𝑃)
815, 7, 8, 9, 10trlat 35950 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
8244, 46, 45, 80, 81syl112anc 1486 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → (𝑅𝐹) ∈ 𝐴)
8354, 39latmcl 17257 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ (𝑄 (𝐹𝑄)) ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ (Base‘𝐾))
8450, 56, 70, 83syl3anc 1483 . . . . . . 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 35090 . . . . . . 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 1485 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ≠ (0.‘𝐾))
8786neneqd 2983 . . . . 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 1338 . . . . . . 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 35306 . . . . . . 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 1497 . . . . . 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 882 . . . . 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 142 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) ∧ (𝐹𝑃) ≠ 𝑃) → ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴)
935, 7atcmp 35091 . . . 4 ((𝐾 ∈ AtLat ∧ (𝑅𝐹) ∈ 𝐴 ∧ ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ∈ 𝐴) → ((𝑅𝐹) ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))) ↔ (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄)))))
9479, 82, 92, 93syl3anc 1483 . . 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 223 . 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 3065 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 197  wa 384  wo 865  w3a 1100   = wceq 1637  wcel 2156  wne 2978   class class class wbr 4844  cfv 6101  (class class class)co 6874  Basecbs 16068  lecple 16160  joincjn 17149  meetcmee 17150  0.cp0 17242  Latclat 17250  Atomscatm 35043  AtLatcal 35044  HLchlt 35130  LHypclh 35764  LTrncltrn 35881  trLctrl 35939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-map 8094  df-proset 17133  df-poset 17151  df-plt 17163  df-lub 17179  df-glb 17180  df-join 17181  df-meet 17182  df-p0 17244  df-p1 17245  df-lat 17251  df-clat 17313  df-oposet 34956  df-ol 34958  df-oml 34959  df-covers 35046  df-ats 35047  df-atl 35078  df-cvlat 35102  df-hlat 35131  df-llines 35278  df-lhyp 35768  df-laut 35769  df-ldil 35884  df-ltrn 35885  df-trl 35940
This theorem is referenced by:  trlval4  35969
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