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Theorem trlval4 34292
Description: The value of the trace of a lattice translation in terms of 2 atoms. (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
trlval4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))

Proof of Theorem trlval4
StepHypRef Expression
1 simp1 1053 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp21 1086 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → 𝐹𝑇)
3 simp22 1087 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
4 simp23 1088 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
5 simp3r 1082 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → ¬ (𝑅𝐹) (𝑃 𝑄))
6 simpl1l 1104 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝐾 ∈ HL)
7 simp23l 1174 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → 𝑄𝐴)
87adantr 479 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑄𝐴)
9 simpl1 1056 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
10 simpl21 1131 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝐹𝑇)
11 trlval3.l . . . . . . . . . 10 = (le‘𝐾)
12 trlval3.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
13 trlval3.h . . . . . . . . . 10 𝐻 = (LHyp‘𝐾)
14 trlval3.t . . . . . . . . . 10 𝑇 = ((LTrn‘𝐾)‘𝑊)
1511, 12, 13, 14ltrnat 34243 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑄𝐴) → (𝐹𝑄) ∈ 𝐴)
169, 10, 8, 15syl3anc 1317 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (𝐹𝑄) ∈ 𝐴)
17 trlval3.j . . . . . . . . 9 = (join‘𝐾)
1811, 17, 12hlatlej1 33478 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴 ∧ (𝐹𝑄) ∈ 𝐴) → 𝑄 (𝑄 (𝐹𝑄)))
196, 8, 16, 18syl3anc 1317 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑄 (𝑄 (𝐹𝑄)))
20 simpl22 1132 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
21 trlval3.r . . . . . . . . . 10 𝑅 = ((trL‘𝐾)‘𝑊)
2211, 17, 12, 13, 14, 21trljat1 34270 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝑅𝐹)) = (𝑃 (𝐹𝑃)))
239, 10, 20, 22syl3anc 1317 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (𝑃 (𝑅𝐹)) = (𝑃 (𝐹𝑃)))
24 simpr 475 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄)))
2523, 24eqtrd 2639 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (𝑃 (𝑅𝐹)) = (𝑄 (𝐹𝑄)))
2619, 25breqtrrd 4601 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑄 (𝑃 (𝑅𝐹)))
27 simpl3r 1109 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → ¬ (𝑅𝐹) (𝑃 𝑄))
28 simpll1 1092 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) ∧ (𝐹𝑃) = 𝑃) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2920adantr 479 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) ∧ (𝐹𝑃) = 𝑃) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3010adantr 479 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) ∧ (𝐹𝑃) = 𝑃) → 𝐹𝑇)
31 simpr 475 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) ∧ (𝐹𝑃) = 𝑃) → (𝐹𝑃) = 𝑃)
32 eqid 2605 . . . . . . . . . . . . . 14 (0.‘𝐾) = (0.‘𝐾)
3311, 32, 12, 13, 14, 21trl0 34274 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑅𝐹) = (0.‘𝐾))
3428, 29, 30, 31, 33syl112anc 1321 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) ∧ (𝐹𝑃) = 𝑃) → (𝑅𝐹) = (0.‘𝐾))
35 hlatl 33464 . . . . . . . . . . . . . . 15 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
366, 35syl 17 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝐾 ∈ AtLat)
37 simp22l 1172 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → 𝑃𝐴)
3837adantr 479 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑃𝐴)
39 eqid 2605 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
4039, 17, 12hlatjcl 33470 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
416, 38, 8, 40syl3anc 1317 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (𝑃 𝑄) ∈ (Base‘𝐾))
4239, 11, 32atl0le 33408 . . . . . . . . . . . . . 14 ((𝐾 ∈ AtLat ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (0.‘𝐾) (𝑃 𝑄))
4336, 41, 42syl2anc 690 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (0.‘𝐾) (𝑃 𝑄))
4443adantr 479 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) ∧ (𝐹𝑃) = 𝑃) → (0.‘𝐾) (𝑃 𝑄))
4534, 44eqbrtrd 4595 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) ∧ (𝐹𝑃) = 𝑃) → (𝑅𝐹) (𝑃 𝑄))
4645ex 448 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → ((𝐹𝑃) = 𝑃 → (𝑅𝐹) (𝑃 𝑄)))
4746necon3bd 2791 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (¬ (𝑅𝐹) (𝑃 𝑄) → (𝐹𝑃) ≠ 𝑃))
4827, 47mpd 15 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (𝐹𝑃) ≠ 𝑃)
4911, 12, 13, 14, 21trlat 34273 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
509, 20, 10, 48, 49syl112anc 1321 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (𝑅𝐹) ∈ 𝐴)
51 simpl3l 1108 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑃𝑄)
5251necomd 2832 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑄𝑃)
5311, 17, 12hlatexch1 33498 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑄𝐴 ∧ (𝑅𝐹) ∈ 𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 (𝑅𝐹)) → (𝑅𝐹) (𝑃 𝑄)))
546, 8, 50, 38, 52, 53syl131anc 1330 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (𝑄 (𝑃 (𝑅𝐹)) → (𝑅𝐹) (𝑃 𝑄)))
5526, 54mpd 15 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (𝑅𝐹) (𝑃 𝑄))
5655ex 448 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → ((𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄)) → (𝑅𝐹) (𝑃 𝑄)))
5756necon3bd 2791 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → (¬ (𝑅𝐹) (𝑃 𝑄) → (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄))))
585, 57mpd 15 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))
59 trlval3.m . . 3 = (meet‘𝐾)
6011, 17, 59, 12, 13, 14, 21trlval3 34291 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))
611, 2, 3, 4, 58, 60syl113anc 1329 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2775   class class class wbr 4573  cfv 5786  (class class class)co 6523  Basecbs 15637  lecple 15717  joincjn 16709  meetcmee 16710  0.cp0 16802  Atomscatm 33367  AtLatcal 33368  HLchlt 33454  LHypclh 34087  LTrncltrn 34204  trLctrl 34262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-iin 4448  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-1st 7032  df-2nd 7033  df-map 7719  df-preset 16693  df-poset 16711  df-plt 16723  df-lub 16739  df-glb 16740  df-join 16741  df-meet 16742  df-p0 16804  df-p1 16805  df-lat 16811  df-clat 16873  df-oposet 33280  df-ol 33282  df-oml 33283  df-covers 33370  df-ats 33371  df-atl 33402  df-cvlat 33426  df-hlat 33455  df-llines 33601  df-psubsp 33606  df-pmap 33607  df-padd 33899  df-lhyp 34091  df-laut 34092  df-ldil 34207  df-ltrn 34208  df-trl 34263
This theorem is referenced by:  cdlemg10a  34745  cdlemg12d  34751
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