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Theorem trlval4 35294
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 1059 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp21 1092 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → 𝐹𝑇)
3 simp22 1093 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
4 simp23 1094 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
5 simp3r 1088 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → ¬ (𝑅𝐹) (𝑃 𝑄))
6 simpl1l 1110 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝐾 ∈ HL)
7 simp23l 1180 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → 𝑄𝐴)
87adantr 481 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑄𝐴)
9 simpl1 1062 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
10 simpl21 1137 . . . . . . . . 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 35245 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑄𝐴) → (𝐹𝑄) ∈ 𝐴)
169, 10, 8, 15syl3anc 1324 . . . . . . . 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 34480 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴 ∧ (𝐹𝑄) ∈ 𝐴) → 𝑄 (𝑄 (𝐹𝑄)))
196, 8, 16, 18syl3anc 1324 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑄 (𝑄 (𝐹𝑄)))
20 simpl22 1138 . . . . . . . . 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 35272 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝑅𝐹)) = (𝑃 (𝐹𝑃)))
239, 10, 20, 22syl3anc 1324 . . . . . . . 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 477 . . . . . . . 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 2654 . . . . . . 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 4672 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑄 (𝑃 (𝑅𝐹)))
27 simpl3r 1115 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → ¬ (𝑅𝐹) (𝑃 𝑄))
28 simpll1 1098 . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) ∧ (𝐹𝑃) = 𝑃) → 𝐹𝑇)
31 simpr 477 . . . . . . . . . . . . 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 2620 . . . . . . . . . . . . . 14 (0.‘𝐾) = (0.‘𝐾)
3311, 32, 12, 13, 14, 21trl0 35276 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑅𝐹) = (0.‘𝐾))
3428, 29, 30, 31, 33syl112anc 1328 . . . . . . . . . . . 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 34466 . . . . . . . . . . . . . . 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 1178 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → 𝑃𝐴)
3837adantr 481 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑃𝐴)
39 eqid 2620 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
4039, 17, 12hlatjcl 34472 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
416, 38, 8, 40syl3anc 1324 . . . . . . . . . . . . . 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 34410 . . . . . . . . . . . . . 14 ((𝐾 ∈ AtLat ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (0.‘𝐾) (𝑃 𝑄))
4336, 41, 42syl2anc 692 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (0.‘𝐾) (𝑃 𝑄))
4443adantr 481 . . . . . . . . . . . 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 4666 . . . . . . . . . . 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 450 . . . . . . . . . 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 2805 . . . . . . . . 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 35275 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
509, 20, 10, 48, 49syl112anc 1328 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (𝑅𝐹) ∈ 𝐴)
51 simpl3l 1114 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑃𝑄)
5251necomd 2846 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑄𝑃)
5311, 17, 12hlatexch1 34500 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑄𝐴 ∧ (𝑅𝐹) ∈ 𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 (𝑅𝐹)) → (𝑅𝐹) (𝑃 𝑄)))
546, 8, 50, 38, 52, 53syl131anc 1337 . . . . . 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 450 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → ((𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄)) → (𝑅𝐹) (𝑃 𝑄)))
5756necon3bd 2805 . . 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 35293 . 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 1336 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 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791   class class class wbr 4644  cfv 5876  (class class class)co 6635  Basecbs 15838  lecple 15929  joincjn 16925  meetcmee 16926  0.cp0 17018  Atomscatm 34369  AtLatcal 34370  HLchlt 34456  LHypclh 35089  LTrncltrn 35206  trLctrl 35264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-map 7844  df-preset 16909  df-poset 16927  df-plt 16939  df-lub 16955  df-glb 16956  df-join 16957  df-meet 16958  df-p0 17020  df-p1 17021  df-lat 17027  df-clat 17089  df-oposet 34282  df-ol 34284  df-oml 34285  df-covers 34372  df-ats 34373  df-atl 34404  df-cvlat 34428  df-hlat 34457  df-llines 34603  df-psubsp 34608  df-pmap 34609  df-padd 34901  df-lhyp 35093  df-laut 35094  df-ldil 35209  df-ltrn 35210  df-trl 35265
This theorem is referenced by:  cdlemg10a  35747  cdlemg12d  35753
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