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Theorem trlval4 35996
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 1131 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp21 1249 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → 𝐹𝑇)
3 simp22 1250 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
4 simp23 1251 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
5 simp3r 1245 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → ¬ (𝑅𝐹) (𝑃 𝑄))
6 simpl1l 1279 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝐾 ∈ HL)
7 simp23l 1379 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → 𝑄𝐴)
87adantr 472 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑄𝐴)
9 simpl1 1228 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
10 simpl21 1321 . . . . . . . . 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 35947 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑄𝐴) → (𝐹𝑄) ∈ 𝐴)
169, 10, 8, 15syl3anc 1477 . . . . . . . 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 35182 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑄𝐴 ∧ (𝐹𝑄) ∈ 𝐴) → 𝑄 (𝑄 (𝐹𝑄)))
196, 8, 16, 18syl3anc 1477 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑄 (𝑄 (𝐹𝑄)))
20 simpl22 1323 . . . . . . . . 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 35974 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝑅𝐹)) = (𝑃 (𝐹𝑃)))
239, 10, 20, 22syl3anc 1477 . . . . . . . 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 479 . . . . . . . 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 2794 . . . . . . 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 4832 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑄 (𝑃 (𝑅𝐹)))
27 simpl3r 1289 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → ¬ (𝑅𝐹) (𝑃 𝑄))
28 simpll1 1255 . . . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 472 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) ∧ (𝐹𝑃) = 𝑃) → 𝐹𝑇)
31 simpr 479 . . . . . . . . . . . . 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 2760 . . . . . . . . . . . . . 14 (0.‘𝐾) = (0.‘𝐾)
3311, 32, 12, 13, 14, 21trl0 35978 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑅𝐹) = (0.‘𝐾))
3428, 29, 30, 31, 33syl112anc 1481 . . . . . . . . . . . 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 35168 . . . . . . . . . . . . . . 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 1377 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → 𝑃𝐴)
3837adantr 472 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑃𝐴)
39 eqid 2760 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
4039, 17, 12hlatjcl 35174 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
416, 38, 8, 40syl3anc 1477 . . . . . . . . . . . . . 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 35112 . . . . . . . . . . . . . 14 ((𝐾 ∈ AtLat ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (0.‘𝐾) (𝑃 𝑄))
4336, 41, 42syl2anc 696 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (0.‘𝐾) (𝑃 𝑄))
4443adantr 472 . . . . . . . . . . . 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 4826 . . . . . . . . . . 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 449 . . . . . . . . . 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 2946 . . . . . . . . 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 35977 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
509, 20, 10, 48, 49syl112anc 1481 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → (𝑅𝐹) ∈ 𝐴)
51 simpl3l 1287 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑃𝑄)
5251necomd 2987 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) ∧ (𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄))) → 𝑄𝑃)
5311, 17, 12hlatexch1 35202 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑄𝐴 ∧ (𝑅𝐹) ∈ 𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 (𝑅𝐹)) → (𝑅𝐹) (𝑃 𝑄)))
546, 8, 50, 38, 52, 53syl131anc 1490 . . . . . 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 449 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → ((𝑃 (𝐹𝑃)) = (𝑄 (𝐹𝑄)) → (𝑅𝐹) (𝑃 𝑄)))
5756necon3bd 2946 . . 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 35995 . 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 1489 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 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932   class class class wbr 4804  cfv 6049  (class class class)co 6814  Basecbs 16079  lecple 16170  joincjn 17165  meetcmee 17166  0.cp0 17258  Atomscatm 35071  AtLatcal 35072  HLchlt 35158  LHypclh 35791  LTrncltrn 35908  trLctrl 35966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027  df-preset 17149  df-poset 17167  df-plt 17179  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p0 17260  df-p1 17261  df-lat 17267  df-clat 17329  df-oposet 34984  df-ol 34986  df-oml 34987  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159  df-llines 35305  df-psubsp 35310  df-pmap 35311  df-padd 35603  df-lhyp 35795  df-laut 35796  df-ldil 35911  df-ltrn 35912  df-trl 35967
This theorem is referenced by:  cdlemg10a  36448  cdlemg12d  36454
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