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Theorem cdlemc5 36004
Description: Lemma for cdlemc 36006. (Contributed by NM, 26-May-2012.)
Hypotheses
Ref Expression
cdlemc3.l = (le‘𝐾)
cdlemc3.j = (join‘𝐾)
cdlemc3.m = (meet‘𝐾)
cdlemc3.a 𝐴 = (Atoms‘𝐾)
cdlemc3.h 𝐻 = (LHyp‘𝐾)
cdlemc3.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemc3.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemc5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) = ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))))

Proof of Theorem cdlemc5
StepHypRef Expression
1 simp1l 1239 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝐾 ∈ HL)
2 simp23l 1378 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑄𝐴)
3 simp1 1130 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
4 simp21 1248 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝐹𝑇)
5 cdlemc3.l . . . . . . 7 = (le‘𝐾)
6 cdlemc3.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
7 cdlemc3.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
8 cdlemc3.t . . . . . . 7 𝑇 = ((LTrn‘𝐾)‘𝑊)
95, 6, 7, 8ltrnat 35948 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑄𝐴) → (𝐹𝑄) ∈ 𝐴)
103, 4, 2, 9syl3anc 1476 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) ∈ 𝐴)
11 cdlemc3.j . . . . . 6 = (join‘𝐾)
125, 11, 6hlatlej2 35184 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴 ∧ (𝐹𝑄) ∈ 𝐴) → (𝐹𝑄) (𝑄 (𝐹𝑄)))
131, 2, 10, 12syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) (𝑄 (𝐹𝑄)))
14 simp23 1250 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
15 cdlemc3.r . . . . . 6 𝑅 = ((trL‘𝐾)‘𝑊)
165, 11, 6, 7, 8, 15trljat1 35975 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄 (𝑅𝐹)) = (𝑄 (𝐹𝑄)))
173, 4, 14, 16syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑄 (𝑅𝐹)) = (𝑄 (𝐹𝑄)))
1813, 17breqtrrd 4814 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) (𝑄 (𝑅𝐹)))
19 simp22 1249 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
20 cdlemc3.m . . . . 5 = (meet‘𝐾)
215, 11, 20, 6, 7, 8cdlemc2 36001 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝐹𝑄) ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
223, 4, 19, 14, 21syl112anc 1480 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
23 hllat 35172 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
241, 23syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝐾 ∈ Lat)
25 eqid 2771 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2625, 6atbase 35098 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
272, 26syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑄 ∈ (Base‘𝐾))
2825, 7, 8ltrncl 35933 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑄 ∈ (Base‘𝐾)) → (𝐹𝑄) ∈ (Base‘𝐾))
293, 4, 27, 28syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) ∈ (Base‘𝐾))
3025, 7, 8, 15trlcl 35973 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) ∈ (Base‘𝐾))
313, 4, 30syl2anc 573 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ (Base‘𝐾))
3225, 11latjcl 17259 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑅𝐹) ∈ (Base‘𝐾)) → (𝑄 (𝑅𝐹)) ∈ (Base‘𝐾))
3324, 27, 31, 32syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑄 (𝑅𝐹)) ∈ (Base‘𝐾))
34 simp22l 1376 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑃𝐴)
3525, 6atbase 35098 . . . . . . 7 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3634, 35syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑃 ∈ (Base‘𝐾))
3725, 7, 8ltrncl 35933 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃 ∈ (Base‘𝐾)) → (𝐹𝑃) ∈ (Base‘𝐾))
383, 4, 36, 37syl3anc 1476 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑃) ∈ (Base‘𝐾))
3925, 11, 6hlatjcl 35175 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
401, 34, 2, 39syl3anc 1476 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑃 𝑄) ∈ (Base‘𝐾))
41 simp1r 1240 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑊𝐻)
4225, 7lhpbase 35806 . . . . . . 7 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
4341, 42syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑊 ∈ (Base‘𝐾))
4425, 20latmcl 17260 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
4524, 40, 43, 44syl3anc 1476 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
4625, 11latjcl 17259 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝑃) ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾)) → ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) ∈ (Base‘𝐾))
4724, 38, 45, 46syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) ∈ (Base‘𝐾))
4825, 5, 20latlem12 17286 . . . 4 ((𝐾 ∈ Lat ∧ ((𝐹𝑄) ∈ (Base‘𝐾) ∧ (𝑄 (𝑅𝐹)) ∈ (Base‘𝐾) ∧ ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) ∈ (Base‘𝐾))) → (((𝐹𝑄) (𝑄 (𝑅𝐹)) ∧ (𝐹𝑄) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ↔ (𝐹𝑄) ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))))
4924, 29, 33, 47, 48syl13anc 1478 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (((𝐹𝑄) (𝑄 (𝑅𝐹)) ∧ (𝐹𝑄) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ↔ (𝐹𝑄) ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))))
5018, 22, 49mpbi2and 691 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))))
51 hlatl 35169 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
521, 51syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝐾 ∈ AtLat)
53 simp3r 1244 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑃) ≠ 𝑃)
545, 6, 7, 8, 15trlat 35978 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
553, 19, 4, 53, 54syl112anc 1480 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
565, 7, 8, 15trlle 35993 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) 𝑊)
573, 4, 56syl2anc 573 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) 𝑊)
58 simp23r 1379 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ¬ 𝑄 𝑊)
59 nbrne2 4806 . . . . . . 7 (((𝑅𝐹) 𝑊 ∧ ¬ 𝑄 𝑊) → (𝑅𝐹) ≠ 𝑄)
6059necomd 2998 . . . . . 6 (((𝑅𝐹) 𝑊 ∧ ¬ 𝑄 𝑊) → 𝑄 ≠ (𝑅𝐹))
6157, 58, 60syl2anc 573 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑄 ≠ (𝑅𝐹))
62 eqid 2771 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
6311, 6, 62llni2 35320 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄𝐴 ∧ (𝑅𝐹) ∈ 𝐴) ∧ 𝑄 ≠ (𝑅𝐹)) → (𝑄 (𝑅𝐹)) ∈ (LLines‘𝐾))
641, 2, 55, 61, 63syl31anc 1479 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑄 (𝑅𝐹)) ∈ (LLines‘𝐾))
655, 6, 7, 8ltrnat 35948 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃𝐴) → (𝐹𝑃) ∈ 𝐴)
663, 4, 34, 65syl3anc 1476 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑃) ∈ 𝐴)
675, 11, 6hlatlej1 35183 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → 𝑃 (𝑃 (𝐹𝑃)))
681, 34, 66, 67syl3anc 1476 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑃 (𝑃 (𝐹𝑃)))
69 simp3l 1243 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ¬ 𝑄 (𝑃 (𝐹𝑃)))
70 nbrne2 4806 . . . . . . 7 ((𝑃 (𝑃 (𝐹𝑃)) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) → 𝑃𝑄)
7168, 69, 70syl2anc 573 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑃𝑄)
725, 11, 20, 6, 7lhpat 35851 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → ((𝑃 𝑄) 𝑊) ∈ 𝐴)
733, 19, 2, 71, 72syl112anc 1480 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝑃 𝑄) 𝑊) ∈ 𝐴)
7425, 5, 20latmle2 17285 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
7524, 40, 43, 74syl3anc 1476 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝑃 𝑄) 𝑊) 𝑊)
765, 6, 7, 8ltrnel 35947 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
7776simprd 483 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ¬ (𝐹𝑃) 𝑊)
783, 4, 19, 77syl3anc 1476 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ¬ (𝐹𝑃) 𝑊)
79 nbrne2 4806 . . . . . . 7 ((((𝑃 𝑄) 𝑊) 𝑊 ∧ ¬ (𝐹𝑃) 𝑊) → ((𝑃 𝑄) 𝑊) ≠ (𝐹𝑃))
8079necomd 2998 . . . . . 6 ((((𝑃 𝑄) 𝑊) 𝑊 ∧ ¬ (𝐹𝑃) 𝑊) → (𝐹𝑃) ≠ ((𝑃 𝑄) 𝑊))
8175, 78, 80syl2anc 573 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑃) ≠ ((𝑃 𝑄) 𝑊))
8211, 6, 62llni2 35320 . . . . 5 (((𝐾 ∈ HL ∧ (𝐹𝑃) ∈ 𝐴 ∧ ((𝑃 𝑄) 𝑊) ∈ 𝐴) ∧ (𝐹𝑃) ≠ ((𝑃 𝑄) 𝑊)) → ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) ∈ (LLines‘𝐾))
831, 66, 73, 81, 82syl31anc 1479 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) ∈ (LLines‘𝐾))
845, 11, 20, 6, 7, 8, 15cdlemc4 36003 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) → (𝑄 (𝑅𝐹)) ≠ ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
85843adant3r 1195 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑄 (𝑅𝐹)) ≠ ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
8625, 20latmcl 17260 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 (𝑅𝐹)) ∈ (Base‘𝐾) ∧ ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) ∈ (Base‘𝐾)) → ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ∈ (Base‘𝐾))
8724, 33, 47, 86syl3anc 1476 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ∈ (Base‘𝐾))
88 eqid 2771 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
8925, 5, 88, 6atlen0 35119 . . . . 5 (((𝐾 ∈ AtLat ∧ ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ∈ (Base‘𝐾) ∧ (𝐹𝑄) ∈ 𝐴) ∧ (𝐹𝑄) ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))) → ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ≠ (0.‘𝐾))
9052, 87, 10, 50, 89syl31anc 1479 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ≠ (0.‘𝐾))
9120, 88, 6, 622llnmat 35332 . . . 4 (((𝐾 ∈ HL ∧ (𝑄 (𝑅𝐹)) ∈ (LLines‘𝐾) ∧ ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) ∈ (LLines‘𝐾)) ∧ ((𝑄 (𝑅𝐹)) ≠ ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) ∧ ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ≠ (0.‘𝐾))) → ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ∈ 𝐴)
921, 64, 83, 85, 90, 91syl32anc 1484 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ∈ 𝐴)
935, 6atcmp 35120 . . 3 ((𝐾 ∈ AtLat ∧ (𝐹𝑄) ∈ 𝐴 ∧ ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ∈ 𝐴) → ((𝐹𝑄) ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ↔ (𝐹𝑄) = ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))))
9452, 10, 92, 93syl3anc 1476 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝐹𝑄) ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ↔ (𝐹𝑄) = ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))))
9550, 94mpbid 222 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) = ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  0.cp0 17245  Latclat 17253  Atomscatm 35072  AtLatcal 35073  HLchlt 35159  LLinesclln 35299  LHypclh 35792  LTrncltrn 35909  trLctrl 35967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-p1 17248  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35306  df-psubsp 35311  df-pmap 35312  df-padd 35604  df-lhyp 35796  df-laut 35797  df-ldil 35912  df-ltrn 35913  df-trl 35968
This theorem is referenced by:  cdlemc  36006
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