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Theorem cdlemc5 34303
Description: Lemma for cdlemc 34305. (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 1077 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝐾 ∈ HL)
2 simp23l 1174 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑄𝐴)
3 simp1 1053 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
4 simp21 1086 . . . . . 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 34247 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑄𝐴) → (𝐹𝑄) ∈ 𝐴)
103, 4, 2, 9syl3anc 1317 . . . . 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 33483 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴 ∧ (𝐹𝑄) ∈ 𝐴) → (𝐹𝑄) (𝑄 (𝐹𝑄)))
131, 2, 10, 12syl3anc 1317 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) (𝑄 (𝐹𝑄)))
14 simp23 1088 . . . . 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 34274 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄 (𝑅𝐹)) = (𝑄 (𝐹𝑄)))
173, 4, 14, 16syl3anc 1317 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑄 (𝑅𝐹)) = (𝑄 (𝐹𝑄)))
1813, 17breqtrrd 4605 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) (𝑄 (𝑅𝐹)))
19 simp22 1087 . . . 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 34300 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝐹𝑄) ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
223, 4, 19, 14, 21syl112anc 1321 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
23 hllat 33471 . . . . 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 2609 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2625, 6atbase 33397 . . . . . 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 34232 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑄 ∈ (Base‘𝐾)) → (𝐹𝑄) ∈ (Base‘𝐾))
293, 4, 27, 28syl3anc 1317 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) ∈ (Base‘𝐾))
3025, 7, 8, 15trlcl 34272 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) ∈ (Base‘𝐾))
313, 4, 30syl2anc 690 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ (Base‘𝐾))
3225, 11latjcl 16820 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑅𝐹) ∈ (Base‘𝐾)) → (𝑄 (𝑅𝐹)) ∈ (Base‘𝐾))
3324, 27, 31, 32syl3anc 1317 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑄 (𝑅𝐹)) ∈ (Base‘𝐾))
34 simp22l 1172 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑃𝐴)
3525, 6atbase 33397 . . . . . . 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 34232 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃 ∈ (Base‘𝐾)) → (𝐹𝑃) ∈ (Base‘𝐾))
383, 4, 36, 37syl3anc 1317 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑃) ∈ (Base‘𝐾))
3925, 11, 6hlatjcl 33474 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
401, 34, 2, 39syl3anc 1317 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑃 𝑄) ∈ (Base‘𝐾))
41 simp1r 1078 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑊𝐻)
4225, 7lhpbase 34105 . . . . . . 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 16821 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
4524, 40, 43, 44syl3anc 1317 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
4625, 11latjcl 16820 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝑃) ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾)) → ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) ∈ (Base‘𝐾))
4724, 38, 45, 46syl3anc 1317 . . . 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 16847 . . . 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 1319 . . 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 957 . 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 33468 . . . 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 1082 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑃) ≠ 𝑃)
545, 6, 7, 8, 15trlat 34277 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
553, 19, 4, 53, 54syl112anc 1321 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
565, 7, 8, 15trlle 34292 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) 𝑊)
573, 4, 56syl2anc 690 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) 𝑊)
58 simp23r 1175 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ¬ 𝑄 𝑊)
59 nbrne2 4597 . . . . . . 7 (((𝑅𝐹) 𝑊 ∧ ¬ 𝑄 𝑊) → (𝑅𝐹) ≠ 𝑄)
6059necomd 2836 . . . . . 6 (((𝑅𝐹) 𝑊 ∧ ¬ 𝑄 𝑊) → 𝑄 ≠ (𝑅𝐹))
6157, 58, 60syl2anc 690 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑄 ≠ (𝑅𝐹))
62 eqid 2609 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
6311, 6, 62llni2 33619 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄𝐴 ∧ (𝑅𝐹) ∈ 𝐴) ∧ 𝑄 ≠ (𝑅𝐹)) → (𝑄 (𝑅𝐹)) ∈ (LLines‘𝐾))
641, 2, 55, 61, 63syl31anc 1320 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑄 (𝑅𝐹)) ∈ (LLines‘𝐾))
655, 6, 7, 8ltrnat 34247 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃𝐴) → (𝐹𝑃) ∈ 𝐴)
663, 4, 34, 65syl3anc 1317 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑃) ∈ 𝐴)
675, 11, 6hlatlej1 33482 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → 𝑃 (𝑃 (𝐹𝑃)))
681, 34, 66, 67syl3anc 1317 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑃 (𝑃 (𝐹𝑃)))
69 simp3l 1081 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ¬ 𝑄 (𝑃 (𝐹𝑃)))
70 nbrne2 4597 . . . . . . 7 ((𝑃 (𝑃 (𝐹𝑃)) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) → 𝑃𝑄)
7168, 69, 70syl2anc 690 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑃𝑄)
725, 11, 20, 6, 7lhpat 34150 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → ((𝑃 𝑄) 𝑊) ∈ 𝐴)
733, 19, 2, 71, 72syl112anc 1321 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝑃 𝑄) 𝑊) ∈ 𝐴)
7425, 5, 20latmle2 16846 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
7524, 40, 43, 74syl3anc 1317 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝑃 𝑄) 𝑊) 𝑊)
765, 6, 7, 8ltrnel 34246 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
7776simprd 477 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ¬ (𝐹𝑃) 𝑊)
783, 4, 19, 77syl3anc 1317 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ¬ (𝐹𝑃) 𝑊)
79 nbrne2 4597 . . . . . . 7 ((((𝑃 𝑄) 𝑊) 𝑊 ∧ ¬ (𝐹𝑃) 𝑊) → ((𝑃 𝑄) 𝑊) ≠ (𝐹𝑃))
8079necomd 2836 . . . . . 6 ((((𝑃 𝑄) 𝑊) 𝑊 ∧ ¬ (𝐹𝑃) 𝑊) → (𝐹𝑃) ≠ ((𝑃 𝑄) 𝑊))
8175, 78, 80syl2anc 690 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑃) ≠ ((𝑃 𝑄) 𝑊))
8211, 6, 62llni2 33619 . . . . 5 (((𝐾 ∈ HL ∧ (𝐹𝑃) ∈ 𝐴 ∧ ((𝑃 𝑄) 𝑊) ∈ 𝐴) ∧ (𝐹𝑃) ≠ ((𝑃 𝑄) 𝑊)) → ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) ∈ (LLines‘𝐾))
831, 66, 73, 81, 82syl31anc 1320 . . . 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 34302 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) → (𝑄 (𝑅𝐹)) ≠ ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
85843adant3r 1314 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑄 (𝑅𝐹)) ≠ ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
8625, 20latmcl 16821 . . . . . 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 1317 . . . . 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 2609 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
8925, 5, 88, 6atlen0 33418 . . . . 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 1320 . . . 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 33631 . . . 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 1325 . . 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 33419 . . 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 1317 . 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 220 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  lecple 15721  joincjn 16713  meetcmee 16714  0.cp0 16806  Latclat 16814  Atomscatm 33371  AtLatcal 33372  HLchlt 33458  LLinesclln 33598  LHypclh 34091  LTrncltrn 34208  trLctrl 34266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
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 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-map 7723  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-p1 16809  df-lat 16815  df-clat 16877  df-oposet 33284  df-ol 33286  df-oml 33287  df-covers 33374  df-ats 33375  df-atl 33406  df-cvlat 33430  df-hlat 33459  df-llines 33605  df-psubsp 33610  df-pmap 33611  df-padd 33903  df-lhyp 34095  df-laut 34096  df-ldil 34211  df-ltrn 34212  df-trl 34267
This theorem is referenced by:  cdlemc  34305
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