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Theorem cdlemc5 36883
Description: Lemma for cdlemc 36885. (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 1190 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝐾 ∈ HL)
2 simp23l 1287 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑄𝐴)
3 simp1 1129 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
4 simp21 1199 . . . . . 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 36828 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑄𝐴) → (𝐹𝑄) ∈ 𝐴)
103, 4, 2, 9syl3anc 1364 . . . . 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 36064 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴 ∧ (𝐹𝑄) ∈ 𝐴) → (𝐹𝑄) (𝑄 (𝐹𝑄)))
131, 2, 10, 12syl3anc 1364 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) (𝑄 (𝐹𝑄)))
14 simp23 1201 . . . . 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 36854 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄 (𝑅𝐹)) = (𝑄 (𝐹𝑄)))
173, 4, 14, 16syl3anc 1364 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑄 (𝑅𝐹)) = (𝑄 (𝐹𝑄)))
1813, 17breqtrrd 4996 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) (𝑄 (𝑅𝐹)))
19 simp22 1200 . . . 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 36880 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝐹𝑄) ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
223, 4, 19, 14, 21syl112anc 1367 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
231hllatd 36052 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝐾 ∈ Lat)
24 eqid 2797 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2524, 6atbase 35977 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
262, 25syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑄 ∈ (Base‘𝐾))
2724, 7, 8ltrncl 36813 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑄 ∈ (Base‘𝐾)) → (𝐹𝑄) ∈ (Base‘𝐾))
283, 4, 26, 27syl3anc 1364 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) ∈ (Base‘𝐾))
2924, 7, 8, 15trlcl 36852 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) ∈ (Base‘𝐾))
303, 4, 29syl2anc 584 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ (Base‘𝐾))
3124, 11latjcl 17494 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑅𝐹) ∈ (Base‘𝐾)) → (𝑄 (𝑅𝐹)) ∈ (Base‘𝐾))
3223, 26, 30, 31syl3anc 1364 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑄 (𝑅𝐹)) ∈ (Base‘𝐾))
33 simp22l 1285 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑃𝐴)
3424, 6atbase 35977 . . . . . . 7 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3533, 34syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑃 ∈ (Base‘𝐾))
3624, 7, 8ltrncl 36813 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃 ∈ (Base‘𝐾)) → (𝐹𝑃) ∈ (Base‘𝐾))
373, 4, 35, 36syl3anc 1364 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑃) ∈ (Base‘𝐾))
3824, 11, 6hlatjcl 36055 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
391, 33, 2, 38syl3anc 1364 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑃 𝑄) ∈ (Base‘𝐾))
40 simp1r 1191 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑊𝐻)
4124, 7lhpbase 36686 . . . . . . 7 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
4240, 41syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑊 ∈ (Base‘𝐾))
4324, 20latmcl 17495 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
4423, 39, 42, 43syl3anc 1364 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
4524, 11latjcl 17494 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝑃) ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾)) → ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) ∈ (Base‘𝐾))
4623, 37, 44, 45syl3anc 1364 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) ∈ (Base‘𝐾))
4724, 5, 20latlem12 17521 . . . 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)))))
4823, 28, 32, 46, 47syl13anc 1365 . . 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)))))
4918, 22, 48mpbi2and 708 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))))
50 hlatl 36048 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
511, 50syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝐾 ∈ AtLat)
52 simp3r 1195 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑃) ≠ 𝑃)
535, 6, 7, 8, 15trlat 36857 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
543, 19, 4, 52, 53syl112anc 1367 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
555, 7, 8, 15trlle 36872 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) 𝑊)
563, 4, 55syl2anc 584 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) 𝑊)
57 simp23r 1288 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ¬ 𝑄 𝑊)
58 nbrne2 4988 . . . . . . 7 (((𝑅𝐹) 𝑊 ∧ ¬ 𝑄 𝑊) → (𝑅𝐹) ≠ 𝑄)
5958necomd 3041 . . . . . 6 (((𝑅𝐹) 𝑊 ∧ ¬ 𝑄 𝑊) → 𝑄 ≠ (𝑅𝐹))
6056, 57, 59syl2anc 584 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑄 ≠ (𝑅𝐹))
61 eqid 2797 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
6211, 6, 61llni2 36200 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄𝐴 ∧ (𝑅𝐹) ∈ 𝐴) ∧ 𝑄 ≠ (𝑅𝐹)) → (𝑄 (𝑅𝐹)) ∈ (LLines‘𝐾))
631, 2, 54, 60, 62syl31anc 1366 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑄 (𝑅𝐹)) ∈ (LLines‘𝐾))
645, 6, 7, 8ltrnat 36828 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃𝐴) → (𝐹𝑃) ∈ 𝐴)
653, 4, 33, 64syl3anc 1364 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑃) ∈ 𝐴)
665, 11, 6hlatlej1 36063 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → 𝑃 (𝑃 (𝐹𝑃)))
671, 33, 65, 66syl3anc 1364 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑃 (𝑃 (𝐹𝑃)))
68 simp3l 1194 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ¬ 𝑄 (𝑃 (𝐹𝑃)))
69 nbrne2 4988 . . . . . . 7 ((𝑃 (𝑃 (𝐹𝑃)) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) → 𝑃𝑄)
7067, 68, 69syl2anc 584 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → 𝑃𝑄)
715, 11, 20, 6, 7lhpat 36731 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → ((𝑃 𝑄) 𝑊) ∈ 𝐴)
723, 19, 2, 70, 71syl112anc 1367 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝑃 𝑄) 𝑊) ∈ 𝐴)
7324, 5, 20latmle2 17520 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
7423, 39, 42, 73syl3anc 1364 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝑃 𝑄) 𝑊) 𝑊)
755, 6, 7, 8ltrnel 36827 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
7675simprd 496 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ¬ (𝐹𝑃) 𝑊)
773, 4, 19, 76syl3anc 1364 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ¬ (𝐹𝑃) 𝑊)
78 nbrne2 4988 . . . . . . 7 ((((𝑃 𝑄) 𝑊) 𝑊 ∧ ¬ (𝐹𝑃) 𝑊) → ((𝑃 𝑄) 𝑊) ≠ (𝐹𝑃))
7978necomd 3041 . . . . . 6 ((((𝑃 𝑄) 𝑊) 𝑊 ∧ ¬ (𝐹𝑃) 𝑊) → (𝐹𝑃) ≠ ((𝑃 𝑄) 𝑊))
8074, 77, 79syl2anc 584 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑃) ≠ ((𝑃 𝑄) 𝑊))
8111, 6, 61llni2 36200 . . . . 5 (((𝐾 ∈ HL ∧ (𝐹𝑃) ∈ 𝐴 ∧ ((𝑃 𝑄) 𝑊) ∈ 𝐴) ∧ (𝐹𝑃) ≠ ((𝑃 𝑄) 𝑊)) → ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) ∈ (LLines‘𝐾))
821, 65, 72, 80, 81syl31anc 1366 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) ∈ (LLines‘𝐾))
835, 11, 20, 6, 7, 8, 15cdlemc4 36882 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) → (𝑄 (𝑅𝐹)) ≠ ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
84833adant3r 1174 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑄 (𝑅𝐹)) ≠ ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
8524, 20latmcl 17495 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 (𝑅𝐹)) ∈ (Base‘𝐾) ∧ ((𝐹𝑃) ((𝑃 𝑄) 𝑊)) ∈ (Base‘𝐾)) → ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ∈ (Base‘𝐾))
8623, 32, 46, 85syl3anc 1364 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ∈ (Base‘𝐾))
87 eqid 2797 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
8824, 5, 87, 6atlen0 35998 . . . . 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))
8951, 86, 10, 49, 88syl31anc 1366 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ≠ (0.‘𝐾))
9020, 87, 6, 612llnmat 36212 . . . 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)
911, 63, 82, 84, 89, 90syl32anc 1371 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))) ∈ 𝐴)
925, 6atcmp 35999 . . 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)))))
9351, 10, 91, 92syl3anc 1364 . 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)))))
9449, 93mpbid 233 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 207  wa 396  w3a 1080   = wceq 1525  wcel 2083  wne 2986   class class class wbr 4968  cfv 6232  (class class class)co 7023  Basecbs 16316  lecple 16405  joincjn 17387  meetcmee 17388  0.cp0 17480  Latclat 17488  Atomscatm 35951  AtLatcal 35952  HLchlt 36038  LLinesclln 36179  LHypclh 36672  LTrncltrn 36789  trLctrl 36846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-iin 4834  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-1st 7552  df-2nd 7553  df-map 8265  df-proset 17371  df-poset 17389  df-plt 17401  df-lub 17417  df-glb 17418  df-join 17419  df-meet 17420  df-p0 17482  df-p1 17483  df-lat 17489  df-clat 17551  df-oposet 35864  df-ol 35866  df-oml 35867  df-covers 35954  df-ats 35955  df-atl 35986  df-cvlat 36010  df-hlat 36039  df-llines 36186  df-psubsp 36191  df-pmap 36192  df-padd 36484  df-lhyp 36676  df-laut 36677  df-ldil 36792  df-ltrn 36793  df-trl 36847
This theorem is referenced by:  cdlemc  36885
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