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Theorem dia2dimlem1 38080
Description: Lemma for dia2dim 38093. Show properties of the auxiliary atom 𝑄. Part of proof of Lemma M in [Crawley] p. 121 line 3. (Contributed by NM, 8-Sep-2014.)
Hypotheses
Ref Expression
dia2dimlem1.l = (le‘𝐾)
dia2dimlem1.j = (join‘𝐾)
dia2dimlem1.m = (meet‘𝐾)
dia2dimlem1.a 𝐴 = (Atoms‘𝐾)
dia2dimlem1.h 𝐻 = (LHyp‘𝐾)
dia2dimlem1.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dia2dimlem1.r 𝑅 = ((trL‘𝐾)‘𝑊)
dia2dimlem1.q 𝑄 = ((𝑃 𝑈) ((𝐹𝑃) 𝑉))
dia2dimlem1.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
dia2dimlem1.u (𝜑 → (𝑈𝐴𝑈 𝑊))
dia2dimlem1.v (𝜑 → (𝑉𝐴𝑉 𝑊))
dia2dimlem1.p (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
dia2dimlem1.f (𝜑 → (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃))
dia2dimlem1.rf (𝜑 → (𝑅𝐹) (𝑈 𝑉))
dia2dimlem1.uv (𝜑𝑈𝑉)
dia2dimlem1.ru (𝜑 → (𝑅𝐹) ≠ 𝑈)
Assertion
Ref Expression
dia2dimlem1 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))

Proof of Theorem dia2dimlem1
StepHypRef Expression
1 dia2dimlem1.q . . 3 𝑄 = ((𝑃 𝑈) ((𝐹𝑃) 𝑉))
2 dia2dimlem1.k . . . . 5 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
32simpld 495 . . . 4 (𝜑𝐾 ∈ HL)
4 dia2dimlem1.p . . . . 5 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
54simpld 495 . . . 4 (𝜑𝑃𝐴)
6 dia2dimlem1.f . . . . 5 (𝜑 → (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃))
7 dia2dimlem1.l . . . . . 6 = (le‘𝐾)
8 dia2dimlem1.a . . . . . 6 𝐴 = (Atoms‘𝐾)
9 dia2dimlem1.h . . . . . 6 𝐻 = (LHyp‘𝐾)
10 dia2dimlem1.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
11 dia2dimlem1.r . . . . . 6 𝑅 = ((trL‘𝐾)‘𝑊)
127, 8, 9, 10, 11trlat 37185 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
132, 4, 6, 12syl3anc 1363 . . . 4 (𝜑 → (𝑅𝐹) ∈ 𝐴)
14 dia2dimlem1.u . . . . 5 (𝜑 → (𝑈𝐴𝑈 𝑊))
1514simpld 495 . . . 4 (𝜑𝑈𝐴)
166simpld 495 . . . . . 6 (𝜑𝐹𝑇)
177, 8, 9, 10ltrnel 37155 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
182, 16, 4, 17syl3anc 1363 . . . . 5 (𝜑 → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
1918simpld 495 . . . 4 (𝜑 → (𝐹𝑃) ∈ 𝐴)
20 dia2dimlem1.v . . . . 5 (𝜑 → (𝑉𝐴𝑉 𝑊))
2120simpld 495 . . . 4 (𝜑𝑉𝐴)
224simprd 496 . . . . . 6 (𝜑 → ¬ 𝑃 𝑊)
237, 9, 10, 11trlle 37200 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) 𝑊)
242, 16, 23syl2anc 584 . . . . . . . 8 (𝜑 → (𝑅𝐹) 𝑊)
2514simprd 496 . . . . . . . 8 (𝜑𝑈 𝑊)
263hllatd 36380 . . . . . . . . 9 (𝜑𝐾 ∈ Lat)
27 eqid 2818 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
2827, 8atbase 36305 . . . . . . . . . 10 ((𝑅𝐹) ∈ 𝐴 → (𝑅𝐹) ∈ (Base‘𝐾))
2913, 28syl 17 . . . . . . . . 9 (𝜑 → (𝑅𝐹) ∈ (Base‘𝐾))
3027, 8atbase 36305 . . . . . . . . . 10 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
3115, 30syl 17 . . . . . . . . 9 (𝜑𝑈 ∈ (Base‘𝐾))
322simprd 496 . . . . . . . . . 10 (𝜑𝑊𝐻)
3327, 9lhpbase 37014 . . . . . . . . . 10 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3432, 33syl 17 . . . . . . . . 9 (𝜑𝑊 ∈ (Base‘𝐾))
35 dia2dimlem1.j . . . . . . . . . 10 = (join‘𝐾)
3627, 7, 35latjle12 17660 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((𝑅𝐹) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((𝑅𝐹) 𝑊𝑈 𝑊) ↔ ((𝑅𝐹) 𝑈) 𝑊))
3726, 29, 31, 34, 36syl13anc 1364 . . . . . . . 8 (𝜑 → (((𝑅𝐹) 𝑊𝑈 𝑊) ↔ ((𝑅𝐹) 𝑈) 𝑊))
3824, 25, 37mpbi2and 708 . . . . . . 7 (𝜑 → ((𝑅𝐹) 𝑈) 𝑊)
3927, 8atbase 36305 . . . . . . . . 9 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
405, 39syl 17 . . . . . . . 8 (𝜑𝑃 ∈ (Base‘𝐾))
4127, 35, 8hlatjcl 36383 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑅𝐹) ∈ 𝐴𝑈𝐴) → ((𝑅𝐹) 𝑈) ∈ (Base‘𝐾))
423, 13, 15, 41syl3anc 1363 . . . . . . . 8 (𝜑 → ((𝑅𝐹) 𝑈) ∈ (Base‘𝐾))
4327, 7lattr 17654 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ ((𝑅𝐹) 𝑈) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑃 ((𝑅𝐹) 𝑈) ∧ ((𝑅𝐹) 𝑈) 𝑊) → 𝑃 𝑊))
4426, 40, 42, 34, 43syl13anc 1364 . . . . . . 7 (𝜑 → ((𝑃 ((𝑅𝐹) 𝑈) ∧ ((𝑅𝐹) 𝑈) 𝑊) → 𝑃 𝑊))
4538, 44mpan2d 690 . . . . . 6 (𝜑 → (𝑃 ((𝑅𝐹) 𝑈) → 𝑃 𝑊))
4622, 45mtod 199 . . . . 5 (𝜑 → ¬ 𝑃 ((𝑅𝐹) 𝑈))
4720simprd 496 . . . . . . 7 (𝜑𝑉 𝑊)
4818simprd 496 . . . . . . 7 (𝜑 → ¬ (𝐹𝑃) 𝑊)
49 nbrne2 5077 . . . . . . 7 ((𝑉 𝑊 ∧ ¬ (𝐹𝑃) 𝑊) → 𝑉 ≠ (𝐹𝑃))
5047, 48, 49syl2anc 584 . . . . . 6 (𝜑𝑉 ≠ (𝐹𝑃))
5150necomd 3068 . . . . 5 (𝜑 → (𝐹𝑃) ≠ 𝑉)
5246, 51jca 512 . . . 4 (𝜑 → (¬ 𝑃 ((𝑅𝐹) 𝑈) ∧ (𝐹𝑃) ≠ 𝑉))
5326adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝐾 ∈ Lat)
5440adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑃 ∈ (Base‘𝐾))
5527, 35, 8hlatjcl 36383 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑉𝐴𝑈𝐴) → (𝑉 𝑈) ∈ (Base‘𝐾))
563, 21, 15, 55syl3anc 1363 . . . . . . . . 9 (𝜑 → (𝑉 𝑈) ∈ (Base‘𝐾))
5756adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → (𝑉 𝑈) ∈ (Base‘𝐾))
5834adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑊 ∈ (Base‘𝐾))
597, 35, 8hlatlej2 36392 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝐹𝑃) ∈ 𝐴𝑉𝐴) → 𝑉 ((𝐹𝑃) 𝑉))
603, 19, 21, 59syl3anc 1363 . . . . . . . . . . 11 (𝜑𝑉 ((𝐹𝑃) 𝑉))
6160adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑉 ((𝐹𝑃) 𝑉))
62 simpr 485 . . . . . . . . . 10 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → (𝑃 𝑈) = ((𝐹𝑃) 𝑉))
6361, 62breqtrrd 5085 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑉 (𝑃 𝑈))
64 dia2dimlem1.uv . . . . . . . . . . . 12 (𝜑𝑈𝑉)
6564necomd 3068 . . . . . . . . . . 11 (𝜑𝑉𝑈)
667, 35, 8hlatexch2 36412 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑉𝐴𝑃𝐴𝑈𝐴) ∧ 𝑉𝑈) → (𝑉 (𝑃 𝑈) → 𝑃 (𝑉 𝑈)))
673, 21, 5, 15, 65, 66syl131anc 1375 . . . . . . . . . 10 (𝜑 → (𝑉 (𝑃 𝑈) → 𝑃 (𝑉 𝑈)))
6867adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → (𝑉 (𝑃 𝑈) → 𝑃 (𝑉 𝑈)))
6963, 68mpd 15 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑃 (𝑉 𝑈))
7027, 8atbase 36305 . . . . . . . . . . . 12 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
7121, 70syl 17 . . . . . . . . . . 11 (𝜑𝑉 ∈ (Base‘𝐾))
7227, 7, 35latjle12 17660 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑉 𝑊𝑈 𝑊) ↔ (𝑉 𝑈) 𝑊))
7326, 71, 31, 34, 72syl13anc 1364 . . . . . . . . . 10 (𝜑 → ((𝑉 𝑊𝑈 𝑊) ↔ (𝑉 𝑈) 𝑊))
7447, 25, 73mpbi2and 708 . . . . . . . . 9 (𝜑 → (𝑉 𝑈) 𝑊)
7574adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → (𝑉 𝑈) 𝑊)
7627, 7, 53, 54, 57, 58, 69, 75lattrd 17656 . . . . . . 7 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑃 𝑊)
7776ex 413 . . . . . 6 (𝜑 → ((𝑃 𝑈) = ((𝐹𝑃) 𝑉) → 𝑃 𝑊))
7877necon3bd 3027 . . . . 5 (𝜑 → (¬ 𝑃 𝑊 → (𝑃 𝑈) ≠ ((𝐹𝑃) 𝑉)))
7922, 78mpd 15 . . . 4 (𝜑 → (𝑃 𝑈) ≠ ((𝐹𝑃) 𝑉))
807, 35, 8hlatlej2 36392 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → (𝐹𝑃) (𝑃 (𝐹𝑃)))
813, 5, 19, 80syl3anc 1363 . . . . . 6 (𝜑 → (𝐹𝑃) (𝑃 (𝐹𝑃)))
82 dia2dimlem1.m . . . . . . . . . 10 = (meet‘𝐾)
837, 35, 82, 8, 9, 10, 11trlval2 37179 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
842, 16, 4, 83syl3anc 1363 . . . . . . . 8 (𝜑 → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
8584oveq2d 7161 . . . . . . 7 (𝜑 → (𝑃 (𝑅𝐹)) = (𝑃 ((𝑃 (𝐹𝑃)) 𝑊)))
8627, 35, 8hlatjcl 36383 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
873, 5, 19, 86syl3anc 1363 . . . . . . . . 9 (𝜑 → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
887, 35, 8hlatlej1 36391 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → 𝑃 (𝑃 (𝐹𝑃)))
893, 5, 19, 88syl3anc 1363 . . . . . . . . 9 (𝜑𝑃 (𝑃 (𝐹𝑃)))
9027, 7, 35, 82, 8atmod3i1 36880 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴 ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑃 (𝑃 (𝐹𝑃))) → (𝑃 ((𝑃 (𝐹𝑃)) 𝑊)) = ((𝑃 (𝐹𝑃)) (𝑃 𝑊)))
913, 5, 87, 34, 89, 90syl131anc 1375 . . . . . . . 8 (𝜑 → (𝑃 ((𝑃 (𝐹𝑃)) 𝑊)) = ((𝑃 (𝐹𝑃)) (𝑃 𝑊)))
92 eqid 2818 . . . . . . . . . . . 12 (1.‘𝐾) = (1.‘𝐾)
937, 35, 92, 8, 9lhpjat2 37037 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = (1.‘𝐾))
942, 4, 93syl2anc 584 . . . . . . . . . 10 (𝜑 → (𝑃 𝑊) = (1.‘𝐾))
9594oveq2d 7161 . . . . . . . . 9 (𝜑 → ((𝑃 (𝐹𝑃)) (𝑃 𝑊)) = ((𝑃 (𝐹𝑃)) (1.‘𝐾)))
96 hlol 36377 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ OL)
973, 96syl 17 . . . . . . . . . 10 (𝜑𝐾 ∈ OL)
9827, 82, 92olm11 36243 . . . . . . . . . 10 ((𝐾 ∈ OL ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) (1.‘𝐾)) = (𝑃 (𝐹𝑃)))
9997, 87, 98syl2anc 584 . . . . . . . . 9 (𝜑 → ((𝑃 (𝐹𝑃)) (1.‘𝐾)) = (𝑃 (𝐹𝑃)))
10095, 99eqtrd 2853 . . . . . . . 8 (𝜑 → ((𝑃 (𝐹𝑃)) (𝑃 𝑊)) = (𝑃 (𝐹𝑃)))
10191, 100eqtrd 2853 . . . . . . 7 (𝜑 → (𝑃 ((𝑃 (𝐹𝑃)) 𝑊)) = (𝑃 (𝐹𝑃)))
10285, 101eqtrd 2853 . . . . . 6 (𝜑 → (𝑃 (𝑅𝐹)) = (𝑃 (𝐹𝑃)))
10381, 102breqtrrd 5085 . . . . 5 (𝜑 → (𝐹𝑃) (𝑃 (𝑅𝐹)))
104 dia2dimlem1.rf . . . . . . 7 (𝜑 → (𝑅𝐹) (𝑈 𝑉))
10535, 8hlatjcom 36384 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑉𝐴) → (𝑈 𝑉) = (𝑉 𝑈))
1063, 15, 21, 105syl3anc 1363 . . . . . . 7 (𝜑 → (𝑈 𝑉) = (𝑉 𝑈))
107104, 106breqtrd 5083 . . . . . 6 (𝜑 → (𝑅𝐹) (𝑉 𝑈))
108 dia2dimlem1.ru . . . . . . 7 (𝜑 → (𝑅𝐹) ≠ 𝑈)
1097, 35, 8hlatexch2 36412 . . . . . . 7 ((𝐾 ∈ HL ∧ ((𝑅𝐹) ∈ 𝐴𝑉𝐴𝑈𝐴) ∧ (𝑅𝐹) ≠ 𝑈) → ((𝑅𝐹) (𝑉 𝑈) → 𝑉 ((𝑅𝐹) 𝑈)))
1103, 13, 21, 15, 108, 109syl131anc 1375 . . . . . 6 (𝜑 → ((𝑅𝐹) (𝑉 𝑈) → 𝑉 ((𝑅𝐹) 𝑈)))
111107, 110mpd 15 . . . . 5 (𝜑𝑉 ((𝑅𝐹) 𝑈))
112103, 111jca 512 . . . 4 (𝜑 → ((𝐹𝑃) (𝑃 (𝑅𝐹)) ∧ 𝑉 ((𝑅𝐹) 𝑈)))
1137, 35, 82, 8ps-2c 36544 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝑅𝐹) ∈ 𝐴) ∧ (𝑈𝐴 ∧ (𝐹𝑃) ∈ 𝐴𝑉𝐴) ∧ ((¬ 𝑃 ((𝑅𝐹) 𝑈) ∧ (𝐹𝑃) ≠ 𝑉) ∧ (𝑃 𝑈) ≠ ((𝐹𝑃) 𝑉) ∧ ((𝐹𝑃) (𝑃 (𝑅𝐹)) ∧ 𝑉 ((𝑅𝐹) 𝑈)))) → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) ∈ 𝐴)
1143, 5, 13, 15, 19, 21, 52, 79, 112, 113syl333anc 1394 . . 3 (𝜑 → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) ∈ 𝐴)
1151, 114eqeltrid 2914 . 2 (𝜑𝑄𝐴)
11627, 35, 8hlatjcl 36383 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑈𝐴) → (𝑃 𝑈) ∈ (Base‘𝐾))
1173, 5, 15, 116syl3anc 1363 . . . . . . . . . . . 12 (𝜑 → (𝑃 𝑈) ∈ (Base‘𝐾))
11827, 35, 8hlatjcl 36383 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝐹𝑃) ∈ 𝐴𝑉𝐴) → ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾))
1193, 19, 21, 118syl3anc 1363 . . . . . . . . . . . 12 (𝜑 → ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾))
12027, 7, 82latmle1 17674 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑃 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾)) → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) (𝑃 𝑈))
12126, 117, 119, 120syl3anc 1363 . . . . . . . . . . 11 (𝜑 → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) (𝑃 𝑈))
1221, 121eqbrtrid 5092 . . . . . . . . . 10 (𝜑𝑄 (𝑃 𝑈))
12327, 8atbase 36305 . . . . . . . . . . . . 13 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
124115, 123syl 17 . . . . . . . . . . . 12 (𝜑𝑄 ∈ (Base‘𝐾))
12527, 7, 82latlem12 17676 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑃 𝑈) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑄 (𝑃 𝑈) ∧ 𝑄 𝑊) ↔ 𝑄 ((𝑃 𝑈) 𝑊)))
12626, 124, 117, 34, 125syl13anc 1364 . . . . . . . . . . 11 (𝜑 → ((𝑄 (𝑃 𝑈) ∧ 𝑄 𝑊) ↔ 𝑄 ((𝑃 𝑈) 𝑊)))
127126biimpd 230 . . . . . . . . . 10 (𝜑 → ((𝑄 (𝑃 𝑈) ∧ 𝑄 𝑊) → 𝑄 ((𝑃 𝑈) 𝑊)))
128122, 127mpand 691 . . . . . . . . 9 (𝜑 → (𝑄 𝑊𝑄 ((𝑃 𝑈) 𝑊)))
129128imp 407 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑄 ((𝑃 𝑈) 𝑊))
130 eqid 2818 . . . . . . . . . . . . 13 (0.‘𝐾) = (0.‘𝐾)
1317, 82, 130, 8, 9lhpmat 37046 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = (0.‘𝐾))
1322, 4, 131syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑊) = (0.‘𝐾))
133132oveq1d 7160 . . . . . . . . . 10 (𝜑 → ((𝑃 𝑊) 𝑈) = ((0.‘𝐾) 𝑈))
13427, 7, 35, 82, 8atmod4i1 36882 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑈𝐴𝑃 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑈 𝑊) → ((𝑃 𝑊) 𝑈) = ((𝑃 𝑈) 𝑊))
1353, 15, 40, 34, 25, 134syl131anc 1375 . . . . . . . . . 10 (𝜑 → ((𝑃 𝑊) 𝑈) = ((𝑃 𝑈) 𝑊))
13627, 35, 130olj02 36242 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑈 ∈ (Base‘𝐾)) → ((0.‘𝐾) 𝑈) = 𝑈)
13797, 31, 136syl2anc 584 . . . . . . . . . 10 (𝜑 → ((0.‘𝐾) 𝑈) = 𝑈)
138133, 135, 1373eqtr3d 2861 . . . . . . . . 9 (𝜑 → ((𝑃 𝑈) 𝑊) = 𝑈)
139138adantr 481 . . . . . . . 8 ((𝜑𝑄 𝑊) → ((𝑃 𝑈) 𝑊) = 𝑈)
140129, 139breqtrd 5083 . . . . . . 7 ((𝜑𝑄 𝑊) → 𝑄 𝑈)
141 hlatl 36376 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
1423, 141syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ AtLat)
143142adantr 481 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝐾 ∈ AtLat)
144115adantr 481 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑄𝐴)
14515adantr 481 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑈𝐴)
1467, 8atcmp 36327 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑄𝐴𝑈𝐴) → (𝑄 𝑈𝑄 = 𝑈))
147143, 144, 145, 146syl3anc 1363 . . . . . . 7 ((𝜑𝑄 𝑊) → (𝑄 𝑈𝑄 = 𝑈))
148140, 147mpbid 233 . . . . . 6 ((𝜑𝑄 𝑊) → 𝑄 = 𝑈)
14927, 7, 82latmle2 17675 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑃 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾)) → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) ((𝐹𝑃) 𝑉))
15026, 117, 119, 149syl3anc 1363 . . . . . . . . . . 11 (𝜑 → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) ((𝐹𝑃) 𝑉))
1511, 150eqbrtrid 5092 . . . . . . . . . 10 (𝜑𝑄 ((𝐹𝑃) 𝑉))
15227, 7, 82latlem12 17676 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑄 ((𝐹𝑃) 𝑉) ∧ 𝑄 𝑊) ↔ 𝑄 (((𝐹𝑃) 𝑉) 𝑊)))
15326, 124, 119, 34, 152syl13anc 1364 . . . . . . . . . . 11 (𝜑 → ((𝑄 ((𝐹𝑃) 𝑉) ∧ 𝑄 𝑊) ↔ 𝑄 (((𝐹𝑃) 𝑉) 𝑊)))
154153biimpd 230 . . . . . . . . . 10 (𝜑 → ((𝑄 ((𝐹𝑃) 𝑉) ∧ 𝑄 𝑊) → 𝑄 (((𝐹𝑃) 𝑉) 𝑊)))
155151, 154mpand 691 . . . . . . . . 9 (𝜑 → (𝑄 𝑊𝑄 (((𝐹𝑃) 𝑉) 𝑊)))
156155imp 407 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑄 (((𝐹𝑃) 𝑉) 𝑊))
1577, 82, 130, 8, 9lhpmat 37046 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊)) → ((𝐹𝑃) 𝑊) = (0.‘𝐾))
1582, 18, 157syl2anc 584 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑃) 𝑊) = (0.‘𝐾))
159158oveq1d 7160 . . . . . . . . . 10 (𝜑 → (((𝐹𝑃) 𝑊) 𝑉) = ((0.‘𝐾) 𝑉))
16027, 8atbase 36305 . . . . . . . . . . . 12 ((𝐹𝑃) ∈ 𝐴 → (𝐹𝑃) ∈ (Base‘𝐾))
16119, 160syl 17 . . . . . . . . . . 11 (𝜑 → (𝐹𝑃) ∈ (Base‘𝐾))
16227, 7, 35, 82, 8atmod4i1 36882 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑉𝐴 ∧ (𝐹𝑃) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑉 𝑊) → (((𝐹𝑃) 𝑊) 𝑉) = (((𝐹𝑃) 𝑉) 𝑊))
1633, 21, 161, 34, 47, 162syl131anc 1375 . . . . . . . . . 10 (𝜑 → (((𝐹𝑃) 𝑊) 𝑉) = (((𝐹𝑃) 𝑉) 𝑊))
16427, 35, 130olj02 36242 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑉 ∈ (Base‘𝐾)) → ((0.‘𝐾) 𝑉) = 𝑉)
16597, 71, 164syl2anc 584 . . . . . . . . . 10 (𝜑 → ((0.‘𝐾) 𝑉) = 𝑉)
166159, 163, 1653eqtr3d 2861 . . . . . . . . 9 (𝜑 → (((𝐹𝑃) 𝑉) 𝑊) = 𝑉)
167166adantr 481 . . . . . . . 8 ((𝜑𝑄 𝑊) → (((𝐹𝑃) 𝑉) 𝑊) = 𝑉)
168156, 167breqtrd 5083 . . . . . . 7 ((𝜑𝑄 𝑊) → 𝑄 𝑉)
16921adantr 481 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑉𝐴)
1707, 8atcmp 36327 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑄𝐴𝑉𝐴) → (𝑄 𝑉𝑄 = 𝑉))
171143, 144, 169, 170syl3anc 1363 . . . . . . 7 ((𝜑𝑄 𝑊) → (𝑄 𝑉𝑄 = 𝑉))
172168, 171mpbid 233 . . . . . 6 ((𝜑𝑄 𝑊) → 𝑄 = 𝑉)
173148, 172eqtr3d 2855 . . . . 5 ((𝜑𝑄 𝑊) → 𝑈 = 𝑉)
174173ex 413 . . . 4 (𝜑 → (𝑄 𝑊𝑈 = 𝑉))
175174necon3ad 3026 . . 3 (𝜑 → (𝑈𝑉 → ¬ 𝑄 𝑊))
17664, 175mpd 15 . 2 (𝜑 → ¬ 𝑄 𝑊)
177115, 176jca 512 1 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  joincjn 17542  meetcmee 17543  0.cp0 17635  1.cp1 17636  Latclat 17643  OLcol 36190  Atomscatm 36279  AtLatcal 36280  HLchlt 36366  LHypclh 37000  LTrncltrn 37117  trLctrl 37174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-p1 17638  df-lat 17644  df-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367  df-llines 36514  df-psubsp 36519  df-pmap 36520  df-padd 36812  df-lhyp 37004  df-laut 37005  df-ldil 37120  df-ltrn 37121  df-trl 37175
This theorem is referenced by:  dia2dimlem3  38082  dia2dimlem6  38085
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