Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dia2dimlem1 Structured version   Visualization version   GIF version

Theorem dia2dimlem1 41067
Description: Lemma for dia2dim 41080. 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 494 . . . 4 (𝜑𝐾 ∈ HL)
4 dia2dimlem1.p . . . . 5 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
54simpld 494 . . . 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 40172 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
132, 4, 6, 12syl3anc 1372 . . . 4 (𝜑 → (𝑅𝐹) ∈ 𝐴)
14 dia2dimlem1.u . . . . 5 (𝜑 → (𝑈𝐴𝑈 𝑊))
1514simpld 494 . . . 4 (𝜑𝑈𝐴)
166simpld 494 . . . . . 6 (𝜑𝐹𝑇)
177, 8, 9, 10ltrnel 40142 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
182, 16, 4, 17syl3anc 1372 . . . . 5 (𝜑 → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
1918simpld 494 . . . 4 (𝜑 → (𝐹𝑃) ∈ 𝐴)
20 dia2dimlem1.v . . . . 5 (𝜑 → (𝑉𝐴𝑉 𝑊))
2120simpld 494 . . . 4 (𝜑𝑉𝐴)
224simprd 495 . . . . . 6 (𝜑 → ¬ 𝑃 𝑊)
237, 9, 10, 11trlle 40187 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) 𝑊)
242, 16, 23syl2anc 584 . . . . . . . 8 (𝜑 → (𝑅𝐹) 𝑊)
2514simprd 495 . . . . . . . 8 (𝜑𝑈 𝑊)
263hllatd 39366 . . . . . . . . 9 (𝜑𝐾 ∈ Lat)
27 eqid 2736 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
2827, 8atbase 39291 . . . . . . . . . 10 ((𝑅𝐹) ∈ 𝐴 → (𝑅𝐹) ∈ (Base‘𝐾))
2913, 28syl 17 . . . . . . . . 9 (𝜑 → (𝑅𝐹) ∈ (Base‘𝐾))
3027, 8atbase 39291 . . . . . . . . . 10 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
3115, 30syl 17 . . . . . . . . 9 (𝜑𝑈 ∈ (Base‘𝐾))
322simprd 495 . . . . . . . . . 10 (𝜑𝑊𝐻)
3327, 9lhpbase 40001 . . . . . . . . . 10 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3432, 33syl 17 . . . . . . . . 9 (𝜑𝑊 ∈ (Base‘𝐾))
35 dia2dimlem1.j . . . . . . . . . 10 = (join‘𝐾)
3627, 7, 35latjle12 18496 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((𝑅𝐹) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((𝑅𝐹) 𝑊𝑈 𝑊) ↔ ((𝑅𝐹) 𝑈) 𝑊))
3726, 29, 31, 34, 36syl13anc 1373 . . . . . . . 8 (𝜑 → (((𝑅𝐹) 𝑊𝑈 𝑊) ↔ ((𝑅𝐹) 𝑈) 𝑊))
3824, 25, 37mpbi2and 712 . . . . . . 7 (𝜑 → ((𝑅𝐹) 𝑈) 𝑊)
3927, 8atbase 39291 . . . . . . . . 9 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
405, 39syl 17 . . . . . . . 8 (𝜑𝑃 ∈ (Base‘𝐾))
4127, 35, 8hlatjcl 39369 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑅𝐹) ∈ 𝐴𝑈𝐴) → ((𝑅𝐹) 𝑈) ∈ (Base‘𝐾))
423, 13, 15, 41syl3anc 1372 . . . . . . . 8 (𝜑 → ((𝑅𝐹) 𝑈) ∈ (Base‘𝐾))
4327, 7lattr 18490 . . . . . . . 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 1373 . . . . . . 7 (𝜑 → ((𝑃 ((𝑅𝐹) 𝑈) ∧ ((𝑅𝐹) 𝑈) 𝑊) → 𝑃 𝑊))
4538, 44mpan2d 694 . . . . . 6 (𝜑 → (𝑃 ((𝑅𝐹) 𝑈) → 𝑃 𝑊))
4622, 45mtod 198 . . . . 5 (𝜑 → ¬ 𝑃 ((𝑅𝐹) 𝑈))
4720simprd 495 . . . . . . 7 (𝜑𝑉 𝑊)
4818simprd 495 . . . . . . 7 (𝜑 → ¬ (𝐹𝑃) 𝑊)
49 nbrne2 5162 . . . . . . 7 ((𝑉 𝑊 ∧ ¬ (𝐹𝑃) 𝑊) → 𝑉 ≠ (𝐹𝑃))
5047, 48, 49syl2anc 584 . . . . . 6 (𝜑𝑉 ≠ (𝐹𝑃))
5150necomd 2995 . . . . 5 (𝜑 → (𝐹𝑃) ≠ 𝑉)
5246, 51jca 511 . . . 4 (𝜑 → (¬ 𝑃 ((𝑅𝐹) 𝑈) ∧ (𝐹𝑃) ≠ 𝑉))
5326adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝐾 ∈ Lat)
5440adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑃 ∈ (Base‘𝐾))
5527, 35, 8hlatjcl 39369 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑉𝐴𝑈𝐴) → (𝑉 𝑈) ∈ (Base‘𝐾))
563, 21, 15, 55syl3anc 1372 . . . . . . . . 9 (𝜑 → (𝑉 𝑈) ∈ (Base‘𝐾))
5756adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → (𝑉 𝑈) ∈ (Base‘𝐾))
5834adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑊 ∈ (Base‘𝐾))
597, 35, 8hlatlej2 39378 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝐹𝑃) ∈ 𝐴𝑉𝐴) → 𝑉 ((𝐹𝑃) 𝑉))
603, 19, 21, 59syl3anc 1372 . . . . . . . . . . 11 (𝜑𝑉 ((𝐹𝑃) 𝑉))
6160adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑉 ((𝐹𝑃) 𝑉))
62 simpr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → (𝑃 𝑈) = ((𝐹𝑃) 𝑉))
6361, 62breqtrrd 5170 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑉 (𝑃 𝑈))
64 dia2dimlem1.uv . . . . . . . . . . . 12 (𝜑𝑈𝑉)
6564necomd 2995 . . . . . . . . . . 11 (𝜑𝑉𝑈)
667, 35, 8hlatexch2 39399 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑉𝐴𝑃𝐴𝑈𝐴) ∧ 𝑉𝑈) → (𝑉 (𝑃 𝑈) → 𝑃 (𝑉 𝑈)))
673, 21, 5, 15, 65, 66syl131anc 1384 . . . . . . . . . 10 (𝜑 → (𝑉 (𝑃 𝑈) → 𝑃 (𝑉 𝑈)))
6867adantr 480 . . . . . . . . 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 39291 . . . . . . . . . . . 12 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
7121, 70syl 17 . . . . . . . . . . 11 (𝜑𝑉 ∈ (Base‘𝐾))
7227, 7, 35latjle12 18496 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑉 𝑊𝑈 𝑊) ↔ (𝑉 𝑈) 𝑊))
7326, 71, 31, 34, 72syl13anc 1373 . . . . . . . . . 10 (𝜑 → ((𝑉 𝑊𝑈 𝑊) ↔ (𝑉 𝑈) 𝑊))
7447, 25, 73mpbi2and 712 . . . . . . . . 9 (𝜑 → (𝑉 𝑈) 𝑊)
7574adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → (𝑉 𝑈) 𝑊)
7627, 7, 53, 54, 57, 58, 69, 75lattrd 18492 . . . . . . 7 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑃 𝑊)
7776ex 412 . . . . . 6 (𝜑 → ((𝑃 𝑈) = ((𝐹𝑃) 𝑉) → 𝑃 𝑊))
7877necon3bd 2953 . . . . 5 (𝜑 → (¬ 𝑃 𝑊 → (𝑃 𝑈) ≠ ((𝐹𝑃) 𝑉)))
7922, 78mpd 15 . . . 4 (𝜑 → (𝑃 𝑈) ≠ ((𝐹𝑃) 𝑉))
807, 35, 8hlatlej2 39378 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → (𝐹𝑃) (𝑃 (𝐹𝑃)))
813, 5, 19, 80syl3anc 1372 . . . . . 6 (𝜑 → (𝐹𝑃) (𝑃 (𝐹𝑃)))
82 dia2dimlem1.m . . . . . . . . . 10 = (meet‘𝐾)
837, 35, 82, 8, 9, 10, 11trlval2 40166 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
842, 16, 4, 83syl3anc 1372 . . . . . . . 8 (𝜑 → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
8584oveq2d 7448 . . . . . . 7 (𝜑 → (𝑃 (𝑅𝐹)) = (𝑃 ((𝑃 (𝐹𝑃)) 𝑊)))
8627, 35, 8hlatjcl 39369 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
873, 5, 19, 86syl3anc 1372 . . . . . . . . 9 (𝜑 → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
887, 35, 8hlatlej1 39377 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → 𝑃 (𝑃 (𝐹𝑃)))
893, 5, 19, 88syl3anc 1372 . . . . . . . . 9 (𝜑𝑃 (𝑃 (𝐹𝑃)))
9027, 7, 35, 82, 8atmod3i1 39867 . . . . . . . . 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 1384 . . . . . . . 8 (𝜑 → (𝑃 ((𝑃 (𝐹𝑃)) 𝑊)) = ((𝑃 (𝐹𝑃)) (𝑃 𝑊)))
92 eqid 2736 . . . . . . . . . . . 12 (1.‘𝐾) = (1.‘𝐾)
937, 35, 92, 8, 9lhpjat2 40024 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = (1.‘𝐾))
942, 4, 93syl2anc 584 . . . . . . . . . 10 (𝜑 → (𝑃 𝑊) = (1.‘𝐾))
9594oveq2d 7448 . . . . . . . . 9 (𝜑 → ((𝑃 (𝐹𝑃)) (𝑃 𝑊)) = ((𝑃 (𝐹𝑃)) (1.‘𝐾)))
96 hlol 39363 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ OL)
973, 96syl 17 . . . . . . . . . 10 (𝜑𝐾 ∈ OL)
9827, 82, 92olm11 39229 . . . . . . . . . 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 2776 . . . . . . . 8 (𝜑 → ((𝑃 (𝐹𝑃)) (𝑃 𝑊)) = (𝑃 (𝐹𝑃)))
10191, 100eqtrd 2776 . . . . . . 7 (𝜑 → (𝑃 ((𝑃 (𝐹𝑃)) 𝑊)) = (𝑃 (𝐹𝑃)))
10285, 101eqtrd 2776 . . . . . 6 (𝜑 → (𝑃 (𝑅𝐹)) = (𝑃 (𝐹𝑃)))
10381, 102breqtrrd 5170 . . . . 5 (𝜑 → (𝐹𝑃) (𝑃 (𝑅𝐹)))
104 dia2dimlem1.rf . . . . . . 7 (𝜑 → (𝑅𝐹) (𝑈 𝑉))
10535, 8hlatjcom 39370 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑉𝐴) → (𝑈 𝑉) = (𝑉 𝑈))
1063, 15, 21, 105syl3anc 1372 . . . . . . 7 (𝜑 → (𝑈 𝑉) = (𝑉 𝑈))
107104, 106breqtrd 5168 . . . . . 6 (𝜑 → (𝑅𝐹) (𝑉 𝑈))
108 dia2dimlem1.ru . . . . . . 7 (𝜑 → (𝑅𝐹) ≠ 𝑈)
1097, 35, 8hlatexch2 39399 . . . . . . 7 ((𝐾 ∈ HL ∧ ((𝑅𝐹) ∈ 𝐴𝑉𝐴𝑈𝐴) ∧ (𝑅𝐹) ≠ 𝑈) → ((𝑅𝐹) (𝑉 𝑈) → 𝑉 ((𝑅𝐹) 𝑈)))
1103, 13, 21, 15, 108, 109syl131anc 1384 . . . . . 6 (𝜑 → ((𝑅𝐹) (𝑉 𝑈) → 𝑉 ((𝑅𝐹) 𝑈)))
111107, 110mpd 15 . . . . 5 (𝜑𝑉 ((𝑅𝐹) 𝑈))
112103, 111jca 511 . . . 4 (𝜑 → ((𝐹𝑃) (𝑃 (𝑅𝐹)) ∧ 𝑉 ((𝑅𝐹) 𝑈)))
1137, 35, 82, 8ps-2c 39531 . . . 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 1403 . . 3 (𝜑 → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) ∈ 𝐴)
1151, 114eqeltrid 2844 . 2 (𝜑𝑄𝐴)
11627, 35, 8hlatjcl 39369 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑈𝐴) → (𝑃 𝑈) ∈ (Base‘𝐾))
1173, 5, 15, 116syl3anc 1372 . . . . . . . . . . . 12 (𝜑 → (𝑃 𝑈) ∈ (Base‘𝐾))
11827, 35, 8hlatjcl 39369 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝐹𝑃) ∈ 𝐴𝑉𝐴) → ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾))
1193, 19, 21, 118syl3anc 1372 . . . . . . . . . . . 12 (𝜑 → ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾))
12027, 7, 82latmle1 18510 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑃 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾)) → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) (𝑃 𝑈))
12126, 117, 119, 120syl3anc 1372 . . . . . . . . . . 11 (𝜑 → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) (𝑃 𝑈))
1221, 121eqbrtrid 5177 . . . . . . . . . 10 (𝜑𝑄 (𝑃 𝑈))
12327, 8atbase 39291 . . . . . . . . . . . . 13 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
124115, 123syl 17 . . . . . . . . . . . 12 (𝜑𝑄 ∈ (Base‘𝐾))
12527, 7, 82latlem12 18512 . . . . . . . . . . . 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 1373 . . . . . . . . . . 11 (𝜑 → ((𝑄 (𝑃 𝑈) ∧ 𝑄 𝑊) ↔ 𝑄 ((𝑃 𝑈) 𝑊)))
127126biimpd 229 . . . . . . . . . 10 (𝜑 → ((𝑄 (𝑃 𝑈) ∧ 𝑄 𝑊) → 𝑄 ((𝑃 𝑈) 𝑊)))
128122, 127mpand 695 . . . . . . . . 9 (𝜑 → (𝑄 𝑊𝑄 ((𝑃 𝑈) 𝑊)))
129128imp 406 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑄 ((𝑃 𝑈) 𝑊))
130 eqid 2736 . . . . . . . . . . . . 13 (0.‘𝐾) = (0.‘𝐾)
1317, 82, 130, 8, 9lhpmat 40033 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = (0.‘𝐾))
1322, 4, 131syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑊) = (0.‘𝐾))
133132oveq1d 7447 . . . . . . . . . 10 (𝜑 → ((𝑃 𝑊) 𝑈) = ((0.‘𝐾) 𝑈))
13427, 7, 35, 82, 8atmod4i1 39869 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑈𝐴𝑃 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑈 𝑊) → ((𝑃 𝑊) 𝑈) = ((𝑃 𝑈) 𝑊))
1353, 15, 40, 34, 25, 134syl131anc 1384 . . . . . . . . . 10 (𝜑 → ((𝑃 𝑊) 𝑈) = ((𝑃 𝑈) 𝑊))
13627, 35, 130olj02 39228 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑈 ∈ (Base‘𝐾)) → ((0.‘𝐾) 𝑈) = 𝑈)
13797, 31, 136syl2anc 584 . . . . . . . . . 10 (𝜑 → ((0.‘𝐾) 𝑈) = 𝑈)
138133, 135, 1373eqtr3d 2784 . . . . . . . . 9 (𝜑 → ((𝑃 𝑈) 𝑊) = 𝑈)
139138adantr 480 . . . . . . . 8 ((𝜑𝑄 𝑊) → ((𝑃 𝑈) 𝑊) = 𝑈)
140129, 139breqtrd 5168 . . . . . . 7 ((𝜑𝑄 𝑊) → 𝑄 𝑈)
141 hlatl 39362 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
1423, 141syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ AtLat)
143142adantr 480 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝐾 ∈ AtLat)
144115adantr 480 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑄𝐴)
14515adantr 480 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑈𝐴)
1467, 8atcmp 39313 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑄𝐴𝑈𝐴) → (𝑄 𝑈𝑄 = 𝑈))
147143, 144, 145, 146syl3anc 1372 . . . . . . 7 ((𝜑𝑄 𝑊) → (𝑄 𝑈𝑄 = 𝑈))
148140, 147mpbid 232 . . . . . 6 ((𝜑𝑄 𝑊) → 𝑄 = 𝑈)
14927, 7, 82latmle2 18511 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑃 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾)) → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) ((𝐹𝑃) 𝑉))
15026, 117, 119, 149syl3anc 1372 . . . . . . . . . . 11 (𝜑 → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) ((𝐹𝑃) 𝑉))
1511, 150eqbrtrid 5177 . . . . . . . . . 10 (𝜑𝑄 ((𝐹𝑃) 𝑉))
15227, 7, 82latlem12 18512 . . . . . . . . . . . 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 1373 . . . . . . . . . . 11 (𝜑 → ((𝑄 ((𝐹𝑃) 𝑉) ∧ 𝑄 𝑊) ↔ 𝑄 (((𝐹𝑃) 𝑉) 𝑊)))
154153biimpd 229 . . . . . . . . . 10 (𝜑 → ((𝑄 ((𝐹𝑃) 𝑉) ∧ 𝑄 𝑊) → 𝑄 (((𝐹𝑃) 𝑉) 𝑊)))
155151, 154mpand 695 . . . . . . . . 9 (𝜑 → (𝑄 𝑊𝑄 (((𝐹𝑃) 𝑉) 𝑊)))
156155imp 406 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑄 (((𝐹𝑃) 𝑉) 𝑊))
1577, 82, 130, 8, 9lhpmat 40033 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊)) → ((𝐹𝑃) 𝑊) = (0.‘𝐾))
1582, 18, 157syl2anc 584 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑃) 𝑊) = (0.‘𝐾))
159158oveq1d 7447 . . . . . . . . . 10 (𝜑 → (((𝐹𝑃) 𝑊) 𝑉) = ((0.‘𝐾) 𝑉))
16027, 8atbase 39291 . . . . . . . . . . . 12 ((𝐹𝑃) ∈ 𝐴 → (𝐹𝑃) ∈ (Base‘𝐾))
16119, 160syl 17 . . . . . . . . . . 11 (𝜑 → (𝐹𝑃) ∈ (Base‘𝐾))
16227, 7, 35, 82, 8atmod4i1 39869 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑉𝐴 ∧ (𝐹𝑃) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑉 𝑊) → (((𝐹𝑃) 𝑊) 𝑉) = (((𝐹𝑃) 𝑉) 𝑊))
1633, 21, 161, 34, 47, 162syl131anc 1384 . . . . . . . . . 10 (𝜑 → (((𝐹𝑃) 𝑊) 𝑉) = (((𝐹𝑃) 𝑉) 𝑊))
16427, 35, 130olj02 39228 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑉 ∈ (Base‘𝐾)) → ((0.‘𝐾) 𝑉) = 𝑉)
16597, 71, 164syl2anc 584 . . . . . . . . . 10 (𝜑 → ((0.‘𝐾) 𝑉) = 𝑉)
166159, 163, 1653eqtr3d 2784 . . . . . . . . 9 (𝜑 → (((𝐹𝑃) 𝑉) 𝑊) = 𝑉)
167166adantr 480 . . . . . . . 8 ((𝜑𝑄 𝑊) → (((𝐹𝑃) 𝑉) 𝑊) = 𝑉)
168156, 167breqtrd 5168 . . . . . . 7 ((𝜑𝑄 𝑊) → 𝑄 𝑉)
16921adantr 480 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑉𝐴)
1707, 8atcmp 39313 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑄𝐴𝑉𝐴) → (𝑄 𝑉𝑄 = 𝑉))
171143, 144, 169, 170syl3anc 1372 . . . . . . 7 ((𝜑𝑄 𝑊) → (𝑄 𝑉𝑄 = 𝑉))
172168, 171mpbid 232 . . . . . 6 ((𝜑𝑄 𝑊) → 𝑄 = 𝑉)
173148, 172eqtr3d 2778 . . . . 5 ((𝜑𝑄 𝑊) → 𝑈 = 𝑉)
174173ex 412 . . . 4 (𝜑 → (𝑄 𝑊𝑈 = 𝑉))
175174necon3ad 2952 . . 3 (𝜑 → (𝑈𝑉 → ¬ 𝑄 𝑊))
17664, 175mpd 15 . 2 (𝜑 → ¬ 𝑄 𝑊)
177115, 176jca 511 1 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wne 2939   class class class wbr 5142  cfv 6560  (class class class)co 7432  Basecbs 17248  lecple 17305  joincjn 18358  meetcmee 18359  0.cp0 18469  1.cp1 18470  Latclat 18477  OLcol 39176  Atomscatm 39265  AtLatcal 39266  HLchlt 39352  LHypclh 39987  LTrncltrn 40104  trLctrl 40161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869  df-proset 18341  df-poset 18360  df-plt 18376  df-lub 18392  df-glb 18393  df-join 18394  df-meet 18395  df-p0 18471  df-p1 18472  df-lat 18478  df-clat 18545  df-oposet 39178  df-ol 39180  df-oml 39181  df-covers 39268  df-ats 39269  df-atl 39300  df-cvlat 39324  df-hlat 39353  df-llines 39501  df-psubsp 39506  df-pmap 39507  df-padd 39799  df-lhyp 39991  df-laut 39992  df-ldil 40107  df-ltrn 40108  df-trl 40162
This theorem is referenced by:  dia2dimlem3  41069  dia2dimlem6  41072
  Copyright terms: Public domain W3C validator