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Theorem dia2dimlem3 38201
Description: Lemma for dia2dim 38212. Define a translation 𝐷 whose trace is atom 𝑉. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
Hypotheses
Ref Expression
dia2dimlem3.l = (le‘𝐾)
dia2dimlem3.j = (join‘𝐾)
dia2dimlem3.m = (meet‘𝐾)
dia2dimlem3.a 𝐴 = (Atoms‘𝐾)
dia2dimlem3.h 𝐻 = (LHyp‘𝐾)
dia2dimlem3.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dia2dimlem3.r 𝑅 = ((trL‘𝐾)‘𝑊)
dia2dimlem3.q 𝑄 = ((𝑃 𝑈) ((𝐹𝑃) 𝑉))
dia2dimlem3.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
dia2dimlem3.u (𝜑 → (𝑈𝐴𝑈 𝑊))
dia2dimlem3.v (𝜑 → (𝑉𝐴𝑉 𝑊))
dia2dimlem3.p (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
dia2dimlem3.f (𝜑 → (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃))
dia2dimlem3.rf (𝜑 → (𝑅𝐹) (𝑈 𝑉))
dia2dimlem3.uv (𝜑𝑈𝑉)
dia2dimlem3.ru (𝜑 → (𝑅𝐹) ≠ 𝑈)
dia2dimlem3.rv (𝜑 → (𝑅𝐹) ≠ 𝑉)
dia2dimlem3.d (𝜑𝐷𝑇)
dia2dimlem3.dv (𝜑 → (𝐷𝑄) = (𝐹𝑃))
Assertion
Ref Expression
dia2dimlem3 (𝜑 → (𝑅𝐷) = 𝑉)

Proof of Theorem dia2dimlem3
StepHypRef Expression
1 dia2dimlem3.k . . . . . . 7 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
21simpld 497 . . . . . 6 (𝜑𝐾 ∈ HL)
3 dia2dimlem3.f . . . . . . . . 9 (𝜑 → (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃))
43simpld 497 . . . . . . . 8 (𝜑𝐹𝑇)
5 dia2dimlem3.p . . . . . . . 8 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
6 dia2dimlem3.l . . . . . . . . 9 = (le‘𝐾)
7 dia2dimlem3.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
8 dia2dimlem3.h . . . . . . . . 9 𝐻 = (LHyp‘𝐾)
9 dia2dimlem3.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
106, 7, 8, 9ltrnel 37274 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
111, 4, 5, 10syl3anc 1367 . . . . . . 7 (𝜑 → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
1211simpld 497 . . . . . 6 (𝜑 → (𝐹𝑃) ∈ 𝐴)
13 dia2dimlem3.v . . . . . . 7 (𝜑 → (𝑉𝐴𝑉 𝑊))
1413simpld 497 . . . . . 6 (𝜑𝑉𝐴)
15 dia2dimlem3.j . . . . . . 7 = (join‘𝐾)
166, 15, 7hlatlej2 36511 . . . . . 6 ((𝐾 ∈ HL ∧ (𝐹𝑃) ∈ 𝐴𝑉𝐴) → 𝑉 ((𝐹𝑃) 𝑉))
172, 12, 14, 16syl3anc 1367 . . . . 5 (𝜑𝑉 ((𝐹𝑃) 𝑉))
182hllatd 36499 . . . . . 6 (𝜑𝐾 ∈ Lat)
19 eqid 2821 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2019, 7atbase 36424 . . . . . . 7 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
2114, 20syl 17 . . . . . 6 (𝜑𝑉 ∈ (Base‘𝐾))
2219, 15, 7hlatjcl 36502 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝐹𝑃) ∈ 𝐴𝑉𝐴) → ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾))
232, 12, 14, 22syl3anc 1367 . . . . . 6 (𝜑 → ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾))
24 dia2dimlem3.r . . . . . . . . 9 𝑅 = ((trL‘𝐾)‘𝑊)
256, 7, 8, 9, 24trlat 37304 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
261, 5, 3, 25syl3anc 1367 . . . . . . 7 (𝜑 → (𝑅𝐹) ∈ 𝐴)
27 dia2dimlem3.u . . . . . . . 8 (𝜑 → (𝑈𝐴𝑈 𝑊))
2827simpld 497 . . . . . . 7 (𝜑𝑈𝐴)
2919, 15, 7hlatjcl 36502 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑅𝐹) ∈ 𝐴𝑈𝐴) → ((𝑅𝐹) 𝑈) ∈ (Base‘𝐾))
302, 26, 28, 29syl3anc 1367 . . . . . 6 (𝜑 → ((𝑅𝐹) 𝑈) ∈ (Base‘𝐾))
31 dia2dimlem3.m . . . . . . 7 = (meet‘𝐾)
3219, 6, 31latmlem2 17691 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑉 ∈ (Base‘𝐾) ∧ ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾) ∧ ((𝑅𝐹) 𝑈) ∈ (Base‘𝐾))) → (𝑉 ((𝐹𝑃) 𝑉) → (((𝑅𝐹) 𝑈) 𝑉) (((𝑅𝐹) 𝑈) ((𝐹𝑃) 𝑉))))
3318, 21, 23, 30, 32syl13anc 1368 . . . . 5 (𝜑 → (𝑉 ((𝐹𝑃) 𝑉) → (((𝑅𝐹) 𝑈) 𝑉) (((𝑅𝐹) 𝑈) ((𝐹𝑃) 𝑉))))
3417, 33mpd 15 . . . 4 (𝜑 → (((𝑅𝐹) 𝑈) 𝑉) (((𝑅𝐹) 𝑈) ((𝐹𝑃) 𝑉)))
35 dia2dimlem3.rf . . . . . . 7 (𝜑 → (𝑅𝐹) (𝑈 𝑉))
3615, 7hlatjcom 36503 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑉𝐴) → (𝑈 𝑉) = (𝑉 𝑈))
372, 28, 14, 36syl3anc 1367 . . . . . . 7 (𝜑 → (𝑈 𝑉) = (𝑉 𝑈))
3835, 37breqtrd 5091 . . . . . 6 (𝜑 → (𝑅𝐹) (𝑉 𝑈))
39 dia2dimlem3.ru . . . . . . 7 (𝜑 → (𝑅𝐹) ≠ 𝑈)
406, 15, 7hlatexch2 36531 . . . . . . 7 ((𝐾 ∈ HL ∧ ((𝑅𝐹) ∈ 𝐴𝑉𝐴𝑈𝐴) ∧ (𝑅𝐹) ≠ 𝑈) → ((𝑅𝐹) (𝑉 𝑈) → 𝑉 ((𝑅𝐹) 𝑈)))
412, 26, 14, 28, 39, 40syl131anc 1379 . . . . . 6 (𝜑 → ((𝑅𝐹) (𝑉 𝑈) → 𝑉 ((𝑅𝐹) 𝑈)))
4238, 41mpd 15 . . . . 5 (𝜑𝑉 ((𝑅𝐹) 𝑈))
4319, 6, 31latleeqm2 17689 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑉 ∈ (Base‘𝐾) ∧ ((𝑅𝐹) 𝑈) ∈ (Base‘𝐾)) → (𝑉 ((𝑅𝐹) 𝑈) ↔ (((𝑅𝐹) 𝑈) 𝑉) = 𝑉))
4418, 21, 30, 43syl3anc 1367 . . . . 5 (𝜑 → (𝑉 ((𝑅𝐹) 𝑈) ↔ (((𝑅𝐹) 𝑈) 𝑉) = 𝑉))
4542, 44mpbid 234 . . . 4 (𝜑 → (((𝑅𝐹) 𝑈) 𝑉) = 𝑉)
46 dia2dimlem3.d . . . . . 6 (𝜑𝐷𝑇)
47 dia2dimlem3.q . . . . . . 7 𝑄 = ((𝑃 𝑈) ((𝐹𝑃) 𝑉))
48 dia2dimlem3.uv . . . . . . 7 (𝜑𝑈𝑉)
496, 15, 31, 7, 8, 9, 24, 47, 1, 27, 13, 5, 3, 35, 48, 39dia2dimlem1 38199 . . . . . 6 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
506, 15, 31, 7, 8, 9, 24trlval2 37298 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐷𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑅𝐷) = ((𝑄 (𝐷𝑄)) 𝑊))
511, 46, 49, 50syl3anc 1367 . . . . 5 (𝜑 → (𝑅𝐷) = ((𝑄 (𝐷𝑄)) 𝑊))
5247a1i 11 . . . . . . . . 9 (𝜑𝑄 = ((𝑃 𝑈) ((𝐹𝑃) 𝑉)))
53 dia2dimlem3.dv . . . . . . . . 9 (𝜑 → (𝐷𝑄) = (𝐹𝑃))
5452, 53oveq12d 7173 . . . . . . . 8 (𝜑 → (𝑄 (𝐷𝑄)) = (((𝑃 𝑈) ((𝐹𝑃) 𝑉)) (𝐹𝑃)))
555simpld 497 . . . . . . . . . 10 (𝜑𝑃𝐴)
5619, 15, 7hlatjcl 36502 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑈𝐴) → (𝑃 𝑈) ∈ (Base‘𝐾))
572, 55, 28, 56syl3anc 1367 . . . . . . . . 9 (𝜑 → (𝑃 𝑈) ∈ (Base‘𝐾))
586, 15, 7hlatlej1 36510 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝐹𝑃) ∈ 𝐴𝑉𝐴) → (𝐹𝑃) ((𝐹𝑃) 𝑉))
592, 12, 14, 58syl3anc 1367 . . . . . . . . 9 (𝜑 → (𝐹𝑃) ((𝐹𝑃) 𝑉))
6019, 6, 15, 31, 7atmod4i1 37001 . . . . . . . . 9 ((𝐾 ∈ HL ∧ ((𝐹𝑃) ∈ 𝐴 ∧ (𝑃 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾)) ∧ (𝐹𝑃) ((𝐹𝑃) 𝑉)) → (((𝑃 𝑈) ((𝐹𝑃) 𝑉)) (𝐹𝑃)) = (((𝑃 𝑈) (𝐹𝑃)) ((𝐹𝑃) 𝑉)))
612, 12, 57, 23, 59, 60syl131anc 1379 . . . . . . . 8 (𝜑 → (((𝑃 𝑈) ((𝐹𝑃) 𝑉)) (𝐹𝑃)) = (((𝑃 𝑈) (𝐹𝑃)) ((𝐹𝑃) 𝑉)))
6215, 7hlatj32 36507 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑈𝐴 ∧ (𝐹𝑃) ∈ 𝐴)) → ((𝑃 𝑈) (𝐹𝑃)) = ((𝑃 (𝐹𝑃)) 𝑈))
632, 55, 28, 12, 62syl13anc 1368 . . . . . . . . 9 (𝜑 → ((𝑃 𝑈) (𝐹𝑃)) = ((𝑃 (𝐹𝑃)) 𝑈))
6463oveq1d 7170 . . . . . . . 8 (𝜑 → (((𝑃 𝑈) (𝐹𝑃)) ((𝐹𝑃) 𝑉)) = (((𝑃 (𝐹𝑃)) 𝑈) ((𝐹𝑃) 𝑉)))
6554, 61, 643eqtrd 2860 . . . . . . 7 (𝜑 → (𝑄 (𝐷𝑄)) = (((𝑃 (𝐹𝑃)) 𝑈) ((𝐹𝑃) 𝑉)))
6665oveq1d 7170 . . . . . 6 (𝜑 → ((𝑄 (𝐷𝑄)) 𝑊) = ((((𝑃 (𝐹𝑃)) 𝑈) ((𝐹𝑃) 𝑉)) 𝑊))
67 hlol 36496 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ OL)
682, 67syl 17 . . . . . . 7 (𝜑𝐾 ∈ OL)
6919, 15, 7hlatjcl 36502 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
702, 55, 12, 69syl3anc 1367 . . . . . . . 8 (𝜑 → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
7119, 7atbase 36424 . . . . . . . . 9 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
7228, 71syl 17 . . . . . . . 8 (𝜑𝑈 ∈ (Base‘𝐾))
7319, 15latjcl 17660 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) 𝑈) ∈ (Base‘𝐾))
7418, 70, 72, 73syl3anc 1367 . . . . . . 7 (𝜑 → ((𝑃 (𝐹𝑃)) 𝑈) ∈ (Base‘𝐾))
751simprd 498 . . . . . . . 8 (𝜑𝑊𝐻)
7619, 8lhpbase 37133 . . . . . . . 8 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
7775, 76syl 17 . . . . . . 7 (𝜑𝑊 ∈ (Base‘𝐾))
7819, 31latm32 36366 . . . . . . 7 ((𝐾 ∈ OL ∧ (((𝑃 (𝐹𝑃)) 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((((𝑃 (𝐹𝑃)) 𝑈) ((𝐹𝑃) 𝑉)) 𝑊) = ((((𝑃 (𝐹𝑃)) 𝑈) 𝑊) ((𝐹𝑃) 𝑉)))
7968, 74, 23, 77, 78syl13anc 1368 . . . . . 6 (𝜑 → ((((𝑃 (𝐹𝑃)) 𝑈) ((𝐹𝑃) 𝑉)) 𝑊) = ((((𝑃 (𝐹𝑃)) 𝑈) 𝑊) ((𝐹𝑃) 𝑉)))
806, 15, 31, 7, 8, 9, 24trlval2 37298 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
811, 4, 5, 80syl3anc 1367 . . . . . . . . 9 (𝜑 → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
8281oveq1d 7170 . . . . . . . 8 (𝜑 → ((𝑅𝐹) 𝑈) = (((𝑃 (𝐹𝑃)) 𝑊) 𝑈))
8327simprd 498 . . . . . . . . 9 (𝜑𝑈 𝑊)
8419, 6, 15, 31, 7atmod4i1 37001 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑈𝐴 ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑈 𝑊) → (((𝑃 (𝐹𝑃)) 𝑊) 𝑈) = (((𝑃 (𝐹𝑃)) 𝑈) 𝑊))
852, 28, 70, 77, 83, 84syl131anc 1379 . . . . . . . 8 (𝜑 → (((𝑃 (𝐹𝑃)) 𝑊) 𝑈) = (((𝑃 (𝐹𝑃)) 𝑈) 𝑊))
8682, 85eqtr2d 2857 . . . . . . 7 (𝜑 → (((𝑃 (𝐹𝑃)) 𝑈) 𝑊) = ((𝑅𝐹) 𝑈))
8786oveq1d 7170 . . . . . 6 (𝜑 → ((((𝑃 (𝐹𝑃)) 𝑈) 𝑊) ((𝐹𝑃) 𝑉)) = (((𝑅𝐹) 𝑈) ((𝐹𝑃) 𝑉)))
8866, 79, 873eqtrd 2860 . . . . 5 (𝜑 → ((𝑄 (𝐷𝑄)) 𝑊) = (((𝑅𝐹) 𝑈) ((𝐹𝑃) 𝑉)))
8951, 88eqtr2d 2857 . . . 4 (𝜑 → (((𝑅𝐹) 𝑈) ((𝐹𝑃) 𝑉)) = (𝑅𝐷))
9034, 45, 893brtr3d 5096 . . 3 (𝜑𝑉 (𝑅𝐷))
91 hlatl 36495 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
922, 91syl 17 . . . 4 (𝜑𝐾 ∈ AtLat)
93 hlop 36497 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ OP)
942, 93syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ OP)
95 eqid 2821 . . . . . . . . . 10 (0.‘𝐾) = (0.‘𝐾)
96 eqid 2821 . . . . . . . . . 10 (lt‘𝐾) = (lt‘𝐾)
9795, 96, 70ltat 36426 . . . . . . . . 9 ((𝐾 ∈ OP ∧ 𝑉𝐴) → (0.‘𝐾)(lt‘𝐾)𝑉)
9894, 14, 97syl2anc 586 . . . . . . . 8 (𝜑 → (0.‘𝐾)(lt‘𝐾)𝑉)
99 hlpos 36501 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ Poset)
1002, 99syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ Poset)
10119, 95op0cl 36319 . . . . . . . . . 10 (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
10294, 101syl 17 . . . . . . . . 9 (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾))
10319, 8, 9, 24trlcl 37299 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐷𝑇) → (𝑅𝐷) ∈ (Base‘𝐾))
1041, 46, 103syl2anc 586 . . . . . . . . 9 (𝜑 → (𝑅𝐷) ∈ (Base‘𝐾))
10519, 6, 96pltletr 17580 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ ((0.‘𝐾) ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ (𝑅𝐷) ∈ (Base‘𝐾))) → (((0.‘𝐾)(lt‘𝐾)𝑉𝑉 (𝑅𝐷)) → (0.‘𝐾)(lt‘𝐾)(𝑅𝐷)))
106100, 102, 21, 104, 105syl13anc 1368 . . . . . . . 8 (𝜑 → (((0.‘𝐾)(lt‘𝐾)𝑉𝑉 (𝑅𝐷)) → (0.‘𝐾)(lt‘𝐾)(𝑅𝐷)))
10798, 90, 106mp2and 697 . . . . . . 7 (𝜑 → (0.‘𝐾)(lt‘𝐾)(𝑅𝐷))
10819, 96, 95opltn0 36325 . . . . . . . 8 ((𝐾 ∈ OP ∧ (𝑅𝐷) ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)(𝑅𝐷) ↔ (𝑅𝐷) ≠ (0.‘𝐾)))
10994, 104, 108syl2anc 586 . . . . . . 7 (𝜑 → ((0.‘𝐾)(lt‘𝐾)(𝑅𝐷) ↔ (𝑅𝐷) ≠ (0.‘𝐾)))
110107, 109mpbid 234 . . . . . 6 (𝜑 → (𝑅𝐷) ≠ (0.‘𝐾))
111110neneqd 3021 . . . . 5 (𝜑 → ¬ (𝑅𝐷) = (0.‘𝐾))
11295, 7, 8, 9, 24trlator0 37306 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐷𝑇) → ((𝑅𝐷) ∈ 𝐴 ∨ (𝑅𝐷) = (0.‘𝐾)))
1131, 46, 112syl2anc 586 . . . . . . 7 (𝜑 → ((𝑅𝐷) ∈ 𝐴 ∨ (𝑅𝐷) = (0.‘𝐾)))
114113orcomd 867 . . . . . 6 (𝜑 → ((𝑅𝐷) = (0.‘𝐾) ∨ (𝑅𝐷) ∈ 𝐴))
115114ord 860 . . . . 5 (𝜑 → (¬ (𝑅𝐷) = (0.‘𝐾) → (𝑅𝐷) ∈ 𝐴))
116111, 115mpd 15 . . . 4 (𝜑 → (𝑅𝐷) ∈ 𝐴)
1176, 7atcmp 36446 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑉𝐴 ∧ (𝑅𝐷) ∈ 𝐴) → (𝑉 (𝑅𝐷) ↔ 𝑉 = (𝑅𝐷)))
11892, 14, 116, 117syl3anc 1367 . . 3 (𝜑 → (𝑉 (𝑅𝐷) ↔ 𝑉 = (𝑅𝐷)))
11990, 118mpbid 234 . 2 (𝜑𝑉 = (𝑅𝐷))
120119eqcomd 2827 1 (𝜑 → (𝑅𝐷) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843   = wceq 1533  wcel 2110  wne 3016   class class class wbr 5065  cfv 6354  (class class class)co 7155  Basecbs 16482  lecple 16571  Posetcpo 17549  ltcplt 17550  joincjn 17553  meetcmee 17554  0.cp0 17646  Latclat 17654  OPcops 36307  OLcol 36309  Atomscatm 36398  AtLatcal 36399  HLchlt 36485  LHypclh 37119  LTrncltrn 37236  trLctrl 37293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-iin 4921  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-map 8407  df-proset 17537  df-poset 17555  df-plt 17567  df-lub 17583  df-glb 17584  df-join 17585  df-meet 17586  df-p0 17648  df-p1 17649  df-lat 17655  df-clat 17717  df-oposet 36311  df-ol 36313  df-oml 36314  df-covers 36401  df-ats 36402  df-atl 36433  df-cvlat 36457  df-hlat 36486  df-llines 36633  df-psubsp 36638  df-pmap 36639  df-padd 36931  df-lhyp 37123  df-laut 37124  df-ldil 37239  df-ltrn 37240  df-trl 37294
This theorem is referenced by:  dia2dimlem5  38203
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