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Theorem dia2dimlem1 35165
Description: Lemma for dia2dim 35178. 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 474 . . . 4 (𝜑𝐾 ∈ HL)
4 dia2dimlem1.p . . . . 5 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
54simpld 474 . . . 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 34268 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)
132, 4, 6, 12syl3anc 1318 . . . 4 (𝜑 → (𝑅𝐹) ∈ 𝐴)
14 dia2dimlem1.u . . . . 5 (𝜑 → (𝑈𝐴𝑈 𝑊))
1514simpld 474 . . . 4 (𝜑𝑈𝐴)
166simpld 474 . . . . . 6 (𝜑𝐹𝑇)
177, 8, 9, 10ltrnel 34237 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
182, 16, 4, 17syl3anc 1318 . . . . 5 (𝜑 → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
1918simpld 474 . . . 4 (𝜑 → (𝐹𝑃) ∈ 𝐴)
20 dia2dimlem1.v . . . . 5 (𝜑 → (𝑉𝐴𝑉 𝑊))
2120simpld 474 . . . 4 (𝜑𝑉𝐴)
224simprd 478 . . . . . 6 (𝜑 → ¬ 𝑃 𝑊)
237, 9, 10, 11trlle 34283 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) 𝑊)
242, 16, 23syl2anc 691 . . . . . . . 8 (𝜑 → (𝑅𝐹) 𝑊)
2514simprd 478 . . . . . . . 8 (𝜑𝑈 𝑊)
26 hllat 33462 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ Lat)
273, 26syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ Lat)
28 eqid 2610 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
2928, 8atbase 33388 . . . . . . . . . 10 ((𝑅𝐹) ∈ 𝐴 → (𝑅𝐹) ∈ (Base‘𝐾))
3013, 29syl 17 . . . . . . . . 9 (𝜑 → (𝑅𝐹) ∈ (Base‘𝐾))
3128, 8atbase 33388 . . . . . . . . . 10 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
3215, 31syl 17 . . . . . . . . 9 (𝜑𝑈 ∈ (Base‘𝐾))
332simprd 478 . . . . . . . . . 10 (𝜑𝑊𝐻)
3428, 9lhpbase 34096 . . . . . . . . . 10 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3533, 34syl 17 . . . . . . . . 9 (𝜑𝑊 ∈ (Base‘𝐾))
36 dia2dimlem1.j . . . . . . . . . 10 = (join‘𝐾)
3728, 7, 36latjle12 16834 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((𝑅𝐹) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((𝑅𝐹) 𝑊𝑈 𝑊) ↔ ((𝑅𝐹) 𝑈) 𝑊))
3827, 30, 32, 35, 37syl13anc 1320 . . . . . . . 8 (𝜑 → (((𝑅𝐹) 𝑊𝑈 𝑊) ↔ ((𝑅𝐹) 𝑈) 𝑊))
3924, 25, 38mpbi2and 958 . . . . . . 7 (𝜑 → ((𝑅𝐹) 𝑈) 𝑊)
4028, 8atbase 33388 . . . . . . . . 9 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
415, 40syl 17 . . . . . . . 8 (𝜑𝑃 ∈ (Base‘𝐾))
4228, 36, 8hlatjcl 33465 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑅𝐹) ∈ 𝐴𝑈𝐴) → ((𝑅𝐹) 𝑈) ∈ (Base‘𝐾))
433, 13, 15, 42syl3anc 1318 . . . . . . . 8 (𝜑 → ((𝑅𝐹) 𝑈) ∈ (Base‘𝐾))
4428, 7lattr 16828 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ ((𝑅𝐹) 𝑈) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑃 ((𝑅𝐹) 𝑈) ∧ ((𝑅𝐹) 𝑈) 𝑊) → 𝑃 𝑊))
4527, 41, 43, 35, 44syl13anc 1320 . . . . . . 7 (𝜑 → ((𝑃 ((𝑅𝐹) 𝑈) ∧ ((𝑅𝐹) 𝑈) 𝑊) → 𝑃 𝑊))
4639, 45mpan2d 706 . . . . . 6 (𝜑 → (𝑃 ((𝑅𝐹) 𝑈) → 𝑃 𝑊))
4722, 46mtod 188 . . . . 5 (𝜑 → ¬ 𝑃 ((𝑅𝐹) 𝑈))
4820simprd 478 . . . . . . 7 (𝜑𝑉 𝑊)
4918simprd 478 . . . . . . 7 (𝜑 → ¬ (𝐹𝑃) 𝑊)
50 nbrne2 4598 . . . . . . 7 ((𝑉 𝑊 ∧ ¬ (𝐹𝑃) 𝑊) → 𝑉 ≠ (𝐹𝑃))
5148, 49, 50syl2anc 691 . . . . . 6 (𝜑𝑉 ≠ (𝐹𝑃))
5251necomd 2837 . . . . 5 (𝜑 → (𝐹𝑃) ≠ 𝑉)
5347, 52jca 553 . . . 4 (𝜑 → (¬ 𝑃 ((𝑅𝐹) 𝑈) ∧ (𝐹𝑃) ≠ 𝑉))
5427adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝐾 ∈ Lat)
5541adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑃 ∈ (Base‘𝐾))
5628, 36, 8hlatjcl 33465 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑉𝐴𝑈𝐴) → (𝑉 𝑈) ∈ (Base‘𝐾))
573, 21, 15, 56syl3anc 1318 . . . . . . . . 9 (𝜑 → (𝑉 𝑈) ∈ (Base‘𝐾))
5857adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → (𝑉 𝑈) ∈ (Base‘𝐾))
5935adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑊 ∈ (Base‘𝐾))
607, 36, 8hlatlej2 33474 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝐹𝑃) ∈ 𝐴𝑉𝐴) → 𝑉 ((𝐹𝑃) 𝑉))
613, 19, 21, 60syl3anc 1318 . . . . . . . . . . 11 (𝜑𝑉 ((𝐹𝑃) 𝑉))
6261adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑉 ((𝐹𝑃) 𝑉))
63 simpr 476 . . . . . . . . . 10 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → (𝑃 𝑈) = ((𝐹𝑃) 𝑉))
6462, 63breqtrrd 4606 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑉 (𝑃 𝑈))
65 dia2dimlem1.uv . . . . . . . . . . . 12 (𝜑𝑈𝑉)
6665necomd 2837 . . . . . . . . . . 11 (𝜑𝑉𝑈)
677, 36, 8hlatexch2 33494 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑉𝐴𝑃𝐴𝑈𝐴) ∧ 𝑉𝑈) → (𝑉 (𝑃 𝑈) → 𝑃 (𝑉 𝑈)))
683, 21, 5, 15, 66, 67syl131anc 1331 . . . . . . . . . 10 (𝜑 → (𝑉 (𝑃 𝑈) → 𝑃 (𝑉 𝑈)))
6968adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → (𝑉 (𝑃 𝑈) → 𝑃 (𝑉 𝑈)))
7064, 69mpd 15 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑃 (𝑉 𝑈))
7128, 8atbase 33388 . . . . . . . . . . . 12 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
7221, 71syl 17 . . . . . . . . . . 11 (𝜑𝑉 ∈ (Base‘𝐾))
7328, 7, 36latjle12 16834 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑉 𝑊𝑈 𝑊) ↔ (𝑉 𝑈) 𝑊))
7427, 72, 32, 35, 73syl13anc 1320 . . . . . . . . . 10 (𝜑 → ((𝑉 𝑊𝑈 𝑊) ↔ (𝑉 𝑈) 𝑊))
7548, 25, 74mpbi2and 958 . . . . . . . . 9 (𝜑 → (𝑉 𝑈) 𝑊)
7675adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → (𝑉 𝑈) 𝑊)
7728, 7, 54, 55, 58, 59, 70, 76lattrd 16830 . . . . . . 7 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑃 𝑊)
7877ex 449 . . . . . 6 (𝜑 → ((𝑃 𝑈) = ((𝐹𝑃) 𝑉) → 𝑃 𝑊))
7978necon3bd 2796 . . . . 5 (𝜑 → (¬ 𝑃 𝑊 → (𝑃 𝑈) ≠ ((𝐹𝑃) 𝑉)))
8022, 79mpd 15 . . . 4 (𝜑 → (𝑃 𝑈) ≠ ((𝐹𝑃) 𝑉))
817, 36, 8hlatlej2 33474 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → (𝐹𝑃) (𝑃 (𝐹𝑃)))
823, 5, 19, 81syl3anc 1318 . . . . . 6 (𝜑 → (𝐹𝑃) (𝑃 (𝐹𝑃)))
83 dia2dimlem1.m . . . . . . . . . 10 = (meet‘𝐾)
847, 36, 83, 8, 9, 10, 11trlval2 34262 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
852, 16, 4, 84syl3anc 1318 . . . . . . . 8 (𝜑 → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
8685oveq2d 6543 . . . . . . 7 (𝜑 → (𝑃 (𝑅𝐹)) = (𝑃 ((𝑃 (𝐹𝑃)) 𝑊)))
8728, 36, 8hlatjcl 33465 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
883, 5, 19, 87syl3anc 1318 . . . . . . . . 9 (𝜑 → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
897, 36, 8hlatlej1 33473 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → 𝑃 (𝑃 (𝐹𝑃)))
903, 5, 19, 89syl3anc 1318 . . . . . . . . 9 (𝜑𝑃 (𝑃 (𝐹𝑃)))
9128, 7, 36, 83, 8atmod3i1 33962 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴 ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑃 (𝑃 (𝐹𝑃))) → (𝑃 ((𝑃 (𝐹𝑃)) 𝑊)) = ((𝑃 (𝐹𝑃)) (𝑃 𝑊)))
923, 5, 88, 35, 90, 91syl131anc 1331 . . . . . . . 8 (𝜑 → (𝑃 ((𝑃 (𝐹𝑃)) 𝑊)) = ((𝑃 (𝐹𝑃)) (𝑃 𝑊)))
93 eqid 2610 . . . . . . . . . . . 12 (1.‘𝐾) = (1.‘𝐾)
947, 36, 93, 8, 9lhpjat2 34119 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = (1.‘𝐾))
952, 4, 94syl2anc 691 . . . . . . . . . 10 (𝜑 → (𝑃 𝑊) = (1.‘𝐾))
9695oveq2d 6543 . . . . . . . . 9 (𝜑 → ((𝑃 (𝐹𝑃)) (𝑃 𝑊)) = ((𝑃 (𝐹𝑃)) (1.‘𝐾)))
97 hlol 33460 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ OL)
983, 97syl 17 . . . . . . . . . 10 (𝜑𝐾 ∈ OL)
9928, 83, 93olm11 33326 . . . . . . . . . 10 ((𝐾 ∈ OL ∧ (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾)) → ((𝑃 (𝐹𝑃)) (1.‘𝐾)) = (𝑃 (𝐹𝑃)))
10098, 88, 99syl2anc 691 . . . . . . . . 9 (𝜑 → ((𝑃 (𝐹𝑃)) (1.‘𝐾)) = (𝑃 (𝐹𝑃)))
10196, 100eqtrd 2644 . . . . . . . 8 (𝜑 → ((𝑃 (𝐹𝑃)) (𝑃 𝑊)) = (𝑃 (𝐹𝑃)))
10292, 101eqtrd 2644 . . . . . . 7 (𝜑 → (𝑃 ((𝑃 (𝐹𝑃)) 𝑊)) = (𝑃 (𝐹𝑃)))
10386, 102eqtrd 2644 . . . . . 6 (𝜑 → (𝑃 (𝑅𝐹)) = (𝑃 (𝐹𝑃)))
10482, 103breqtrrd 4606 . . . . 5 (𝜑 → (𝐹𝑃) (𝑃 (𝑅𝐹)))
105 dia2dimlem1.rf . . . . . . 7 (𝜑 → (𝑅𝐹) (𝑈 𝑉))
10636, 8hlatjcom 33466 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑉𝐴) → (𝑈 𝑉) = (𝑉 𝑈))
1073, 15, 21, 106syl3anc 1318 . . . . . . 7 (𝜑 → (𝑈 𝑉) = (𝑉 𝑈))
108105, 107breqtrd 4604 . . . . . 6 (𝜑 → (𝑅𝐹) (𝑉 𝑈))
109 dia2dimlem1.ru . . . . . . 7 (𝜑 → (𝑅𝐹) ≠ 𝑈)
1107, 36, 8hlatexch2 33494 . . . . . . 7 ((𝐾 ∈ HL ∧ ((𝑅𝐹) ∈ 𝐴𝑉𝐴𝑈𝐴) ∧ (𝑅𝐹) ≠ 𝑈) → ((𝑅𝐹) (𝑉 𝑈) → 𝑉 ((𝑅𝐹) 𝑈)))
1113, 13, 21, 15, 109, 110syl131anc 1331 . . . . . 6 (𝜑 → ((𝑅𝐹) (𝑉 𝑈) → 𝑉 ((𝑅𝐹) 𝑈)))
112108, 111mpd 15 . . . . 5 (𝜑𝑉 ((𝑅𝐹) 𝑈))
113104, 112jca 553 . . . 4 (𝜑 → ((𝐹𝑃) (𝑃 (𝑅𝐹)) ∧ 𝑉 ((𝑅𝐹) 𝑈)))
1147, 36, 83, 8ps-2c 33626 . . . 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)
1153, 5, 13, 15, 19, 21, 53, 80, 113, 114syl333anc 1350 . . 3 (𝜑 → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) ∈ 𝐴)
1161, 115syl5eqel 2692 . 2 (𝜑𝑄𝐴)
11728, 36, 8hlatjcl 33465 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑈𝐴) → (𝑃 𝑈) ∈ (Base‘𝐾))
1183, 5, 15, 117syl3anc 1318 . . . . . . . . . . . 12 (𝜑 → (𝑃 𝑈) ∈ (Base‘𝐾))
11928, 36, 8hlatjcl 33465 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝐹𝑃) ∈ 𝐴𝑉𝐴) → ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾))
1203, 19, 21, 119syl3anc 1318 . . . . . . . . . . . 12 (𝜑 → ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾))
12128, 7, 83latmle1 16848 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑃 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾)) → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) (𝑃 𝑈))
12227, 118, 120, 121syl3anc 1318 . . . . . . . . . . 11 (𝜑 → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) (𝑃 𝑈))
1231, 122syl5eqbr 4613 . . . . . . . . . 10 (𝜑𝑄 (𝑃 𝑈))
12428, 8atbase 33388 . . . . . . . . . . . . 13 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
125116, 124syl 17 . . . . . . . . . . . 12 (𝜑𝑄 ∈ (Base‘𝐾))
12628, 7, 83latlem12 16850 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑃 𝑈) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑄 (𝑃 𝑈) ∧ 𝑄 𝑊) ↔ 𝑄 ((𝑃 𝑈) 𝑊)))
12727, 125, 118, 35, 126syl13anc 1320 . . . . . . . . . . 11 (𝜑 → ((𝑄 (𝑃 𝑈) ∧ 𝑄 𝑊) ↔ 𝑄 ((𝑃 𝑈) 𝑊)))
128127biimpd 218 . . . . . . . . . 10 (𝜑 → ((𝑄 (𝑃 𝑈) ∧ 𝑄 𝑊) → 𝑄 ((𝑃 𝑈) 𝑊)))
129123, 128mpand 707 . . . . . . . . 9 (𝜑 → (𝑄 𝑊𝑄 ((𝑃 𝑈) 𝑊)))
130129imp 444 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑄 ((𝑃 𝑈) 𝑊))
131 eqid 2610 . . . . . . . . . . . . 13 (0.‘𝐾) = (0.‘𝐾)
1327, 83, 131, 8, 9lhpmat 34128 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = (0.‘𝐾))
1332, 4, 132syl2anc 691 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑊) = (0.‘𝐾))
134133oveq1d 6542 . . . . . . . . . 10 (𝜑 → ((𝑃 𝑊) 𝑈) = ((0.‘𝐾) 𝑈))
13528, 7, 36, 83, 8atmod4i1 33964 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑈𝐴𝑃 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑈 𝑊) → ((𝑃 𝑊) 𝑈) = ((𝑃 𝑈) 𝑊))
1363, 15, 41, 35, 25, 135syl131anc 1331 . . . . . . . . . 10 (𝜑 → ((𝑃 𝑊) 𝑈) = ((𝑃 𝑈) 𝑊))
13728, 36, 131olj02 33325 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑈 ∈ (Base‘𝐾)) → ((0.‘𝐾) 𝑈) = 𝑈)
13898, 32, 137syl2anc 691 . . . . . . . . . 10 (𝜑 → ((0.‘𝐾) 𝑈) = 𝑈)
139134, 136, 1383eqtr3d 2652 . . . . . . . . 9 (𝜑 → ((𝑃 𝑈) 𝑊) = 𝑈)
140139adantr 480 . . . . . . . 8 ((𝜑𝑄 𝑊) → ((𝑃 𝑈) 𝑊) = 𝑈)
141130, 140breqtrd 4604 . . . . . . 7 ((𝜑𝑄 𝑊) → 𝑄 𝑈)
142 hlatl 33459 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
1433, 142syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ AtLat)
144143adantr 480 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝐾 ∈ AtLat)
145116adantr 480 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑄𝐴)
14615adantr 480 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑈𝐴)
1477, 8atcmp 33410 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑄𝐴𝑈𝐴) → (𝑄 𝑈𝑄 = 𝑈))
148144, 145, 146, 147syl3anc 1318 . . . . . . 7 ((𝜑𝑄 𝑊) → (𝑄 𝑈𝑄 = 𝑈))
149141, 148mpbid 221 . . . . . 6 ((𝜑𝑄 𝑊) → 𝑄 = 𝑈)
15028, 7, 83latmle2 16849 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑃 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾)) → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) ((𝐹𝑃) 𝑉))
15127, 118, 120, 150syl3anc 1318 . . . . . . . . . . 11 (𝜑 → ((𝑃 𝑈) ((𝐹𝑃) 𝑉)) ((𝐹𝑃) 𝑉))
1521, 151syl5eqbr 4613 . . . . . . . . . 10 (𝜑𝑄 ((𝐹𝑃) 𝑉))
15328, 7, 83latlem12 16850 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑄 ((𝐹𝑃) 𝑉) ∧ 𝑄 𝑊) ↔ 𝑄 (((𝐹𝑃) 𝑉) 𝑊)))
15427, 125, 120, 35, 153syl13anc 1320 . . . . . . . . . . 11 (𝜑 → ((𝑄 ((𝐹𝑃) 𝑉) ∧ 𝑄 𝑊) ↔ 𝑄 (((𝐹𝑃) 𝑉) 𝑊)))
155154biimpd 218 . . . . . . . . . 10 (𝜑 → ((𝑄 ((𝐹𝑃) 𝑉) ∧ 𝑄 𝑊) → 𝑄 (((𝐹𝑃) 𝑉) 𝑊)))
156152, 155mpand 707 . . . . . . . . 9 (𝜑 → (𝑄 𝑊𝑄 (((𝐹𝑃) 𝑉) 𝑊)))
157156imp 444 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑄 (((𝐹𝑃) 𝑉) 𝑊))
1587, 83, 131, 8, 9lhpmat 34128 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊)) → ((𝐹𝑃) 𝑊) = (0.‘𝐾))
1592, 18, 158syl2anc 691 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑃) 𝑊) = (0.‘𝐾))
160159oveq1d 6542 . . . . . . . . . 10 (𝜑 → (((𝐹𝑃) 𝑊) 𝑉) = ((0.‘𝐾) 𝑉))
16128, 8atbase 33388 . . . . . . . . . . . 12 ((𝐹𝑃) ∈ 𝐴 → (𝐹𝑃) ∈ (Base‘𝐾))
16219, 161syl 17 . . . . . . . . . . 11 (𝜑 → (𝐹𝑃) ∈ (Base‘𝐾))
16328, 7, 36, 83, 8atmod4i1 33964 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑉𝐴 ∧ (𝐹𝑃) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑉 𝑊) → (((𝐹𝑃) 𝑊) 𝑉) = (((𝐹𝑃) 𝑉) 𝑊))
1643, 21, 162, 35, 48, 163syl131anc 1331 . . . . . . . . . 10 (𝜑 → (((𝐹𝑃) 𝑊) 𝑉) = (((𝐹𝑃) 𝑉) 𝑊))
16528, 36, 131olj02 33325 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑉 ∈ (Base‘𝐾)) → ((0.‘𝐾) 𝑉) = 𝑉)
16698, 72, 165syl2anc 691 . . . . . . . . . 10 (𝜑 → ((0.‘𝐾) 𝑉) = 𝑉)
167160, 164, 1663eqtr3d 2652 . . . . . . . . 9 (𝜑 → (((𝐹𝑃) 𝑉) 𝑊) = 𝑉)
168167adantr 480 . . . . . . . 8 ((𝜑𝑄 𝑊) → (((𝐹𝑃) 𝑉) 𝑊) = 𝑉)
169157, 168breqtrd 4604 . . . . . . 7 ((𝜑𝑄 𝑊) → 𝑄 𝑉)
17021adantr 480 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑉𝐴)
1717, 8atcmp 33410 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑄𝐴𝑉𝐴) → (𝑄 𝑉𝑄 = 𝑉))
172144, 145, 170, 171syl3anc 1318 . . . . . . 7 ((𝜑𝑄 𝑊) → (𝑄 𝑉𝑄 = 𝑉))
173169, 172mpbid 221 . . . . . 6 ((𝜑𝑄 𝑊) → 𝑄 = 𝑉)
174149, 173eqtr3d 2646 . . . . 5 ((𝜑𝑄 𝑊) → 𝑈 = 𝑉)
175174ex 449 . . . 4 (𝜑 → (𝑄 𝑊𝑈 = 𝑉))
176175necon3ad 2795 . . 3 (𝜑 → (𝑈𝑉 → ¬ 𝑄 𝑊))
17765, 176mpd 15 . 2 (𝜑 → ¬ 𝑄 𝑊)
178116, 177jca 553 1 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wne 2780   class class class wbr 4578  cfv 5790  (class class class)co 6527  Basecbs 15644  lecple 15724  joincjn 16716  meetcmee 16717  0.cp0 16809  1.cp1 16810  Latclat 16817  OLcol 33273  Atomscatm 33362  AtLatcal 33363  HLchlt 33449  LHypclh 34082  LTrncltrn 34199  trLctrl 34257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-iin 4453  df-br 4579  df-opab 4639  df-mpt 4640  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 7037  df-2nd 7038  df-map 7724  df-preset 16700  df-poset 16718  df-plt 16730  df-lub 16746  df-glb 16747  df-join 16748  df-meet 16749  df-p0 16811  df-p1 16812  df-lat 16818  df-clat 16880  df-oposet 33275  df-ol 33277  df-oml 33278  df-covers 33365  df-ats 33366  df-atl 33397  df-cvlat 33421  df-hlat 33450  df-llines 33596  df-psubsp 33601  df-pmap 33602  df-padd 33894  df-lhyp 34086  df-laut 34087  df-ldil 34202  df-ltrn 34203  df-trl 34258
This theorem is referenced by:  dia2dimlem3  35167  dia2dimlem6  35170
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