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Theorem dia2dimlem1 35371
 Description: Lemma for dia2dim 35384. 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 34474 . . . . 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 34443 . . . . . 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 34489 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) 𝑊)
242, 16, 23syl2anc 691 . . . . . . . 8 (𝜑 → (𝑅𝐹) 𝑊)
2514simprd 478 . . . . . . . 8 (𝜑𝑈 𝑊)
26 hllat 33668 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ Lat)
273, 26syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ Lat)
28 eqid 2610 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
2928, 8atbase 33594 . . . . . . . . . 10 ((𝑅𝐹) ∈ 𝐴 → (𝑅𝐹) ∈ (Base‘𝐾))
3013, 29syl 17 . . . . . . . . 9 (𝜑 → (𝑅𝐹) ∈ (Base‘𝐾))
3128, 8atbase 33594 . . . . . . . . . 10 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
3215, 31syl 17 . . . . . . . . 9 (𝜑𝑈 ∈ (Base‘𝐾))
332simprd 478 . . . . . . . . . 10 (𝜑𝑊𝐻)
3428, 9lhpbase 34302 . . . . . . . . . 10 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3533, 34syl 17 . . . . . . . . 9 (𝜑𝑊 ∈ (Base‘𝐾))
36 dia2dimlem1.j . . . . . . . . . 10 = (join‘𝐾)
3728, 7, 36latjle12 16885 . . . . . . . . 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 33594 . . . . . . . . 9 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
415, 40syl 17 . . . . . . . 8 (𝜑𝑃 ∈ (Base‘𝐾))
4228, 36, 8hlatjcl 33671 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑅𝐹) ∈ 𝐴𝑈𝐴) → ((𝑅𝐹) 𝑈) ∈ (Base‘𝐾))
433, 13, 15, 42syl3anc 1318 . . . . . . . 8 (𝜑 → ((𝑅𝐹) 𝑈) ∈ (Base‘𝐾))
4428, 7lattr 16879 . . . . . . . 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 4603 . . . . . . 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 33671 . . . . . . . . . 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 33680 . . . . . . . . . . . 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 4611 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝑈) = ((𝐹𝑃) 𝑉)) → 𝑉 (𝑃 𝑈))
65 dia2dimlem1.uv . . . . . . . . . . . 12 (𝜑𝑈𝑉)
6665necomd 2837 . . . . . . . . . . 11 (𝜑𝑉𝑈)
677, 36, 8hlatexch2 33700 . . . . . . . . . . 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 33594 . . . . . . . . . . . 12 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
7221, 71syl 17 . . . . . . . . . . 11 (𝜑𝑉 ∈ (Base‘𝐾))
7328, 7, 36latjle12 16885 . . . . . . . . . . 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 16881 . . . . . . 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 33680 . . . . . . 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 34468 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
852, 16, 4, 84syl3anc 1318 . . . . . . . 8 (𝜑 → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))
8685oveq2d 6565 . . . . . . 7 (𝜑 → (𝑃 (𝑅𝐹)) = (𝑃 ((𝑃 (𝐹𝑃)) 𝑊)))
8728, 36, 8hlatjcl 33671 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
883, 5, 19, 87syl3anc 1318 . . . . . . . . 9 (𝜑 → (𝑃 (𝐹𝑃)) ∈ (Base‘𝐾))
897, 36, 8hlatlej1 33679 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐹𝑃) ∈ 𝐴) → 𝑃 (𝑃 (𝐹𝑃)))
903, 5, 19, 89syl3anc 1318 . . . . . . . . 9 (𝜑𝑃 (𝑃 (𝐹𝑃)))
9128, 7, 36, 83, 8atmod3i1 34168 . . . . . . . . 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 34325 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = (1.‘𝐾))
952, 4, 94syl2anc 691 . . . . . . . . . 10 (𝜑 → (𝑃 𝑊) = (1.‘𝐾))
9695oveq2d 6565 . . . . . . . . 9 (𝜑 → ((𝑃 (𝐹𝑃)) (𝑃 𝑊)) = ((𝑃 (𝐹𝑃)) (1.‘𝐾)))
97 hlol 33666 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ OL)
983, 97syl 17 . . . . . . . . . 10 (𝜑𝐾 ∈ OL)
9928, 83, 93olm11 33532 . . . . . . . . . 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 4611 . . . . 5 (𝜑 → (𝐹𝑃) (𝑃 (𝑅𝐹)))
105 dia2dimlem1.rf . . . . . . 7 (𝜑 → (𝑅𝐹) (𝑈 𝑉))
10636, 8hlatjcom 33672 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑉𝐴) → (𝑈 𝑉) = (𝑉 𝑈))
1073, 15, 21, 106syl3anc 1318 . . . . . . 7 (𝜑 → (𝑈 𝑉) = (𝑉 𝑈))
108105, 107breqtrd 4609 . . . . . 6 (𝜑 → (𝑅𝐹) (𝑉 𝑈))
109 dia2dimlem1.ru . . . . . . 7 (𝜑 → (𝑅𝐹) ≠ 𝑈)
1107, 36, 8hlatexch2 33700 . . . . . . 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 33832 . . . 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 33671 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑈𝐴) → (𝑃 𝑈) ∈ (Base‘𝐾))
1183, 5, 15, 117syl3anc 1318 . . . . . . . . . . . 12 (𝜑 → (𝑃 𝑈) ∈ (Base‘𝐾))
11928, 36, 8hlatjcl 33671 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝐹𝑃) ∈ 𝐴𝑉𝐴) → ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾))
1203, 19, 21, 119syl3anc 1318 . . . . . . . . . . . 12 (𝜑 → ((𝐹𝑃) 𝑉) ∈ (Base‘𝐾))
12128, 7, 83latmle1 16899 . . . . . . . . . . . 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 4618 . . . . . . . . . 10 (𝜑𝑄 (𝑃 𝑈))
12428, 8atbase 33594 . . . . . . . . . . . . 13 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
125116, 124syl 17 . . . . . . . . . . . 12 (𝜑𝑄 ∈ (Base‘𝐾))
12628, 7, 83latlem12 16901 . . . . . . . . . . . 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 34334 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = (0.‘𝐾))
1332, 4, 132syl2anc 691 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑊) = (0.‘𝐾))
134133oveq1d 6564 . . . . . . . . . 10 (𝜑 → ((𝑃 𝑊) 𝑈) = ((0.‘𝐾) 𝑈))
13528, 7, 36, 83, 8atmod4i1 34170 . . . . . . . . . . 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 33531 . . . . . . . . . . 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 4609 . . . . . . 7 ((𝜑𝑄 𝑊) → 𝑄 𝑈)
142 hlatl 33665 . . . . . . . . . 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 33616 . . . . . . . 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 16900 . . . . . . . . . . . 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 4618 . . . . . . . . . 10 (𝜑𝑄 ((𝐹𝑃) 𝑉))
15328, 7, 83latlem12 16901 . . . . . . . . . . . 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 34334 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊)) → ((𝐹𝑃) 𝑊) = (0.‘𝐾))
1592, 18, 158syl2anc 691 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑃) 𝑊) = (0.‘𝐾))
160159oveq1d 6564 . . . . . . . . . 10 (𝜑 → (((𝐹𝑃) 𝑊) 𝑉) = ((0.‘𝐾) 𝑉))
16128, 8atbase 33594 . . . . . . . . . . . 12 ((𝐹𝑃) ∈ 𝐴 → (𝐹𝑃) ∈ (Base‘𝐾))
16219, 161syl 17 . . . . . . . . . . 11 (𝜑 → (𝐹𝑃) ∈ (Base‘𝐾))
16328, 7, 36, 83, 8atmod4i1 34170 . . . . . . . . . . 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 33531 . . . . . . . . . . 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 4609 . . . . . . 7 ((𝜑𝑄 𝑊) → 𝑄 𝑉)
17021adantr 480 . . . . . . . 8 ((𝜑𝑄 𝑊) → 𝑉𝐴)
1717, 8atcmp 33616 . . . . . . . 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 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  0.cp0 16860  1.cp1 16861  Latclat 16868  OLcol 33479  Atomscatm 33568  AtLatcal 33569  HLchlt 33655  LHypclh 34288  LTrncltrn 34405  trLctrl 34463 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 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 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 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292  df-laut 34293  df-ldil 34408  df-ltrn 34409  df-trl 34464 This theorem is referenced by:  dia2dimlem3  35373  dia2dimlem6  35376
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