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Theorem cdlemg4c 40595
Description: TODO: FIX COMMENT. (Contributed by NM, 24-Apr-2013.)
Hypotheses
Ref Expression
cdlemg4.l = (le‘𝐾)
cdlemg4.a 𝐴 = (Atoms‘𝐾)
cdlemg4.h 𝐻 = (LHyp‘𝐾)
cdlemg4.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg4.r 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemg4.j = (join‘𝐾)
cdlemg4b.v 𝑉 = (𝑅𝐺)
Assertion
Ref Expression
cdlemg4c (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇) ∧ ¬ 𝑄 (𝑃 𝑉)) → ¬ (𝐺𝑄) (𝑃 𝑉))

Proof of Theorem cdlemg4c
StepHypRef Expression
1 simpll 767 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simplr2 1215 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
3 simplr3 1216 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝐺𝑇)
4 cdlemg4.l . . . . . . . . 9 = (le‘𝐾)
5 cdlemg4.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
6 cdlemg4.h . . . . . . . . 9 𝐻 = (LHyp‘𝐾)
7 cdlemg4.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
8 cdlemg4.r . . . . . . . . 9 𝑅 = ((trL‘𝐾)‘𝑊)
9 cdlemg4.j . . . . . . . . 9 = (join‘𝐾)
10 cdlemg4b.v . . . . . . . . 9 𝑉 = (𝑅𝐺)
114, 5, 6, 7, 8, 9, 10cdlemg4b2 40593 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇) → ((𝐺𝑄) 𝑉) = (𝑄 (𝐺𝑄)))
121, 2, 3, 11syl3anc 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → ((𝐺𝑄) 𝑉) = (𝑄 (𝐺𝑄)))
13 simpr 484 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝐺𝑄) (𝑃 𝑉))
14 simpll 767 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝐾 ∈ HL)
1514hllatd 39346 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝐾 ∈ Lat)
16 simpr1l 1229 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝑃𝐴)
17 eqid 2735 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝐾)
1817, 5atbase 39271 . . . . . . . . . . 11 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1916, 18syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝑃 ∈ (Base‘𝐾))
20 simpl 482 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
21 simpr3 1195 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝐺𝑇)
2217, 6, 7, 8trlcl 40147 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇) → (𝑅𝐺) ∈ (Base‘𝐾))
2320, 21, 22syl2anc 584 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → (𝑅𝐺) ∈ (Base‘𝐾))
2410, 23eqeltrid 2843 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝑉 ∈ (Base‘𝐾))
2517, 4, 9latlej2 18507 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → 𝑉 (𝑃 𝑉))
2615, 19, 24, 25syl3anc 1370 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝑉 (𝑃 𝑉))
2726adantr 480 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝑉 (𝑃 𝑉))
28 simpr2l 1231 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝑄𝐴)
2917, 5atbase 39271 . . . . . . . . . . . 12 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3028, 29syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝑄 ∈ (Base‘𝐾))
3117, 6, 7ltrncl 40108 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑄 ∈ (Base‘𝐾)) → (𝐺𝑄) ∈ (Base‘𝐾))
3220, 21, 30, 31syl3anc 1370 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → (𝐺𝑄) ∈ (Base‘𝐾))
3317, 9latjcl 18497 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → (𝑃 𝑉) ∈ (Base‘𝐾))
3415, 19, 24, 33syl3anc 1370 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → (𝑃 𝑉) ∈ (Base‘𝐾))
3517, 4, 9latjle12 18508 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ ((𝐺𝑄) ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ (𝑃 𝑉) ∈ (Base‘𝐾))) → (((𝐺𝑄) (𝑃 𝑉) ∧ 𝑉 (𝑃 𝑉)) ↔ ((𝐺𝑄) 𝑉) (𝑃 𝑉)))
3615, 32, 24, 34, 35syl13anc 1371 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → (((𝐺𝑄) (𝑃 𝑉) ∧ 𝑉 (𝑃 𝑉)) ↔ ((𝐺𝑄) 𝑉) (𝑃 𝑉)))
3736adantr 480 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (((𝐺𝑄) (𝑃 𝑉) ∧ 𝑉 (𝑃 𝑉)) ↔ ((𝐺𝑄) 𝑉) (𝑃 𝑉)))
3813, 27, 37mpbi2and 712 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → ((𝐺𝑄) 𝑉) (𝑃 𝑉))
3912, 38eqbrtrrd 5172 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝑄 (𝐺𝑄)) (𝑃 𝑉))
4015adantr 480 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝐾 ∈ Lat)
4130adantr 480 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝑄 ∈ (Base‘𝐾))
4232adantr 480 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝐺𝑄) ∈ (Base‘𝐾))
4319adantr 480 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝑃 ∈ (Base‘𝐾))
441, 3, 22syl2anc 584 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝑅𝐺) ∈ (Base‘𝐾))
4510, 44eqeltrid 2843 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝑉 ∈ (Base‘𝐾))
4640, 43, 45, 33syl3anc 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝑃 𝑉) ∈ (Base‘𝐾))
4717, 4, 9latjle12 18508 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝐺𝑄) ∈ (Base‘𝐾) ∧ (𝑃 𝑉) ∈ (Base‘𝐾))) → ((𝑄 (𝑃 𝑉) ∧ (𝐺𝑄) (𝑃 𝑉)) ↔ (𝑄 (𝐺𝑄)) (𝑃 𝑉)))
4840, 41, 42, 46, 47syl13anc 1371 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → ((𝑄 (𝑃 𝑉) ∧ (𝐺𝑄) (𝑃 𝑉)) ↔ (𝑄 (𝐺𝑄)) (𝑃 𝑉)))
4939, 48mpbird 257 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝑄 (𝑃 𝑉) ∧ (𝐺𝑄) (𝑃 𝑉)))
5049simpld 494 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝑄 (𝑃 𝑉))
5150ex 412 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → ((𝐺𝑄) (𝑃 𝑉) → 𝑄 (𝑃 𝑉)))
5251con3d 152 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → (¬ 𝑄 (𝑃 𝑉) → ¬ (𝐺𝑄) (𝑃 𝑉)))
53523impia 1116 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇) ∧ ¬ 𝑄 (𝑃 𝑉)) → ¬ (𝐺𝑄) (𝑃 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  Latclat 18489  Atomscatm 39245  HLchlt 39332  LHypclh 39967  LTrncltrn 40084  trLctrl 40141
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142
This theorem is referenced by:  cdlemg4d  40596
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