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Theorem cdlemg4c 37190
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 754 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simplr2 1196 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
3 simplr3 1197 . . . . . . . 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 37188 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇) → ((𝐺𝑄) 𝑉) = (𝑄 (𝐺𝑄)))
121, 2, 3, 11syl3anc 1351 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → ((𝐺𝑄) 𝑉) = (𝑄 (𝐺𝑄)))
13 simpr 477 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝐺𝑄) (𝑃 𝑉))
14 simpll 754 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝐾 ∈ HL)
1514hllatd 35942 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝐾 ∈ Lat)
16 simpr1l 1210 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝑃𝐴)
17 eqid 2779 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝐾)
1817, 5atbase 35867 . . . . . . . . . . 11 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1916, 18syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝑃 ∈ (Base‘𝐾))
20 simpl 475 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
21 simpr3 1176 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝐺𝑇)
2217, 6, 7, 8trlcl 36742 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇) → (𝑅𝐺) ∈ (Base‘𝐾))
2320, 21, 22syl2anc 576 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → (𝑅𝐺) ∈ (Base‘𝐾))
2410, 23syl5eqel 2871 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝑉 ∈ (Base‘𝐾))
2517, 4, 9latlej2 17529 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → 𝑉 (𝑃 𝑉))
2615, 19, 24, 25syl3anc 1351 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝑉 (𝑃 𝑉))
2726adantr 473 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝑉 (𝑃 𝑉))
28 simpr2l 1212 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝑄𝐴)
2917, 5atbase 35867 . . . . . . . . . . . 12 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3028, 29syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝑄 ∈ (Base‘𝐾))
3117, 6, 7ltrncl 36703 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑄 ∈ (Base‘𝐾)) → (𝐺𝑄) ∈ (Base‘𝐾))
3220, 21, 30, 31syl3anc 1351 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → (𝐺𝑄) ∈ (Base‘𝐾))
3317, 9latjcl 17519 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → (𝑃 𝑉) ∈ (Base‘𝐾))
3415, 19, 24, 33syl3anc 1351 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → (𝑃 𝑉) ∈ (Base‘𝐾))
3517, 4, 9latjle12 17530 . . . . . . . . . 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 1352 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → (((𝐺𝑄) (𝑃 𝑉) ∧ 𝑉 (𝑃 𝑉)) ↔ ((𝐺𝑄) 𝑉) (𝑃 𝑉)))
3736adantr 473 . . . . . . . 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 699 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → ((𝐺𝑄) 𝑉) (𝑃 𝑉))
3912, 38eqbrtrrd 4953 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝑄 (𝐺𝑄)) (𝑃 𝑉))
4015adantr 473 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝐾 ∈ Lat)
4130adantr 473 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝑄 ∈ (Base‘𝐾))
4232adantr 473 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝐺𝑄) ∈ (Base‘𝐾))
4319adantr 473 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝑃 ∈ (Base‘𝐾))
441, 3, 22syl2anc 576 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝑅𝐺) ∈ (Base‘𝐾))
4510, 44syl5eqel 2871 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝑉 ∈ (Base‘𝐾))
4640, 43, 45, 33syl3anc 1351 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝑃 𝑉) ∈ (Base‘𝐾))
4717, 4, 9latjle12 17530 . . . . . . 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 1352 . . . . . 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 249 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝑄 (𝑃 𝑉) ∧ (𝐺𝑄) (𝑃 𝑉)))
5049simpld 487 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝑄 (𝑃 𝑉))
5150ex 405 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → ((𝐺𝑄) (𝑃 𝑉) → 𝑄 (𝑃 𝑉)))
5251con3d 150 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → (¬ 𝑄 (𝑃 𝑉) → ¬ (𝐺𝑄) (𝑃 𝑉)))
53523impia 1097 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 198  wa 387  w3a 1068   = wceq 1507  wcel 2050   class class class wbr 4929  cfv 6188  (class class class)co 6976  Basecbs 16339  lecple 16428  joincjn 17412  Latclat 17513  Atomscatm 35841  HLchlt 35928  LHypclh 36562  LTrncltrn 36679  trLctrl 36736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-iin 4795  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-map 8208  df-proset 17396  df-poset 17414  df-plt 17426  df-lub 17442  df-glb 17443  df-join 17444  df-meet 17445  df-p0 17507  df-p1 17508  df-lat 17514  df-clat 17576  df-oposet 35754  df-ol 35756  df-oml 35757  df-covers 35844  df-ats 35845  df-atl 35876  df-cvlat 35900  df-hlat 35929  df-psubsp 36081  df-pmap 36082  df-padd 36374  df-lhyp 36566  df-laut 36567  df-ldil 36682  df-ltrn 36683  df-trl 36737
This theorem is referenced by:  cdlemg4d  37191
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