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Theorem cdlemg4c 38622
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 764 . . . . . . . 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 38620 . . . . . . . 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 485 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝐺𝑄) (𝑃 𝑉))
14 simpll 764 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝐾 ∈ HL)
1514hllatd 37374 . . . . . . . . . 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 2740 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝐾)
1817, 5atbase 37299 . . . . . . . . . . 11 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1916, 18syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝑃 ∈ (Base‘𝐾))
20 simpl 483 . . . . . . . . . . . 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 38174 . . . . . . . . . . . 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 2845 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝑉 ∈ (Base‘𝐾))
2517, 4, 9latlej2 18165 . . . . . . . . . 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 481 . . . . . . . 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 37299 . . . . . . . . . . . 12 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3028, 29syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) → 𝑄 ∈ (Base‘𝐾))
3117, 6, 7ltrncl 38135 . . . . . . . . . . 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 18155 . . . . . . . . . . 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 18166 . . . . . . . . . 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 481 . . . . . . . 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 709 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → ((𝐺𝑄) 𝑉) (𝑃 𝑉))
3912, 38eqbrtrrd 5103 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝑄 (𝐺𝑄)) (𝑃 𝑉))
4015adantr 481 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝐾 ∈ Lat)
4130adantr 481 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝑄 ∈ (Base‘𝐾))
4232adantr 481 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝐺𝑄) ∈ (Base‘𝐾))
4319adantr 481 . . . . . . . 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 2845 . . . . . . . 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 18166 . . . . . . 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 256 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → (𝑄 (𝑃 𝑉) ∧ (𝐺𝑄) (𝑃 𝑉)))
5049simpld 495 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇)) ∧ (𝐺𝑄) (𝑃 𝑉)) → 𝑄 (𝑃 𝑉))
5150ex 413 . . 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 205  wa 396  w3a 1086   = wceq 1542  wcel 2110   class class class wbr 5079  cfv 6432  (class class class)co 7271  Basecbs 16910  lecple 16967  joincjn 18027  Latclat 18147  Atomscatm 37273  HLchlt 37360  LHypclh 37994  LTrncltrn 38111  trLctrl 38168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-iin 4933  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-1st 7824  df-2nd 7825  df-map 8600  df-proset 18011  df-poset 18029  df-plt 18046  df-lub 18062  df-glb 18063  df-join 18064  df-meet 18065  df-p0 18141  df-p1 18142  df-lat 18148  df-clat 18215  df-oposet 37186  df-ol 37188  df-oml 37189  df-covers 37276  df-ats 37277  df-atl 37308  df-cvlat 37332  df-hlat 37361  df-psubsp 37513  df-pmap 37514  df-padd 37806  df-lhyp 37998  df-laut 37999  df-ldil 38114  df-ltrn 38115  df-trl 38169
This theorem is referenced by:  cdlemg4d  38623
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