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Theorem cdlemb3 36415
Description: Given two atoms not under the fiducial co-atom 𝑊, there is a third. Lemma B in [Crawley] p. 112. TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle 35813? Then replace cdlemb2 35849 with it. This is a more general version of cdlemb2 35849 without 𝑃𝑄 condition. (Contributed by NM, 27-Apr-2013.)
Hypotheses
Ref Expression
cdlemg5.l = (le‘𝐾)
cdlemg5.j = (join‘𝐾)
cdlemg5.a 𝐴 = (Atoms‘𝐾)
cdlemg5.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
cdlemb3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ∃𝑟𝐴𝑟 𝑊 ∧ ¬ 𝑟 (𝑃 𝑄)))
Distinct variable groups:   𝐴,𝑟   𝐻,𝑟   𝐾,𝑟   ,𝑟   𝑃,𝑟   𝑊,𝑟   ,𝑟   𝑄,𝑟

Proof of Theorem cdlemb3
StepHypRef Expression
1 simpl1 1227 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl2 1229 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 cdlemg5.l . . . . 5 = (le‘𝐾)
4 cdlemg5.j . . . . 5 = (join‘𝐾)
5 cdlemg5.a . . . . 5 𝐴 = (Atoms‘𝐾)
6 cdlemg5.h . . . . 5 𝐻 = (LHyp‘𝐾)
73, 4, 5, 6cdlemg5 36414 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ∃𝑟𝐴 (𝑃𝑟 ∧ ¬ 𝑟 𝑊))
81, 2, 7syl2anc 573 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄) → ∃𝑟𝐴 (𝑃𝑟 ∧ ¬ 𝑟 𝑊))
9 ancom 452 . . . . . 6 ((𝑃𝑟 ∧ ¬ 𝑟 𝑊) ↔ (¬ 𝑟 𝑊𝑃𝑟))
10 eqcom 2778 . . . . . . . . 9 (𝑃 = 𝑟𝑟 = 𝑃)
11 simp2 1131 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → 𝑃 = 𝑄)
1211oveq2d 6809 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → (𝑃 𝑃) = (𝑃 𝑄))
13 simp11l 1368 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → 𝐾 ∈ HL)
14 simp12l 1370 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → 𝑃𝐴)
154, 5hlatjidm 35177 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑃 𝑃) = 𝑃)
1613, 14, 15syl2anc 573 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → (𝑃 𝑃) = 𝑃)
1712, 16eqtr3d 2807 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → (𝑃 𝑄) = 𝑃)
1817breq2d 4798 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → (𝑟 (𝑃 𝑄) ↔ 𝑟 𝑃))
19 hlatl 35169 . . . . . . . . . . . 12 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2013, 19syl 17 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → 𝐾 ∈ AtLat)
21 simp3 1132 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → 𝑟𝐴)
223, 5atcmp 35120 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ 𝑟𝐴𝑃𝐴) → (𝑟 𝑃𝑟 = 𝑃))
2320, 21, 14, 22syl3anc 1476 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → (𝑟 𝑃𝑟 = 𝑃))
2418, 23bitr2d 269 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → (𝑟 = 𝑃𝑟 (𝑃 𝑄)))
2510, 24syl5bb 272 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → (𝑃 = 𝑟𝑟 (𝑃 𝑄)))
2625necon3abid 2979 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → (𝑃𝑟 ↔ ¬ 𝑟 (𝑃 𝑄)))
2726anbi2d 614 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → ((¬ 𝑟 𝑊𝑃𝑟) ↔ (¬ 𝑟 𝑊 ∧ ¬ 𝑟 (𝑃 𝑄))))
289, 27syl5bb 272 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → ((𝑃𝑟 ∧ ¬ 𝑟 𝑊) ↔ (¬ 𝑟 𝑊 ∧ ¬ 𝑟 (𝑃 𝑄))))
29283expa 1111 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄) ∧ 𝑟𝐴) → ((𝑃𝑟 ∧ ¬ 𝑟 𝑊) ↔ (¬ 𝑟 𝑊 ∧ ¬ 𝑟 (𝑃 𝑄))))
3029rexbidva 3197 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄) → (∃𝑟𝐴 (𝑃𝑟 ∧ ¬ 𝑟 𝑊) ↔ ∃𝑟𝐴𝑟 𝑊 ∧ ¬ 𝑟 (𝑃 𝑄))))
318, 30mpbid 222 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄) → ∃𝑟𝐴𝑟 𝑊 ∧ ¬ 𝑟 (𝑃 𝑄)))
32 simpl1 1227 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → (𝐾 ∈ HL ∧ 𝑊𝐻))
33 simpl2 1229 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
34 simpl3 1231 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
35 simpr 471 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → 𝑃𝑄)
363, 4, 5, 6cdlemb2 35849 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ∃𝑟𝐴𝑟 𝑊 ∧ ¬ 𝑟 (𝑃 𝑄)))
3732, 33, 34, 35, 36syl121anc 1481 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ∃𝑟𝐴𝑟 𝑊 ∧ ¬ 𝑟 (𝑃 𝑄)))
3831, 37pm2.61dane 3030 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ∃𝑟𝐴𝑟 𝑊 ∧ ¬ 𝑟 (𝑃 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wrex 3062   class class class wbr 4786  cfv 6031  (class class class)co 6793  lecple 16156  joincjn 17152  Atomscatm 35072  AtLatcal 35073  HLchlt 35159  LHypclh 35792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-p1 17248  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-lhyp 35796
This theorem is referenced by:  cdlemg6e  36431
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