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Theorem cdlemb3 36211
 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 35609? Then replace cdlemb2 35645 with it. This is a more general version of cdlemb2 35645 without 𝑃 ≠ 𝑄 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 1084 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl2 1085 . . . 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 36210 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ∃𝑟𝐴 (𝑃𝑟 ∧ ¬ 𝑟 𝑊))
81, 2, 7syl2anc 694 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄) → ∃𝑟𝐴 (𝑃𝑟 ∧ ¬ 𝑟 𝑊))
9 ancom 465 . . . . . 6 ((𝑃𝑟 ∧ ¬ 𝑟 𝑊) ↔ (¬ 𝑟 𝑊𝑃𝑟))
10 eqcom 2658 . . . . . . . . 9 (𝑃 = 𝑟𝑟 = 𝑃)
11 simp2 1082 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → 𝑃 = 𝑄)
1211oveq2d 6706 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → (𝑃 𝑃) = (𝑃 𝑄))
13 simp11l 1192 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → 𝐾 ∈ HL)
14 simp12l 1194 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → 𝑃𝐴)
154, 5hlatjidm 34973 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑃 𝑃) = 𝑃)
1613, 14, 15syl2anc 694 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → (𝑃 𝑃) = 𝑃)
1712, 16eqtr3d 2687 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → (𝑃 𝑄) = 𝑃)
1817breq2d 4697 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → (𝑟 (𝑃 𝑄) ↔ 𝑟 𝑃))
19 hlatl 34965 . . . . . . . . . . . 12 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2013, 19syl 17 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → 𝐾 ∈ AtLat)
21 simp3 1083 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → 𝑟𝐴)
223, 5atcmp 34916 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ 𝑟𝐴𝑃𝐴) → (𝑟 𝑃𝑟 = 𝑃))
2320, 21, 14, 22syl3anc 1366 . . . . . . . . . 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 2859 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄𝑟𝐴) → (𝑃𝑟 ↔ ¬ 𝑟 (𝑃 𝑄)))
2726anbi2d 740 . . . . . 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 1284 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄) ∧ 𝑟𝐴) → ((𝑃𝑟 ∧ ¬ 𝑟 𝑊) ↔ (¬ 𝑟 𝑊 ∧ ¬ 𝑟 (𝑃 𝑄))))
3029rexbidva 3078 . . 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 1084 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → (𝐾 ∈ HL ∧ 𝑊𝐻))
33 simpl2 1085 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
34 simpl3 1086 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
35 simpr 476 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → 𝑃𝑄)
363, 4, 5, 6cdlemb2 35645 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ∃𝑟𝐴𝑟 𝑊 ∧ ¬ 𝑟 (𝑃 𝑄)))
3732, 33, 34, 35, 36syl121anc 1371 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ∃𝑟𝐴𝑟 𝑊 ∧ ¬ 𝑟 (𝑃 𝑄)))
3831, 37pm2.61dane 2910 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 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∃wrex 2942   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  lecple 15995  joincjn 16991  Atomscatm 34868  AtLatcal 34869  HLchlt 34955  LHypclh 35588 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-lhyp 35592 This theorem is referenced by:  cdlemg6e  36227
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