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Theorem cdlemg17b 38673
Description: Part of proof of Lemma G in [Crawley] p. 117, 4th line. Whenever (in their terminology) p q/0 (i.e. the sublattice from 0 to p q) contains precisely three atoms and g is not the identity, g(p) = q. See also comments under cdleme0nex 38301. (Contributed by NM, 8-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l = (le‘𝐾)
cdlemg12.j = (join‘𝐾)
cdlemg12.m = (meet‘𝐾)
cdlemg12.a 𝐴 = (Atoms‘𝐾)
cdlemg12.h 𝐻 = (LHyp‘𝐾)
cdlemg12.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg12b.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemg17b ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐺𝑃) = 𝑄)
Distinct variable groups:   𝐴,𝑟   𝐺,𝑟   ,𝑟   ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑊,𝑟
Allowed substitution hints:   𝑅(𝑟)   𝑇(𝑟)   𝐻(𝑟)   𝐾(𝑟)   (𝑟)

Proof of Theorem cdlemg17b
StepHypRef Expression
1 simp31 1208 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐺𝑃) ≠ 𝑃)
21neneqd 2948 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ (𝐺𝑃) = 𝑃)
3 simp11l 1283 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐾 ∈ HL)
4 simp11 1202 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
5 simp12 1203 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
6 simp13 1204 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
7 simp2l 1198 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐺𝑇)
8 simp32 1209 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑅𝐺) (𝑃 𝑄))
9 cdlemg12.l . . . . . 6 = (le‘𝐾)
10 cdlemg12.j . . . . . 6 = (join‘𝐾)
11 cdlemg12.m . . . . . 6 = (meet‘𝐾)
12 cdlemg12.a . . . . . 6 𝐴 = (Atoms‘𝐾)
13 cdlemg12.h . . . . . 6 𝐻 = (LHyp‘𝐾)
14 cdlemg12.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
15 cdlemg12b.r . . . . . 6 𝑅 = ((trL‘𝐾)‘𝑊)
169, 10, 11, 12, 13, 14, 15cdlemg17a 38672 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇 ∧ (𝑅𝐺) (𝑃 𝑄))) → (𝐺𝑃) (𝑃 𝑄))
174, 5, 6, 7, 8, 16syl122anc 1378 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐺𝑃) (𝑃 𝑄))
18 simp33 1210 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
19 simp12l 1285 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑃𝐴)
20 simp13l 1287 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑄𝐴)
21 simp2r 1199 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑃𝑄)
229, 12, 13, 14ltrnel 38150 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐺𝑃) ∈ 𝐴 ∧ ¬ (𝐺𝑃) 𝑊))
234, 7, 5, 22syl3anc 1370 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝐺𝑃) ∈ 𝐴 ∧ ¬ (𝐺𝑃) 𝑊))
249, 10, 12cdleme0nex 38301 . . . 4 (((𝐾 ∈ HL ∧ (𝐺𝑃) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ ((𝐺𝑃) ∈ 𝐴 ∧ ¬ (𝐺𝑃) 𝑊)) → ((𝐺𝑃) = 𝑃 ∨ (𝐺𝑃) = 𝑄))
253, 17, 18, 19, 20, 21, 23, 24syl331anc 1394 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝐺𝑃) = 𝑃 ∨ (𝐺𝑃) = 𝑄))
2625ord 861 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (¬ (𝐺𝑃) = 𝑃 → (𝐺𝑃) = 𝑄))
272, 26mpd 15 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐺𝑃) = 𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 844  w3a 1086   = wceq 1539  wcel 2106  wne 2943  wrex 3065   class class class wbr 5076  cfv 6435  (class class class)co 7277  lecple 16967  joincjn 18027  meetcmee 18028  Atomscatm 37274  HLchlt 37361  LHypclh 37995  LTrncltrn 38112  trLctrl 38169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5211  ax-sep 5225  ax-nul 5232  ax-pow 5290  ax-pr 5354  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-iin 4929  df-br 5077  df-opab 5139  df-mpt 5160  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6393  df-fun 6437  df-fn 6438  df-f 6439  df-f1 6440  df-fo 6441  df-f1o 6442  df-fv 6443  df-riota 7234  df-ov 7280  df-oprab 7281  df-mpo 7282  df-1st 7831  df-2nd 7832  df-map 8615  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 37187  df-ol 37189  df-oml 37190  df-covers 37277  df-ats 37278  df-atl 37309  df-cvlat 37333  df-hlat 37362  df-psubsp 37514  df-pmap 37515  df-padd 37807  df-lhyp 37999  df-laut 38000  df-ldil 38115  df-ltrn 38116  df-trl 38170
This theorem is referenced by:  cdlemg17dN  38674  cdlemg17e  38676  cdlemg17ir  38681  cdlemg17bq  38684  cdlemg17  38688  cdlemg18d  38692
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