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Theorem cdleme0nex 34595
 Description: Part of proof of Lemma E in [Crawley] p. 114, 4th line of 4th paragraph. Whenever (in their terminology) p ∨ q/0 (i.e. the sublattice from 0 to p ∨ q) contains precisely three atoms, any atom not under w must equal either p or q. (In case of 3 atoms, one of them must be u - see cdleme0a 34516- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 33648, our (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) is a shorter way to express 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄). Thus, the negated existential condition states there are no atoms different from p or q that are also not under w. (Contributed by NM, 12-Nov-2012.)
Hypotheses
Ref Expression
cdleme0nex.l = (le‘𝐾)
cdleme0nex.j = (join‘𝐾)
cdleme0nex.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
cdleme0nex (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 = 𝑃𝑅 = 𝑄))
Distinct variable groups:   𝐴,𝑟   ,𝑟   ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑅,𝑟   𝑊,𝑟
Allowed substitution hint:   𝐾(𝑟)

Proof of Theorem cdleme0nex
StepHypRef Expression
1 simp3r 1083 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ 𝑅 𝑊)
2 simp12 1085 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑅 (𝑃 𝑄))
31, 2jca 553 . . 3 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (¬ 𝑅 𝑊𝑅 (𝑃 𝑄)))
4 simp3l 1082 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑅𝐴)
5 simp13 1086 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
6 ralnex 2975 . . . . . . 7 (∀𝑟𝐴 ¬ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
75, 6sylibr 223 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ∀𝑟𝐴 ¬ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
8 breq1 4586 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑟 𝑊𝑅 𝑊))
98notbid 307 . . . . . . . . 9 (𝑟 = 𝑅 → (¬ 𝑟 𝑊 ↔ ¬ 𝑅 𝑊))
10 oveq2 6557 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑃 𝑟) = (𝑃 𝑅))
11 oveq2 6557 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
1210, 11eqeq12d 2625 . . . . . . . . 9 (𝑟 = 𝑅 → ((𝑃 𝑟) = (𝑄 𝑟) ↔ (𝑃 𝑅) = (𝑄 𝑅)))
139, 12anbi12d 743 . . . . . . . 8 (𝑟 = 𝑅 → ((¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅))))
1413notbid 307 . . . . . . 7 (𝑟 = 𝑅 → (¬ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ ¬ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅))))
1514rspcva 3280 . . . . . 6 ((𝑅𝐴 ∧ ∀𝑟𝐴 ¬ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → ¬ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)))
164, 7, 15syl2anc 691 . . . . 5 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)))
17 simp11 1084 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝐾 ∈ HL)
18 hlcvl 33664 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
1917, 18syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝐾 ∈ CvLat)
20 simp21 1087 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑃𝐴)
21 simp22 1088 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑄𝐴)
22 simp23 1089 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑃𝑄)
23 cdleme0nex.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
24 cdleme0nex.l . . . . . . . 8 = (le‘𝐾)
25 cdleme0nex.j . . . . . . . 8 = (join‘𝐾)
2623, 24, 25cvlsupr2 33648 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
2719, 20, 21, 4, 22, 26syl131anc 1331 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
2827anbi2d 736 . . . . 5 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ((¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ↔ (¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)))))
2916, 28mtbid 313 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ (¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
30 ianor 508 . . . . 5 (¬ ((𝑅𝑃𝑅𝑄) ∧ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))) ↔ (¬ (𝑅𝑃𝑅𝑄) ∨ ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
31 df-3an 1033 . . . . . . . 8 ((𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)) ↔ ((𝑅𝑃𝑅𝑄) ∧ 𝑅 (𝑃 𝑄)))
3231anbi2i 726 . . . . . . 7 ((¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) ↔ (¬ 𝑅 𝑊 ∧ ((𝑅𝑃𝑅𝑄) ∧ 𝑅 (𝑃 𝑄))))
33 an12 834 . . . . . . 7 ((¬ 𝑅 𝑊 ∧ ((𝑅𝑃𝑅𝑄) ∧ 𝑅 (𝑃 𝑄))) ↔ ((𝑅𝑃𝑅𝑄) ∧ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
3432, 33bitri 263 . . . . . 6 ((¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) ↔ ((𝑅𝑃𝑅𝑄) ∧ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
3534notbii 309 . . . . 5 (¬ (¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) ↔ ¬ ((𝑅𝑃𝑅𝑄) ∧ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
36 pm4.62 434 . . . . 5 (((𝑅𝑃𝑅𝑄) → ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))) ↔ (¬ (𝑅𝑃𝑅𝑄) ∨ ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
3730, 35, 363bitr4ri 292 . . . 4 (((𝑅𝑃𝑅𝑄) → ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))) ↔ ¬ (¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
3829, 37sylibr 223 . . 3 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ((𝑅𝑃𝑅𝑄) → ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
393, 38mt2d 130 . 2 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ (𝑅𝑃𝑅𝑄))
40 neanior 2874 . . 3 ((𝑅𝑃𝑅𝑄) ↔ ¬ (𝑅 = 𝑃𝑅 = 𝑄))
4140con2bii 346 . 2 ((𝑅 = 𝑃𝑅 = 𝑄) ↔ ¬ (𝑅𝑃𝑅𝑄))
4239, 41sylibr 223 1 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 = 𝑃𝑅 = 𝑄))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  lecple 15775  joincjn 16767  Atomscatm 33568  CvLatclc 33570  HLchlt 33655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656 This theorem is referenced by:  cdleme18c  34598  cdleme18d  34600  cdlemg17b  34968  cdlemg17h  34974
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