Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme0nex Structured version   Visualization version   GIF version

Theorem cdleme0nex 37420
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 37341- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 36473, our (𝑃 𝑟) = (𝑄 𝑟) is a shorter way to express 𝑟𝑃𝑟𝑄𝑟 (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 1198 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ 𝑅 𝑊)
2 simp12 1200 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑅 (𝑃 𝑄))
31, 2jca 514 . . 3 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (¬ 𝑅 𝑊𝑅 (𝑃 𝑄)))
4 simp3l 1197 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑅𝐴)
5 simp13 1201 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
6 ralnex 3236 . . . . . . 7 (∀𝑟𝐴 ¬ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
75, 6sylibr 236 . . . . . 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 5061 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑟 𝑊𝑅 𝑊))
98notbid 320 . . . . . . . . 9 (𝑟 = 𝑅 → (¬ 𝑟 𝑊 ↔ ¬ 𝑅 𝑊))
10 oveq2 7158 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑃 𝑟) = (𝑃 𝑅))
11 oveq2 7158 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
1210, 11eqeq12d 2837 . . . . . . . . 9 (𝑟 = 𝑅 → ((𝑃 𝑟) = (𝑄 𝑟) ↔ (𝑃 𝑅) = (𝑄 𝑅)))
139, 12anbi12d 632 . . . . . . . 8 (𝑟 = 𝑅 → ((¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅))))
1413notbid 320 . . . . . . 7 (𝑟 = 𝑅 → (¬ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ ¬ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅))))
1514rspcva 3620 . . . . . 6 ((𝑅𝐴 ∧ ∀𝑟𝐴 ¬ (¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) → ¬ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)))
164, 7, 15syl2anc 586 . . . . 5 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ (¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)))
17 simp11 1199 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝐾 ∈ HL)
18 hlcvl 36489 . . . . . . . 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 1202 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑃𝐴)
21 simp22 1203 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑄𝐴)
22 simp23 1204 . . . . . . 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 36473 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
2719, 20, 21, 4, 22, 26syl131anc 1379 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
2827anbi2d 630 . . . . 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 326 . . . 4 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ (¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
30 ianor 978 . . . . 5 (¬ ((𝑅𝑃𝑅𝑄) ∧ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))) ↔ (¬ (𝑅𝑃𝑅𝑄) ∨ ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
31 df-3an 1085 . . . . . . . 8 ((𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)) ↔ ((𝑅𝑃𝑅𝑄) ∧ 𝑅 (𝑃 𝑄)))
3231anbi2i 624 . . . . . . 7 ((¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) ↔ (¬ 𝑅 𝑊 ∧ ((𝑅𝑃𝑅𝑄) ∧ 𝑅 (𝑃 𝑄))))
33 an12 643 . . . . . . 7 ((¬ 𝑅 𝑊 ∧ ((𝑅𝑃𝑅𝑄) ∧ 𝑅 (𝑃 𝑄))) ↔ ((𝑅𝑃𝑅𝑄) ∧ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
3432, 33bitri 277 . . . . . 6 ((¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) ↔ ((𝑅𝑃𝑅𝑄) ∧ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
3534notbii 322 . . . . 5 (¬ (¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))) ↔ ¬ ((𝑅𝑃𝑅𝑄) ∧ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
36 pm4.62 852 . . . . 5 (((𝑅𝑃𝑅𝑄) → ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))) ↔ (¬ (𝑅𝑃𝑅𝑄) ∨ ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
3730, 35, 363bitr4ri 306 . . . 4 (((𝑅𝑃𝑅𝑄) → ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))) ↔ ¬ (¬ 𝑅 𝑊 ∧ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
3829, 37sylibr 236 . . 3 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ((𝑅𝑃𝑅𝑄) → ¬ (¬ 𝑅 𝑊𝑅 (𝑃 𝑄))))
393, 38mt2d 138 . 2 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ¬ (𝑅𝑃𝑅𝑄))
40 neanior 3109 . . 3 ((𝑅𝑃𝑅𝑄) ↔ ¬ (𝑅 = 𝑃𝑅 = 𝑄))
4140con2bii 360 . 2 ((𝑅 = 𝑃𝑅 = 𝑄) ↔ ¬ (𝑅𝑃𝑅𝑄))
4239, 41sylibr 236 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 208  wa 398  wo 843  w3a 1083   = wceq 1533  wcel 2110  wne 3016  wral 3138  wrex 3139   class class class wbr 5058  cfv 6349  (class class class)co 7150  lecple 16566  joincjn 17548  Atomscatm 36393  CvLatclc 36395  HLchlt 36480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-proset 17532  df-poset 17550  df-plt 17562  df-lub 17578  df-glb 17579  df-join 17580  df-meet 17581  df-p0 17643  df-lat 17650  df-covers 36396  df-ats 36397  df-atl 36428  df-cvlat 36452  df-hlat 36481
This theorem is referenced by:  cdleme18c  37423  cdleme18d  37425  cdlemg17b  37792  cdlemg17h  37798
  Copyright terms: Public domain W3C validator