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

Theorem cdleme0nex 33768
 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 33689- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 32821, our is a shorter way to express . 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
cdleme0nex.j
cdleme0nex.a
Assertion
Ref Expression
cdleme0nex
Distinct variable groups:   ,   ,   ,   ,   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem cdleme0nex
StepHypRef Expression
1 simp3r 1034 . . . 4
2 simp12 1036 . . . 4
31, 2jca 534 . . 3
4 simp3l 1033 . . . . . 6
5 simp13 1037 . . . . . . 7
6 ralnex 2811 . . . . . . 7
75, 6sylibr 215 . . . . . 6
8 breq1 4369 . . . . . . . . . 10
98notbid 295 . . . . . . . . 9
10 oveq2 6257 . . . . . . . . . 10
11 oveq2 6257 . . . . . . . . . 10
1210, 11eqeq12d 2443 . . . . . . . . 9
139, 12anbi12d 715 . . . . . . . 8
1413notbid 295 . . . . . . 7
1514rspcva 3123 . . . . . 6
164, 7, 15syl2anc 665 . . . . 5
17 simp11 1035 . . . . . . . 8
18 hlcvl 32837 . . . . . . . 8
1917, 18syl 17 . . . . . . 7
20 simp21 1038 . . . . . . 7
21 simp22 1039 . . . . . . 7
22 simp23 1040 . . . . . . 7
23 cdleme0nex.a . . . . . . . 8
24 cdleme0nex.l . . . . . . . 8
25 cdleme0nex.j . . . . . . . 8
2623, 24, 25cvlsupr2 32821 . . . . . . 7
2719, 20, 21, 4, 22, 26syl131anc 1277 . . . . . 6
2827anbi2d 708 . . . . 5
2916, 28mtbid 301 . . . 4
30 ianor 490 . . . . 5
31 df-3an 984 . . . . . . . 8
3231anbi2i 698 . . . . . . 7
33 an12 804 . . . . . . 7
3432, 33bitri 252 . . . . . 6
3534notbii 297 . . . . 5
36 pm4.62 420 . . . . 5
3730, 35, 363bitr4ri 281 . . . 4
3829, 37sylibr 215 . . 3
393, 38mt2d 120 . 2
40 neanior 2693 . . 3
4140con2bii 333 . 2
4239, 41sylibr 215 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 187   wo 369   wa 370   w3a 982   wceq 1437   wcel 1872   wne 2599  wral 2714  wrex 2715   class class class wbr 4366  cfv 5544  (class class class)co 6249  cple 15140  cjn 16132  catm 32741  clc 32743  chlt 32828 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-preset 16116  df-poset 16134  df-plt 16147  df-lub 16163  df-glb 16164  df-join 16165  df-meet 16166  df-p0 16228  df-lat 16235  df-covers 32744  df-ats 32745  df-atl 32776  df-cvlat 32800  df-hlat 32829 This theorem is referenced by:  cdleme18c  33771  cdleme18d  33773  cdlemg17b  34141  cdlemg17h  34147
 Copyright terms: Public domain W3C validator