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Theorem cdleme7 30973
 Description: Part of proof of Lemma E in [Crawley] p. 113. and represent fs(r) and f(s) respectively. is the fiducial co-atom (hyperplane) that they call w. Here and in cdleme7ga 30972 above, we show that fs(r) W (top of p. 114), meaning it is an atom and not under w, which in our notation is expressed as . (Note that we do not have a symbol for their W.) Their proof provides no details of our cdleme7aa 30966 through cdleme7 30973, so there may be a simpler proof that we have overlooked. (Contributed by NM, 9-Jun-2012.)
Hypotheses
Ref Expression
cdleme4.l
cdleme4.j
cdleme4.m
cdleme4.a
cdleme4.h
cdleme4.u
cdleme4.f
cdleme4.g
Assertion
Ref Expression
cdleme7

Proof of Theorem cdleme7
StepHypRef Expression
1 cdleme4.l . . 3
2 cdleme4.j . . 3
3 cdleme4.m . . 3
4 cdleme4.a . . 3
5 cdleme4.h . . 3
6 cdleme4.u . . 3
7 cdleme4.f . . 3
8 cdleme4.g . . 3
9 eqid 2435 . . 3
101, 2, 3, 4, 5, 6, 7, 8, 9cdleme7d 30970 . 2
11 simp11l 1068 . . . . . 6
12 simp2ll 1024 . . . . . 6
131, 2, 3, 4, 5, 6, 7, 8cdleme7ga 30972 . . . . . 6
141, 2, 4hlatlej2 30100 . . . . . 6
1511, 12, 13, 14syl3anc 1184 . . . . 5
1615biantrurd 495 . . . 4
17 hllat 30088 . . . . . . 7
1811, 17syl 16 . . . . . 6
19 eqid 2435 . . . . . . . 8
2019, 4atbase 30014 . . . . . . 7
2113, 20syl 16 . . . . . 6
2219, 2, 4hlatjcl 30091 . . . . . . 7
2311, 12, 13, 22syl3anc 1184 . . . . . 6
24 simp11r 1069 . . . . . . 7
2519, 5lhpbase 30722 . . . . . . 7
2624, 25syl 16 . . . . . 6
2719, 1, 3latlem12 14499 . . . . . 6
2818, 21, 23, 26, 27syl13anc 1186 . . . . 5
29 simp11 987 . . . . . . 7
30 simp12l 1070 . . . . . . 7
31 simp13l 1072 . . . . . . 7
32 simp2l 983 . . . . . . 7
33 simp2r 984 . . . . . . 7
34 simp32 994 . . . . . . 7
351, 2, 3, 4, 5, 6, 7, 8cdleme6 30965 . . . . . . 7
3629, 30, 31, 32, 33, 34, 35syl132anc 1202 . . . . . 6
3736breq2d 4216 . . . . 5
3828, 37bitrd 245 . . . 4
39 hlatl 30085 . . . . . 6
4011, 39syl 16 . . . . 5
41 simp12 988 . . . . . 6
42 simp31 993 . . . . . 6
431, 2, 3, 4, 5, 6lhpat2 30769 . . . . . 6
4429, 41, 31, 42, 43syl112anc 1188 . . . . 5
451, 4atcmp 30036 . . . . 5
4640, 13, 44, 45syl3anc 1184 . . . 4
4716, 38, 463bitrd 271 . . 3
4847necon3bbid 2632 . 2
4910, 48mpbird 224 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725   wne 2598   class class class wbr 4204  cfv 5446  (class class class)co 6073  cbs 13461  cple 13528  cjn 14393  cmee 14394  clat 14466  catm 29988  cal 29989  chlt 30075  clh 30708 This theorem is referenced by:  cdleme18a  31015  cdleme22f2  31071  cdlemefs32sn1aw  31138 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29901  df-ol 29903  df-oml 29904  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-lines 30225  df-psubsp 30227  df-pmap 30228  df-padd 30520  df-lhyp 30712
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