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Theorem cdleme7 29127
 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 29126 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 29120 through cdleme7 29127, 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 2253 . . 3
101, 2, 3, 4, 5, 6, 7, 8, 9cdleme7d 29124 . 2
11 simp11l 1071 . . . . . 6
12 simp2ll 1027 . . . . . 6
131, 2, 3, 4, 5, 6, 7, 8cdleme7ga 29126 . . . . . 6
141, 2, 4hlatlej2 28254 . . . . . 6
1511, 12, 13, 14syl3anc 1187 . . . . 5
1615biantrurd 496 . . . 4
17 hllat 28242 . . . . . . 7
1811, 17syl 17 . . . . . 6
19 eqid 2253 . . . . . . . 8
2019, 4atbase 28168 . . . . . . 7
2113, 20syl 17 . . . . . 6
2219, 2, 4hlatjcl 28245 . . . . . . 7
2311, 12, 13, 22syl3anc 1187 . . . . . 6
24 simp11r 1072 . . . . . . 7
2519, 5lhpbase 28876 . . . . . . 7
2624, 25syl 17 . . . . . 6
2719, 1, 3latlem12 14028 . . . . . 6
2818, 21, 23, 26, 27syl13anc 1189 . . . . 5
29 simp11 990 . . . . . . 7
30 simp12l 1073 . . . . . . 7
31 simp13l 1075 . . . . . . 7
32 simp2l 986 . . . . . . 7
33 simp2r 987 . . . . . . 7
34 simp32 997 . . . . . . 7
351, 2, 3, 4, 5, 6, 7, 8cdleme6 29119 . . . . . . 7
3629, 30, 31, 32, 33, 34, 35syl132anc 1205 . . . . . 6
3736breq2d 3932 . . . . 5
3828, 37bitrd 246 . . . 4
39 hlatl 28239 . . . . . 6
4011, 39syl 17 . . . . 5
41 simp12 991 . . . . . 6
42 simp31 996 . . . . . 6
431, 2, 3, 4, 5, 6lhpat2 28923 . . . . . 6
4429, 41, 31, 42, 43syl112anc 1191 . . . . 5
451, 4atcmp 28190 . . . . 5
4640, 13, 44, 45syl3anc 1187 . . . 4
4716, 38, 463bitrd 272 . . 3
4847necon3bbid 2446 . 2
4910, 48mpbird 225 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wb 178   wa 360   w3a 939   wceq 1619   wcel 1621   wne 2412   class class class wbr 3920  cfv 4592  (class class class)co 5710  cbs 13022  cple 13089  cjn 13922  cmee 13923  clat 13995  catm 28142  cal 28143  chlt 28229  clh 28862 This theorem is referenced by:  cdleme18a  29169  cdleme22f2  29225  cdlemefs32sn1aw  29292 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-lines 28379  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866
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