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Theorem 1cvrat 35364
Description: Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 30-Apr-2012.)
Hypotheses
Ref Expression
1cvrat.b 𝐵 = (Base‘𝐾)
1cvrat.l = (le‘𝐾)
1cvrat.j = (join‘𝐾)
1cvrat.m = (meet‘𝐾)
1cvrat.u 1 = (1.‘𝐾)
1cvrat.c 𝐶 = ( ⋖ ‘𝐾)
1cvrat.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
1cvrat ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑃 𝑄) 𝑋) ∈ 𝐴)

Proof of Theorem 1cvrat
StepHypRef Expression
1 hllat 35251 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1163 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝐾 ∈ Lat)
3 simp21 1263 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑃𝐴)
4 1cvrat.b . . . . . . 7 𝐵 = (Base‘𝐾)
5 1cvrat.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
64, 5atbase 35177 . . . . . 6 (𝑃𝐴𝑃𝐵)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑃𝐵)
8 simp22 1264 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑄𝐴)
94, 5atbase 35177 . . . . . 6 (𝑄𝐴𝑄𝐵)
108, 9syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑄𝐵)
11 1cvrat.j . . . . . 6 = (join‘𝐾)
124, 11latjcom 17326 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) = (𝑄 𝑃))
132, 7, 10, 12syl3anc 1490 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑃 𝑄) = (𝑄 𝑃))
1413oveq1d 6856 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑃 𝑄) 𝑋) = ((𝑄 𝑃) 𝑋))
154, 11latjcl 17318 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑃𝐵) → (𝑄 𝑃) ∈ 𝐵)
162, 10, 7, 15syl3anc 1490 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑄 𝑃) ∈ 𝐵)
17 simp23 1265 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑋𝐵)
18 1cvrat.m . . . . 5 = (meet‘𝐾)
194, 18latmcom 17342 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 𝑃) ∈ 𝐵𝑋𝐵) → ((𝑄 𝑃) 𝑋) = (𝑋 (𝑄 𝑃)))
202, 16, 17, 19syl3anc 1490 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑄 𝑃) 𝑋) = (𝑋 (𝑄 𝑃)))
2114, 20eqtrd 2798 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑃 𝑄) 𝑋) = (𝑋 (𝑄 𝑃)))
22 simp1 1166 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝐾 ∈ HL)
2317, 8, 33jca 1158 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑋𝐵𝑄𝐴𝑃𝐴))
24 simp31 1266 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑃𝑄)
2524necomd 2991 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑄𝑃)
26 simp33 1268 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ¬ 𝑃 𝑋)
27 hlop 35250 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OP)
28273ad2ant1 1163 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝐾 ∈ OP)
29 1cvrat.l . . . . . 6 = (le‘𝐾)
30 1cvrat.u . . . . . 6 1 = (1.‘𝐾)
314, 29, 30ople1 35079 . . . . 5 ((𝐾 ∈ OP ∧ 𝑄𝐵) → 𝑄 1 )
3228, 10, 31syl2anc 579 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑄 1 )
33 simp32 1267 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑋𝐶 1 )
34 1cvrat.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
354, 29, 11, 30, 34, 51cvrjat 35363 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑋 𝑃) = 1 )
3622, 17, 3, 33, 26, 35syl32anc 1497 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑋 𝑃) = 1 )
3732, 36breqtrrd 4836 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑄 (𝑋 𝑃))
384, 29, 11, 18, 5cvrat3 35330 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑄𝐴𝑃𝐴)) → ((𝑄𝑃 ∧ ¬ 𝑃 𝑋𝑄 (𝑋 𝑃)) → (𝑋 (𝑄 𝑃)) ∈ 𝐴))
3938imp 395 . . 3 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑄𝐴𝑃𝐴)) ∧ (𝑄𝑃 ∧ ¬ 𝑃 𝑋𝑄 (𝑋 𝑃))) → (𝑋 (𝑄 𝑃)) ∈ 𝐴)
4022, 23, 25, 26, 37, 39syl23anc 1496 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑋 (𝑄 𝑃)) ∈ 𝐴)
4121, 40eqeltrd 2843 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑃 𝑄) 𝑋) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2936   class class class wbr 4808  cfv 6067  (class class class)co 6841  Basecbs 16131  lecple 16222  joincjn 17211  meetcmee 17212  1.cp1 17305  Latclat 17312  OPcops 35060  ccvr 35150  Atomscatm 35151  HLchlt 35238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-proset 17195  df-poset 17213  df-plt 17225  df-lub 17241  df-glb 17242  df-join 17243  df-meet 17244  df-p0 17306  df-p1 17307  df-lat 17313  df-clat 17375  df-oposet 35064  df-ol 35066  df-oml 35067  df-covers 35154  df-ats 35155  df-atl 35186  df-cvlat 35210  df-hlat 35239
This theorem is referenced by:  cdlemblem  35681  cdlemb  35682  lhpat  35931
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