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Theorem 1cvrat 33578
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 33466 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1074 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝐾 ∈ Lat)
3 simp21 1086 . . . . . 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 33392 . . . . . 6 (𝑃𝐴𝑃𝐵)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑃𝐵)
8 simp22 1087 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑄𝐴)
94, 5atbase 33392 . . . . . 6 (𝑄𝐴𝑄𝐵)
108, 9syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑄𝐵)
11 1cvrat.j . . . . . 6 = (join‘𝐾)
124, 11latjcom 16823 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) = (𝑄 𝑃))
132, 7, 10, 12syl3anc 1317 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑃 𝑄) = (𝑄 𝑃))
1413oveq1d 6537 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑃 𝑄) 𝑋) = ((𝑄 𝑃) 𝑋))
154, 11latjcl 16815 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑃𝐵) → (𝑄 𝑃) ∈ 𝐵)
162, 10, 7, 15syl3anc 1317 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑄 𝑃) ∈ 𝐵)
17 simp23 1088 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑋𝐵)
18 1cvrat.m . . . . 5 = (meet‘𝐾)
194, 18latmcom 16839 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 𝑃) ∈ 𝐵𝑋𝐵) → ((𝑄 𝑃) 𝑋) = (𝑋 (𝑄 𝑃)))
202, 16, 17, 19syl3anc 1317 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑄 𝑃) 𝑋) = (𝑋 (𝑄 𝑃)))
2114, 20eqtrd 2638 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑃 𝑄) 𝑋) = (𝑋 (𝑄 𝑃)))
22 simp1 1053 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝐾 ∈ HL)
2317, 8, 33jca 1234 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑋𝐵𝑄𝐴𝑃𝐴))
24 simp31 1089 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑃𝑄)
2524necomd 2831 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑄𝑃)
26 simp33 1091 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ¬ 𝑃 𝑋)
27 hlop 33465 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OP)
28273ad2ant1 1074 . . . . 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 33294 . . . . 5 ((𝐾 ∈ OP ∧ 𝑄𝐵) → 𝑄 1 )
3228, 10, 31syl2anc 690 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑄 1 )
33 simp32 1090 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑋𝐶 1 )
34 1cvrat.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
354, 29, 11, 30, 34, 51cvrjat 33577 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑋 𝑃) = 1 )
3622, 17, 3, 33, 26, 35syl32anc 1325 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑋 𝑃) = 1 )
3732, 36breqtrrd 4600 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑄 (𝑋 𝑃))
384, 29, 11, 18, 5cvrat3 33544 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑄𝐴𝑃𝐴)) → ((𝑄𝑃 ∧ ¬ 𝑃 𝑋𝑄 (𝑋 𝑃)) → (𝑋 (𝑄 𝑃)) ∈ 𝐴))
3938imp 443 . . 3 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑄𝐴𝑃𝐴)) ∧ (𝑄𝑃 ∧ ¬ 𝑃 𝑋𝑄 (𝑋 𝑃))) → (𝑋 (𝑄 𝑃)) ∈ 𝐴)
4022, 23, 25, 26, 37, 39syl23anc 1324 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑋 (𝑄 𝑃)) ∈ 𝐴)
4121, 40eqeltrd 2682 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 382  w3a 1030   = wceq 1474  wcel 1975  wne 2774   class class class wbr 4572  cfv 5785  (class class class)co 6522  Basecbs 15636  lecple 15716  joincjn 16708  meetcmee 16709  1.cp1 16802  Latclat 16809  OPcops 33275  ccvr 33365  Atomscatm 33366  HLchlt 33453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-preset 16692  df-poset 16710  df-plt 16722  df-lub 16738  df-glb 16739  df-join 16740  df-meet 16741  df-p0 16803  df-p1 16804  df-lat 16810  df-clat 16872  df-oposet 33279  df-ol 33281  df-oml 33282  df-covers 33369  df-ats 33370  df-atl 33401  df-cvlat 33425  df-hlat 33454
This theorem is referenced by:  cdlemblem  33895  cdlemb  33896  lhpat  34145
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