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Theorem 1cvrat 38335
Description: Create an atom under an element covered by the lattice unity. 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 38221 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1133 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝐾 ∈ Lat)
3 simp21 1206 . . . . . 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 38147 . . . . . 6 (𝑃𝐴𝑃𝐵)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑃𝐵)
8 simp22 1207 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑄𝐴)
94, 5atbase 38147 . . . . . 6 (𝑄𝐴𝑄𝐵)
108, 9syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑄𝐵)
11 1cvrat.j . . . . . 6 = (join‘𝐾)
124, 11latjcom 18396 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) = (𝑄 𝑃))
132, 7, 10, 12syl3anc 1371 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑃 𝑄) = (𝑄 𝑃))
1413oveq1d 7420 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑃 𝑄) 𝑋) = ((𝑄 𝑃) 𝑋))
154, 11latjcl 18388 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑃𝐵) → (𝑄 𝑃) ∈ 𝐵)
162, 10, 7, 15syl3anc 1371 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑄 𝑃) ∈ 𝐵)
17 simp23 1208 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑋𝐵)
18 1cvrat.m . . . . 5 = (meet‘𝐾)
194, 18latmcom 18412 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 𝑃) ∈ 𝐵𝑋𝐵) → ((𝑄 𝑃) 𝑋) = (𝑋 (𝑄 𝑃)))
202, 16, 17, 19syl3anc 1371 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑄 𝑃) 𝑋) = (𝑋 (𝑄 𝑃)))
2114, 20eqtrd 2772 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑃 𝑄) 𝑋) = (𝑋 (𝑄 𝑃)))
22 simp1 1136 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝐾 ∈ HL)
2317, 8, 33jca 1128 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑋𝐵𝑄𝐴𝑃𝐴))
24 simp31 1209 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑃𝑄)
2524necomd 2996 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑄𝑃)
26 simp33 1211 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ¬ 𝑃 𝑋)
27 hlop 38220 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OP)
28273ad2ant1 1133 . . . . 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 38049 . . . . 5 ((𝐾 ∈ OP ∧ 𝑄𝐵) → 𝑄 1 )
3228, 10, 31syl2anc 584 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑄 1 )
33 simp32 1210 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑋𝐶 1 )
34 1cvrat.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
354, 29, 11, 30, 34, 51cvrjat 38334 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑋 𝑃) = 1 )
3622, 17, 3, 33, 26, 35syl32anc 1378 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑋 𝑃) = 1 )
3732, 36breqtrrd 5175 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → 𝑄 (𝑋 𝑃))
384, 29, 11, 18, 5cvrat3 38301 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑄𝐴𝑃𝐴)) → ((𝑄𝑃 ∧ ¬ 𝑃 𝑋𝑄 (𝑋 𝑃)) → (𝑋 (𝑄 𝑃)) ∈ 𝐴))
3938imp 407 . . 3 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑄𝐴𝑃𝐴)) ∧ (𝑄𝑃 ∧ ¬ 𝑃 𝑋𝑄 (𝑋 𝑃))) → (𝑋 (𝑄 𝑃)) ∈ 𝐴)
4022, 23, 25, 26, 37, 39syl23anc 1377 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → (𝑋 (𝑄 𝑃)) ∈ 𝐴)
4121, 40eqeltrd 2833 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 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940   class class class wbr 5147  cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  joincjn 18260  meetcmee 18261  1.cp1 18373  Latclat 18380  OPcops 38030  ccvr 38120  Atomscatm 38121  HLchlt 38208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-p1 18375  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209
This theorem is referenced by:  cdlemblem  38652  cdlemb  38653  lhpat  38902
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