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Theorem 1cvrat 38650
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 𝐡 = (Baseβ€˜πΎ)
1cvrat.l ≀ = (leβ€˜πΎ)
1cvrat.j ∨ = (joinβ€˜πΎ)
1cvrat.m ∧ = (meetβ€˜πΎ)
1cvrat.u 1 = (1.β€˜πΎ)
1cvrat.c 𝐢 = ( β‹– β€˜πΎ)
1cvrat.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
1cvrat ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴)
Proof of Theorem 1cvrat
StepHypRef Expression
1 hllat 38536 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
213ad2ant1 1133 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ 𝐾 ∈ Lat)
3 simp21 1206 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ 𝑃 ∈ 𝐴)
4 1cvrat.b . . . . . . 7 𝐡 = (Baseβ€˜πΎ)
5 1cvrat.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
64, 5atbase 38462 . . . . . 6 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ 𝑃 ∈ 𝐡)
8 simp22 1207 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ 𝑄 ∈ 𝐴)
94, 5atbase 38462 . . . . . 6 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
108, 9syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ 𝑄 ∈ 𝐡)
11 1cvrat.j . . . . . 6 ∨ = (joinβ€˜πΎ)
124, 11latjcom 18404 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
132, 7, 10, 12syl3anc 1371 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
1413oveq1d 7426 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄) ∧ 𝑋) = ((𝑄 ∨ 𝑃) ∧ 𝑋))
154, 11latjcl 18396 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐡 ∧ 𝑃 ∈ 𝐡) β†’ (𝑄 ∨ 𝑃) ∈ 𝐡)
162, 10, 7, 15syl3anc 1371 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ (𝑄 ∨ 𝑃) ∈ 𝐡)
17 simp23 1208 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ 𝑋 ∈ 𝐡)
18 1cvrat.m . . . . 5 ∧ = (meetβ€˜πΎ)
194, 18latmcom 18420 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑃) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ ((𝑄 ∨ 𝑃) ∧ 𝑋) = (𝑋 ∧ (𝑄 ∨ 𝑃)))
202, 16, 17, 19syl3anc 1371 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ ((𝑄 ∨ 𝑃) ∧ 𝑋) = (𝑋 ∧ (𝑄 ∨ 𝑃)))
2114, 20eqtrd 2772 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄) ∧ 𝑋) = (𝑋 ∧ (𝑄 ∨ 𝑃)))
22 simp1 1136 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ 𝐾 ∈ HL)
2317, 8, 33jca 1128 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴))
24 simp31 1209 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ 𝑃 β‰  𝑄)
2524necomd 2996 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ 𝑄 β‰  𝑃)
26 simp33 1211 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ Β¬ 𝑃 ≀ 𝑋)
27 hlop 38535 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
28273ad2ant1 1133 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ 𝐾 ∈ OP)
29 1cvrat.l . . . . . 6 ≀ = (leβ€˜πΎ)
30 1cvrat.u . . . . . 6 1 = (1.β€˜πΎ)
314, 29, 30ople1 38364 . . . . 5 ((𝐾 ∈ OP ∧ 𝑄 ∈ 𝐡) β†’ 𝑄 ≀ 1 )
3228, 10, 31syl2anc 584 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ 𝑄 ≀ 1 )
33 simp32 1210 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ 𝑋𝐢 1 )
34 1cvrat.c . . . . . 6 𝐢 = ( β‹– β€˜πΎ)
354, 29, 11, 30, 34, 51cvrjat 38649 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ (𝑋 ∨ 𝑃) = 1 )
3622, 17, 3, 33, 26, 35syl32anc 1378 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ (𝑋 ∨ 𝑃) = 1 )
3732, 36breqtrrd 5176 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃))
384, 29, 11, 18, 5cvrat3 38616 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴)) β†’ ((𝑄 β‰  𝑃 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ 𝑄 ≀ (𝑋 ∨ 𝑃)) β†’ (𝑋 ∧ (𝑄 ∨ 𝑃)) ∈ 𝐴))
3938imp 407 . . 3 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴)) ∧ (𝑄 β‰  𝑃 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ 𝑄 ≀ (𝑋 ∨ 𝑃))) β†’ (𝑋 ∧ (𝑄 ∨ 𝑃)) ∈ 𝐴)
4022, 23, 25, 26, 37, 39syl23anc 1377 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ (𝑋 ∧ (𝑄 ∨ 𝑃)) ∈ 𝐴)
4121, 40eqeltrd 2833 1 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  Basecbs 17148  lecple 17208  joincjn 18268  meetcmee 18269  1.cp1 18381  Latclat 18388  OPcops 38345   β‹– ccvr 38435  Atomscatm 38436  HLchlt 38523
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-proset 18252  df-poset 18270  df-plt 18287  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-p0 18382  df-p1 18383  df-lat 18389  df-clat 18456  df-oposet 38349  df-ol 38351  df-oml 38352  df-covers 38439  df-ats 38440  df-atl 38471  df-cvlat 38495  df-hlat 38524
This theorem is referenced by:  cdlemblem  38967  cdlemb  38968  lhpat  39217
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