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Theorem 2atjm 37706
Description: The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.)
Hypotheses
Ref Expression
2atjm.b 𝐵 = (Base‘𝐾)
2atjm.l = (le‘𝐾)
2atjm.j = (join‘𝐾)
2atjm.m = (meet‘𝐾)
2atjm.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
2atjm ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄) 𝑋) = 𝑃)

Proof of Theorem 2atjm
StepHypRef Expression
1 hllat 37623 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1132 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝐾 ∈ Lat)
3 simp21 1205 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝐴)
4 2atjm.b . . . . . . 7 𝐵 = (Base‘𝐾)
5 2atjm.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
64, 5atbase 37549 . . . . . 6 (𝑃𝐴𝑃𝐵)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝐵)
8 simp22 1206 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑄𝐴)
94, 5atbase 37549 . . . . . 6 (𝑄𝐴𝑄𝐵)
108, 9syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑄𝐵)
11 2atjm.l . . . . . 6 = (le‘𝐾)
12 2atjm.j . . . . . 6 = (join‘𝐾)
134, 11, 12latlej1 18255 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → 𝑃 (𝑃 𝑄))
142, 7, 10, 13syl3anc 1370 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 (𝑃 𝑄))
15 simp3l 1200 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 𝑋)
16 simp1 1135 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝐾 ∈ HL)
174, 12, 5hlatjcl 37627 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
1816, 3, 8, 17syl3anc 1370 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑃 𝑄) ∈ 𝐵)
19 simp23 1207 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑋𝐵)
20 2atjm.m . . . . . 6 = (meet‘𝐾)
214, 11, 20latlem12 18273 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃𝐵 ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵)) → ((𝑃 (𝑃 𝑄) ∧ 𝑃 𝑋) ↔ 𝑃 ((𝑃 𝑄) 𝑋)))
222, 7, 18, 19, 21syl13anc 1371 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 (𝑃 𝑄) ∧ 𝑃 𝑋) ↔ 𝑃 ((𝑃 𝑄) 𝑋)))
2314, 15, 22mpbi2and 709 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 ((𝑃 𝑄) 𝑋))
24 hlatl 37620 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
25243ad2ant1 1132 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝐾 ∈ AtLat)
264, 20latmcom 18270 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵) → ((𝑃 𝑄) 𝑋) = (𝑋 (𝑃 𝑄)))
272, 18, 19, 26syl3anc 1370 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄) 𝑋) = (𝑋 (𝑃 𝑄)))
2819, 3, 83jca 1127 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑋𝐵𝑃𝐴𝑄𝐴))
29 nbrne2 5109 . . . . . . 7 ((𝑃 𝑋 ∧ ¬ 𝑄 𝑋) → 𝑃𝑄)
30293ad2ant3 1134 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝑄)
31 simp3r 1201 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ¬ 𝑄 𝑋)
324, 12latjcl 18246 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → (𝑋 𝑄) ∈ 𝐵)
332, 19, 10, 32syl3anc 1370 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑋 𝑄) ∈ 𝐵)
344, 11, 12latlej1 18255 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → 𝑋 (𝑋 𝑄))
352, 19, 10, 34syl3anc 1370 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑋 (𝑋 𝑄))
364, 11, 2, 7, 19, 33, 15, 35lattrd 18253 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 (𝑋 𝑄))
374, 11, 12, 20, 5cvrat3 37703 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))
3837imp 407 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → (𝑋 (𝑃 𝑄)) ∈ 𝐴)
3916, 28, 30, 31, 36, 38syl23anc 1376 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴)
4027, 39eqeltrd 2837 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄) 𝑋) ∈ 𝐴)
4111, 5atcmp 37571 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴 ∧ ((𝑃 𝑄) 𝑋) ∈ 𝐴) → (𝑃 ((𝑃 𝑄) 𝑋) ↔ 𝑃 = ((𝑃 𝑄) 𝑋)))
4225, 3, 40, 41syl3anc 1370 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑃 ((𝑃 𝑄) 𝑋) ↔ 𝑃 = ((𝑃 𝑄) 𝑋)))
4323, 42mpbid 231 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 = ((𝑃 𝑄) 𝑋))
4443eqcomd 2742 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄) 𝑋) = 𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wne 2940   class class class wbr 5089  cfv 6473  (class class class)co 7329  Basecbs 17001  lecple 17058  joincjn 18118  meetcmee 18119  Latclat 18238  Atomscatm 37523  AtLatcal 37524  HLchlt 37610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-proset 18102  df-poset 18120  df-plt 18137  df-lub 18153  df-glb 18154  df-join 18155  df-meet 18156  df-p0 18232  df-lat 18239  df-clat 18306  df-oposet 37436  df-ol 37438  df-oml 37439  df-covers 37526  df-ats 37527  df-atl 37558  df-cvlat 37582  df-hlat 37611
This theorem is referenced by:  atbtwn  37707  dalem24  37958  dalem25  37959
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