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Theorem 2atjm 33552
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 33471 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1074 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝐾 ∈ Lat)
3 simp21 1086 . . . . . 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 33397 . . . . . 6 (𝑃𝐴𝑃𝐵)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝐵)
8 simp22 1087 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑄𝐴)
94, 5atbase 33397 . . . . . 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 16829 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → 𝑃 (𝑃 𝑄))
142, 7, 10, 13syl3anc 1317 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 (𝑃 𝑄))
15 simp3l 1081 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 𝑋)
16 simp1 1053 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝐾 ∈ HL)
174, 12, 5hlatjcl 33474 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
1816, 3, 8, 17syl3anc 1317 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑃 𝑄) ∈ 𝐵)
19 simp23 1088 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑋𝐵)
20 2atjm.m . . . . . 6 = (meet‘𝐾)
214, 11, 20latlem12 16847 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃𝐵 ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵)) → ((𝑃 (𝑃 𝑄) ∧ 𝑃 𝑋) ↔ 𝑃 ((𝑃 𝑄) 𝑋)))
222, 7, 18, 19, 21syl13anc 1319 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 (𝑃 𝑄) ∧ 𝑃 𝑋) ↔ 𝑃 ((𝑃 𝑄) 𝑋)))
2314, 15, 22mpbi2and 957 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 ((𝑃 𝑄) 𝑋))
24 hlatl 33468 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
25243ad2ant1 1074 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝐾 ∈ AtLat)
264, 20latmcom 16844 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵) → ((𝑃 𝑄) 𝑋) = (𝑋 (𝑃 𝑄)))
272, 18, 19, 26syl3anc 1317 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄) 𝑋) = (𝑋 (𝑃 𝑄)))
2819, 3, 83jca 1234 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑋𝐵𝑃𝐴𝑄𝐴))
29 nbrne2 4597 . . . . . . 7 ((𝑃 𝑋 ∧ ¬ 𝑄 𝑋) → 𝑃𝑄)
30293ad2ant3 1076 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝑄)
31 simp3r 1082 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ¬ 𝑄 𝑋)
324, 12latjcl 16820 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → (𝑋 𝑄) ∈ 𝐵)
332, 19, 10, 32syl3anc 1317 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑋 𝑄) ∈ 𝐵)
344, 11, 12latlej1 16829 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → 𝑋 (𝑋 𝑄))
352, 19, 10, 34syl3anc 1317 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑋 (𝑋 𝑄))
364, 11, 2, 7, 19, 33, 15, 35lattrd 16827 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 (𝑋 𝑄))
374, 11, 12, 20, 5cvrat3 33549 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))
3837imp 443 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → (𝑋 (𝑃 𝑄)) ∈ 𝐴)
3916, 28, 30, 31, 36, 38syl23anc 1324 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴)
4027, 39eqeltrd 2687 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄) 𝑋) ∈ 𝐴)
4111, 5atcmp 33419 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴 ∧ ((𝑃 𝑄) 𝑋) ∈ 𝐴) → (𝑃 ((𝑃 𝑄) 𝑋) ↔ 𝑃 = ((𝑃 𝑄) 𝑋)))
4225, 3, 40, 41syl3anc 1317 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑃 ((𝑃 𝑄) 𝑋) ↔ 𝑃 = ((𝑃 𝑄) 𝑋)))
4323, 42mpbid 220 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 = ((𝑃 𝑄) 𝑋))
4443eqcomd 2615 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  lecple 15721  joincjn 16713  meetcmee 16714  Latclat 16814  Atomscatm 33371  AtLatcal 33372  HLchlt 33458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
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 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-lat 16815  df-clat 16877  df-oposet 33284  df-ol 33286  df-oml 33287  df-covers 33374  df-ats 33375  df-atl 33406  df-cvlat 33430  df-hlat 33459
This theorem is referenced by:  atbtwn  33553  dalem24  33804  dalem25  33805
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