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Theorem exatleN 39407
Description: A condition for an atom to be less than or equal to a lattice element. Part of proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
atomle.b 𝐵 = (Base‘𝐾)
atomle.l = (le‘𝐾)
atomle.j = (join‘𝐾)
atomle.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
exatleN (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅 𝑋𝑅 = 𝑃))

Proof of Theorem exatleN
StepHypRef Expression
1 simpl32 1255 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃) → ¬ 𝑄 𝑋)
2 atomle.b . . . . . . 7 𝐵 = (Base‘𝐾)
3 atomle.l . . . . . . 7 = (le‘𝐾)
4 simp11l 1284 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝐾 ∈ HL)
54hllatd 39366 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝐾 ∈ Lat)
6 simp122 1306 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑄𝐴)
7 atomle.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
82, 7atbase 39291 . . . . . . . 8 (𝑄𝐴𝑄𝐵)
96, 8syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑄𝐵)
10 simp121 1305 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑃𝐴)
112, 7atbase 39291 . . . . . . . . 9 (𝑃𝐴𝑃𝐵)
1210, 11syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑃𝐵)
13 simp123 1307 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑅𝐴)
142, 7atbase 39291 . . . . . . . . 9 (𝑅𝐴𝑅𝐵)
1513, 14syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑅𝐵)
16 atomle.j . . . . . . . . 9 = (join‘𝐾)
172, 16latjcl 18485 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑅𝐵) → (𝑃 𝑅) ∈ 𝐵)
185, 12, 15, 17syl3anc 1372 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → (𝑃 𝑅) ∈ 𝐵)
19 simp11r 1285 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑋𝐵)
2013, 6, 103jca 1128 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → (𝑅𝐴𝑄𝐴𝑃𝐴))
21 simp2 1137 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑅𝑃)
224, 20, 213jca 1128 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → (𝐾 ∈ HL ∧ (𝑅𝐴𝑄𝐴𝑃𝐴) ∧ 𝑅𝑃))
23 simp133 1310 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑅 (𝑃 𝑄))
243, 16, 7hlatexch1 39398 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑄𝐴𝑃𝐴) ∧ 𝑅𝑃) → (𝑅 (𝑃 𝑄) → 𝑄 (𝑃 𝑅)))
2522, 23, 24sylc 65 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑄 (𝑃 𝑅))
26 simp131 1308 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑃 𝑋)
27 simp3 1138 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑅 𝑋)
282, 3, 16latjle12 18496 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑅𝐵𝑋𝐵)) → ((𝑃 𝑋𝑅 𝑋) ↔ (𝑃 𝑅) 𝑋))
295, 12, 15, 19, 28syl13anc 1373 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → ((𝑃 𝑋𝑅 𝑋) ↔ (𝑃 𝑅) 𝑋))
3026, 27, 29mpbi2and 712 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → (𝑃 𝑅) 𝑋)
312, 3, 5, 9, 18, 19, 25, 30lattrd 18492 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑄 𝑋)
32313expia 1121 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃) → (𝑅 𝑋𝑄 𝑋))
331, 32mtod 198 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃) → ¬ 𝑅 𝑋)
3433ex 412 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅𝑃 → ¬ 𝑅 𝑋))
3534necon4ad 2958 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅 𝑋𝑅 = 𝑃))
36 simp31 1209 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → 𝑃 𝑋)
37 breq1 5145 . . 3 (𝑅 = 𝑃 → (𝑅 𝑋𝑃 𝑋))
3836, 37syl5ibrcom 247 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅 = 𝑃𝑅 𝑋))
3935, 38impbid 212 1 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅 𝑋𝑅 = 𝑃))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wne 2939   class class class wbr 5142  cfv 6560  (class class class)co 7432  Basecbs 17248  lecple 17305  joincjn 18358  Latclat 18477  Atomscatm 39265  HLchlt 39352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-proset 18341  df-poset 18360  df-plt 18376  df-lub 18392  df-glb 18393  df-join 18394  df-meet 18395  df-p0 18471  df-lat 18478  df-covers 39268  df-ats 39269  df-atl 39300  df-cvlat 39324  df-hlat 39353
This theorem is referenced by:  cdlema2N  39795
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