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Theorem exatleN 36420
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 1247 . . . . 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 1276 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝐾 ∈ HL)
54hllatd 36380 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝐾 ∈ Lat)
6 simp122 1298 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑄𝐴)
7 atomle.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
82, 7atbase 36305 . . . . . . . 8 (𝑄𝐴𝑄𝐵)
96, 8syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑄𝐵)
10 simp121 1297 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑃𝐴)
112, 7atbase 36305 . . . . . . . . 9 (𝑃𝐴𝑃𝐵)
1210, 11syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑃𝐵)
13 simp123 1299 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑅𝐴)
142, 7atbase 36305 . . . . . . . . 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 17649 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑅𝐵) → (𝑃 𝑅) ∈ 𝐵)
185, 12, 15, 17syl3anc 1363 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → (𝑃 𝑅) ∈ 𝐵)
19 simp11r 1277 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑋𝐵)
2013, 6, 103jca 1120 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → (𝑅𝐴𝑄𝐴𝑃𝐴))
21 simp2 1129 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑅𝑃)
224, 20, 213jca 1120 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → (𝐾 ∈ HL ∧ (𝑅𝐴𝑄𝐴𝑃𝐴) ∧ 𝑅𝑃))
23 simp133 1302 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑅 (𝑃 𝑄))
243, 16, 7hlatexch1 36411 . . . . . . . 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 1300 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑃 𝑋)
27 simp3 1130 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑅 𝑋)
282, 3, 16latjle12 17660 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑅𝐵𝑋𝐵)) → ((𝑃 𝑋𝑅 𝑋) ↔ (𝑃 𝑅) 𝑋))
295, 12, 15, 19, 28syl13anc 1364 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → ((𝑃 𝑋𝑅 𝑋) ↔ (𝑃 𝑅) 𝑋))
3026, 27, 29mpbi2and 708 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → (𝑃 𝑅) 𝑋)
312, 3, 5, 9, 18, 19, 25, 30lattrd 17656 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑄 𝑋)
32313expia 1113 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃) → (𝑅 𝑋𝑄 𝑋))
331, 32mtod 199 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃) → ¬ 𝑅 𝑋)
3433ex 413 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅𝑃 → ¬ 𝑅 𝑋))
3534necon4ad 3032 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅 𝑋𝑅 = 𝑃))
36 simp31 1201 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → 𝑃 𝑋)
37 breq1 5060 . . 3 (𝑅 = 𝑃 → (𝑅 𝑋𝑃 𝑋))
3836, 37syl5ibrcom 248 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅 = 𝑃𝑅 𝑋))
3935, 38impbid 213 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  joincjn 17542  Latclat 17643  Atomscatm 36279  HLchlt 36366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-lat 17644  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367
This theorem is referenced by:  cdlema2N  36808
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