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Theorem exatleN 35212
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 1326 . . . . 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 1366 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝐾 ∈ HL)
5 hllat 35172 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ Lat)
64, 5syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝐾 ∈ Lat)
7 simp122 1388 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑄𝐴)
8 atomle.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
92, 8atbase 35098 . . . . . . . 8 (𝑄𝐴𝑄𝐵)
107, 9syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑄𝐵)
11 simp121 1387 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑃𝐴)
122, 8atbase 35098 . . . . . . . . 9 (𝑃𝐴𝑃𝐵)
1311, 12syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑃𝐵)
14 simp123 1389 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑅𝐴)
152, 8atbase 35098 . . . . . . . . 9 (𝑅𝐴𝑅𝐵)
1614, 15syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑅𝐵)
17 atomle.j . . . . . . . . 9 = (join‘𝐾)
182, 17latjcl 17265 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑅𝐵) → (𝑃 𝑅) ∈ 𝐵)
196, 13, 16, 18syl3anc 1474 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → (𝑃 𝑅) ∈ 𝐵)
20 simp11r 1367 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑋𝐵)
2114, 7, 113jca 1120 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → (𝑅𝐴𝑄𝐴𝑃𝐴))
22 simp2 1129 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑅𝑃)
234, 21, 223jca 1120 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → (𝐾 ∈ HL ∧ (𝑅𝐴𝑄𝐴𝑃𝐴) ∧ 𝑅𝑃))
24 simp133 1392 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑅 (𝑃 𝑄))
253, 17, 8hlatexch1 35203 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑄𝐴𝑃𝐴) ∧ 𝑅𝑃) → (𝑅 (𝑃 𝑄) → 𝑄 (𝑃 𝑅)))
2623, 24, 25sylc 65 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑄 (𝑃 𝑅))
27 simp131 1390 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑃 𝑋)
28 simp3 1130 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑅 𝑋)
292, 3, 17latjle12 17276 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑅𝐵𝑋𝐵)) → ((𝑃 𝑋𝑅 𝑋) ↔ (𝑃 𝑅) 𝑋))
306, 13, 16, 20, 29syl13anc 1476 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → ((𝑃 𝑋𝑅 𝑋) ↔ (𝑃 𝑅) 𝑋))
3127, 28, 30mpbi2and 991 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → (𝑃 𝑅) 𝑋)
322, 3, 6, 10, 19, 20, 26, 31lattrd 17272 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃𝑅 𝑋) → 𝑄 𝑋)
33323expia 1112 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃) → (𝑅 𝑋𝑄 𝑋))
341, 33mtod 189 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) ∧ 𝑅𝑃) → ¬ 𝑅 𝑋)
3534ex 448 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅𝑃 → ¬ 𝑅 𝑋))
3635necon4ad 2960 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅 𝑋𝑅 = 𝑃))
37 simp31 1249 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → 𝑃 𝑋)
38 breq1 4786 . . 3 (𝑅 = 𝑃 → (𝑅 𝑋𝑃 𝑋))
3937, 38syl5ibrcom 237 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅 = 𝑃𝑅 𝑋))
4036, 39impbid 202 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 196  wa 383  w3a 1069   = wceq 1629  wcel 2143  wne 2941   class class class wbr 4783  cfv 6030  (class class class)co 6791  Basecbs 16070  lecple 16162  joincjn 17158  Latclat 17259  Atomscatm 35072  HLchlt 35159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-rep 4901  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-reu 3066  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-iun 4653  df-br 4784  df-opab 4844  df-mpt 4861  df-id 5156  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6752  df-ov 6794  df-oprab 6795  df-preset 17142  df-poset 17160  df-plt 17172  df-lub 17188  df-glb 17189  df-join 17190  df-meet 17191  df-p0 17253  df-lat 17260  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160
This theorem is referenced by:  cdlema2N  35600
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