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Theorem cvrat3 35046
Description: A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 29383 analog.) (Contributed by NM, 30-Nov-2011.)
Hypotheses
Ref Expression
cvrat3.b 𝐵 = (Base‘𝐾)
cvrat3.l = (le‘𝐾)
cvrat3.j = (join‘𝐾)
cvrat3.m = (meet‘𝐾)
cvrat3.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
cvrat3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))

Proof of Theorem cvrat3
StepHypRef Expression
1 cvrat3.b . . . . . . . . . . . 12 𝐵 = (Base‘𝐾)
2 cvrat3.l . . . . . . . . . . . 12 = (le‘𝐾)
3 cvrat3.j . . . . . . . . . . . 12 = (join‘𝐾)
4 eqid 2651 . . . . . . . . . . . 12 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
5 cvrat3.a . . . . . . . . . . . 12 𝐴 = (Atoms‘𝐾)
61, 2, 3, 4, 5cvr1 35014 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑄𝐴) → (¬ 𝑄 𝑋𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
763adant3r2 1296 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ 𝑄 𝑋𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
87biimpa 500 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ¬ 𝑄 𝑋) → 𝑋( ⋖ ‘𝐾)(𝑋 𝑄))
98adantrr 753 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → 𝑋( ⋖ ‘𝐾)(𝑋 𝑄))
10 hllat 34968 . . . . . . . . . . . . . . . . . 18 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1110adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ Lat)
12 simpr2 1088 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐴)
131, 5atbase 34894 . . . . . . . . . . . . . . . . . 18 (𝑃𝐴𝑃𝐵)
1412, 13syl 17 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐵)
15 simpr3 1089 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐴)
161, 5atbase 34894 . . . . . . . . . . . . . . . . . 18 (𝑄𝐴𝑄𝐵)
1715, 16syl 17 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐵)
181, 3latjcom 17106 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) = (𝑄 𝑃))
1911, 14, 17, 18syl3anc 1366 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
2019oveq2d 6706 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) = (𝑋 (𝑄 𝑃)))
21 simpr1 1087 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑋𝐵)
221, 3latjass 17142 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑄𝐵𝑃𝐵)) → ((𝑋 𝑄) 𝑃) = (𝑋 (𝑄 𝑃)))
2311, 21, 17, 14, 22syl13anc 1368 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋 𝑄) 𝑃) = (𝑋 (𝑄 𝑃)))
2420, 23eqtr4d 2688 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) = ((𝑋 𝑄) 𝑃))
2524adantr 480 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) = ((𝑋 𝑄) 𝑃))
261, 3latjcl 17098 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → (𝑋 𝑄) ∈ 𝐵)
2711, 21, 17, 26syl3anc 1366 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 𝑄) ∈ 𝐵)
281, 2, 3latjlej2 17113 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ (𝑃𝐵 ∧ (𝑋 𝑄) ∈ 𝐵 ∧ (𝑋 𝑄) ∈ 𝐵)) → (𝑃 (𝑋 𝑄) → ((𝑋 𝑄) 𝑃) ((𝑋 𝑄) (𝑋 𝑄))))
2911, 14, 27, 27, 28syl13anc 1368 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 (𝑋 𝑄) → ((𝑋 𝑄) 𝑃) ((𝑋 𝑄) (𝑋 𝑄))))
3029imp 444 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋 𝑄) 𝑃) ((𝑋 𝑄) (𝑋 𝑄)))
3125, 30eqbrtrd 4707 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) ((𝑋 𝑄) (𝑋 𝑄)))
321, 3latjidm 17121 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑋 𝑄) ∈ 𝐵) → ((𝑋 𝑄) (𝑋 𝑄)) = (𝑋 𝑄))
3311, 27, 32syl2anc 694 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋 𝑄) (𝑋 𝑄)) = (𝑋 𝑄))
3433adantr 480 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋 𝑄) (𝑋 𝑄)) = (𝑋 𝑄))
3531, 34breqtrd 4711 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) (𝑋 𝑄))
36 simpl 472 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ HL)
372, 3, 5hlatlej2 34980 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄 (𝑃 𝑄))
3836, 12, 15, 37syl3anc 1366 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄 (𝑃 𝑄))
391, 3latjcl 17098 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
4011, 14, 17, 39syl3anc 1366 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) ∈ 𝐵)
411, 2, 3latjlej2 17113 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑄𝐵 ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵)) → (𝑄 (𝑃 𝑄) → (𝑋 𝑄) (𝑋 (𝑃 𝑄))))
4211, 17, 40, 21, 41syl13anc 1368 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄 (𝑃 𝑄) → (𝑋 𝑄) (𝑋 (𝑃 𝑄))))
4338, 42mpd 15 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 𝑄) (𝑋 (𝑃 𝑄)))
4443adantr 480 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 𝑄) (𝑋 (𝑃 𝑄)))
451, 3latjcl 17098 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
4611, 21, 40, 45syl3anc 1366 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
471, 2latasymb 17101 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ (𝑋 (𝑃 𝑄)) ∈ 𝐵 ∧ (𝑋 𝑄) ∈ 𝐵) → (((𝑋 (𝑃 𝑄)) (𝑋 𝑄) ∧ (𝑋 𝑄) (𝑋 (𝑃 𝑄))) ↔ (𝑋 (𝑃 𝑄)) = (𝑋 𝑄)))
4811, 46, 27, 47syl3anc 1366 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (((𝑋 (𝑃 𝑄)) (𝑋 𝑄) ∧ (𝑋 𝑄) (𝑋 (𝑃 𝑄))) ↔ (𝑋 (𝑃 𝑄)) = (𝑋 𝑄)))
4948adantr 480 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (((𝑋 (𝑃 𝑄)) (𝑋 𝑄) ∧ (𝑋 𝑄) (𝑋 (𝑃 𝑄))) ↔ (𝑋 (𝑃 𝑄)) = (𝑋 𝑄)))
5035, 44, 49mpbi2and 976 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) = (𝑋 𝑄))
5150breq2d 4697 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄)) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
5251adantrl 752 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → (𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄)) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
539, 52mpbird 247 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄)))
5453ex 449 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄))))
55 cvrat3.m . . . . . . . 8 = (meet‘𝐾)
561, 3, 55, 4cvrexch 35024 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → ((𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄))))
5736, 21, 40, 56syl3anc 1366 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄))))
5854, 57sylibrd 249 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄)))
5958adantr 480 . . . 4 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃𝑄) → ((¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄)))
601, 55latmcl 17099 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
6111, 21, 40, 60syl3anc 1366 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
621, 3, 4, 5cvrat2 35033 . . . . . . 7 ((𝐾 ∈ HL ∧ ((𝑋 (𝑃 𝑄)) ∈ 𝐵𝑃𝐴𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄))) → (𝑋 (𝑃 𝑄)) ∈ 𝐴)
63623expia 1286 . . . . . 6 ((𝐾 ∈ HL ∧ ((𝑋 (𝑃 𝑄)) ∈ 𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))
6436, 61, 12, 15, 63syl13anc 1368 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))
6564expdimp 452 . . . 4 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃𝑄) → ((𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))
6659, 65syld 47 . . 3 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃𝑄) → ((¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))
6766exp4b 631 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃𝑄 → (¬ 𝑄 𝑋 → (𝑃 (𝑋 𝑄) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))))
68673impd 1303 1 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  Latclat 17092  ccvr 34867  Atomscatm 34868  HLchlt 34955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956
This theorem is referenced by:  cvrat4  35047  2atjm  35049  1cvrat  35080  2llnma1b  35390
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