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Theorem cvrat3 33746
 Description: A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 28639 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 2610 . . . . . . . . . . . 12 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
5 cvrat3.a . . . . . . . . . . . 12 𝐴 = (Atoms‘𝐾)
61, 2, 3, 4, 5cvr1 33714 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑄𝐴) → (¬ 𝑄 𝑋𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
763adant3r2 1267 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ 𝑄 𝑋𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
87biimpa 500 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ¬ 𝑄 𝑋) → 𝑋( ⋖ ‘𝐾)(𝑋 𝑄))
98adantrr 749 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → 𝑋( ⋖ ‘𝐾)(𝑋 𝑄))
10 hllat 33668 . . . . . . . . . . . . . . . . . 18 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1110adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ Lat)
12 simpr2 1061 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐴)
131, 5atbase 33594 . . . . . . . . . . . . . . . . . 18 (𝑃𝐴𝑃𝐵)
1412, 13syl 17 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐵)
15 simpr3 1062 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐴)
161, 5atbase 33594 . . . . . . . . . . . . . . . . . 18 (𝑄𝐴𝑄𝐵)
1715, 16syl 17 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐵)
181, 3latjcom 16882 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) = (𝑄 𝑃))
1911, 14, 17, 18syl3anc 1318 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
2019oveq2d 6565 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) = (𝑋 (𝑄 𝑃)))
21 simpr1 1060 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑋𝐵)
221, 3latjass 16918 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑄𝐵𝑃𝐵)) → ((𝑋 𝑄) 𝑃) = (𝑋 (𝑄 𝑃)))
2311, 21, 17, 14, 22syl13anc 1320 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋 𝑄) 𝑃) = (𝑋 (𝑄 𝑃)))
2420, 23eqtr4d 2647 . . . . . . . . . . . . . 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 16874 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → (𝑋 𝑄) ∈ 𝐵)
2711, 21, 17, 26syl3anc 1318 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 𝑄) ∈ 𝐵)
281, 2, 3latjlej2 16889 . . . . . . . . . . . . . . 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 1320 . . . . . . . . . . . . . 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 4605 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) ((𝑋 𝑄) (𝑋 𝑄)))
321, 3latjidm 16897 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑋 𝑄) ∈ 𝐵) → ((𝑋 𝑄) (𝑋 𝑄)) = (𝑋 𝑄))
3311, 27, 32syl2anc 691 . . . . . . . . . . . . 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 4609 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) (𝑋 𝑄))
36 simpl 472 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ HL)
372, 3, 5hlatlej2 33680 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄 (𝑃 𝑄))
3836, 12, 15, 37syl3anc 1318 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄 (𝑃 𝑄))
391, 3latjcl 16874 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
4011, 14, 17, 39syl3anc 1318 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) ∈ 𝐵)
411, 2, 3latjlej2 16889 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑄𝐵 ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵)) → (𝑄 (𝑃 𝑄) → (𝑋 𝑄) (𝑋 (𝑃 𝑄))))
4211, 17, 40, 21, 41syl13anc 1320 . . . . . . . . . . . . 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 16874 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
4611, 21, 40, 45syl3anc 1318 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
471, 2latasymb 16877 . . . . . . . . . . . . 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 1318 . . . . . . . . . . . 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 958 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) = (𝑋 𝑄))
5150breq2d 4595 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄)) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
5251adantrl 748 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → (𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄)) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
539, 52mpbird 246 . . . . . . 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 33724 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → ((𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄))))
5736, 21, 40, 56syl3anc 1318 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄))))
5854, 57sylibrd 248 . . . . 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 16875 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
6111, 21, 40, 60syl3anc 1318 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
621, 3, 4, 5cvrat2 33733 . . . . . . 7 ((𝐾 ∈ HL ∧ ((𝑋 (𝑃 𝑄)) ∈ 𝐵𝑃𝐴𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄))) → (𝑋 (𝑃 𝑄)) ∈ 𝐴)
63623expia 1259 . . . . . 6 ((𝐾 ∈ HL ∧ ((𝑋 (𝑃 𝑄)) ∈ 𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))
6436, 61, 12, 15, 63syl13anc 1320 . . . . 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 46 . . 3 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃𝑄) → ((¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))
6766exp4b 630 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃𝑄 → (¬ 𝑄 𝑋 → (𝑃 (𝑋 𝑄) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))))
68673impd 1273 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Latclat 16868   ⋖ ccvr 33567  Atomscatm 33568  HLchlt 33655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656 This theorem is referenced by:  cvrat4  33747  2atjm  33749  1cvrat  33780  2llnma1b  34090
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