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Theorem cvrat3 36458
Description: A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 30100 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 2818 . . . . . . . . . . . 12 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
5 cvrat3.a . . . . . . . . . . . 12 𝐴 = (Atoms‘𝐾)
61, 2, 3, 4, 5cvr1 36426 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑄𝐴) → (¬ 𝑄 𝑋𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
763adant3r2 1175 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ 𝑄 𝑋𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
87biimpa 477 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ¬ 𝑄 𝑋) → 𝑋( ⋖ ‘𝐾)(𝑋 𝑄))
98adantrr 713 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → 𝑋( ⋖ ‘𝐾)(𝑋 𝑄))
10 hllat 36379 . . . . . . . . . . . . . . . . . 18 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1110adantr 481 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ Lat)
12 simpr2 1187 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐴)
131, 5atbase 36305 . . . . . . . . . . . . . . . . . 18 (𝑃𝐴𝑃𝐵)
1412, 13syl 17 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐵)
15 simpr3 1188 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐴)
161, 5atbase 36305 . . . . . . . . . . . . . . . . . 18 (𝑄𝐴𝑄𝐵)
1715, 16syl 17 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐵)
181, 3latjcom 17657 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) = (𝑄 𝑃))
1911, 14, 17, 18syl3anc 1363 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
2019oveq2d 7161 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) = (𝑋 (𝑄 𝑃)))
21 simpr1 1186 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑋𝐵)
221, 3latjass 17693 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑄𝐵𝑃𝐵)) → ((𝑋 𝑄) 𝑃) = (𝑋 (𝑄 𝑃)))
2311, 21, 17, 14, 22syl13anc 1364 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋 𝑄) 𝑃) = (𝑋 (𝑄 𝑃)))
2420, 23eqtr4d 2856 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) = ((𝑋 𝑄) 𝑃))
2524adantr 481 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) = ((𝑋 𝑄) 𝑃))
261, 3latjcl 17649 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → (𝑋 𝑄) ∈ 𝐵)
2711, 21, 17, 26syl3anc 1363 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 𝑄) ∈ 𝐵)
281, 2, 3latjlej2 17664 . . . . . . . . . . . . . . 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 1364 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 (𝑋 𝑄) → ((𝑋 𝑄) 𝑃) ((𝑋 𝑄) (𝑋 𝑄))))
3029imp 407 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋 𝑄) 𝑃) ((𝑋 𝑄) (𝑋 𝑄)))
3125, 30eqbrtrd 5079 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) ((𝑋 𝑄) (𝑋 𝑄)))
321, 3latjidm 17672 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑋 𝑄) ∈ 𝐵) → ((𝑋 𝑄) (𝑋 𝑄)) = (𝑋 𝑄))
3311, 27, 32syl2anc 584 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋 𝑄) (𝑋 𝑄)) = (𝑋 𝑄))
3433adantr 481 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋 𝑄) (𝑋 𝑄)) = (𝑋 𝑄))
3531, 34breqtrd 5083 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) (𝑋 𝑄))
36 simpl 483 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ HL)
372, 3, 5hlatlej2 36392 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄 (𝑃 𝑄))
3836, 12, 15, 37syl3anc 1363 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄 (𝑃 𝑄))
391, 3latjcl 17649 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
4011, 14, 17, 39syl3anc 1363 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) ∈ 𝐵)
411, 2, 3latjlej2 17664 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑄𝐵 ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵)) → (𝑄 (𝑃 𝑄) → (𝑋 𝑄) (𝑋 (𝑃 𝑄))))
4211, 17, 40, 21, 41syl13anc 1364 . . . . . . . . . . . . 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 481 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 𝑄) (𝑋 (𝑃 𝑄)))
451, 3latjcl 17649 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
4611, 21, 40, 45syl3anc 1363 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
471, 2latasymb 17652 . . . . . . . . . . . . 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 1363 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (((𝑋 (𝑃 𝑄)) (𝑋 𝑄) ∧ (𝑋 𝑄) (𝑋 (𝑃 𝑄))) ↔ (𝑋 (𝑃 𝑄)) = (𝑋 𝑄)))
4948adantr 481 . . . . . . . . . . 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 708 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) = (𝑋 𝑄))
5150breq2d 5069 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄)) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
5251adantrl 712 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → (𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄)) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
539, 52mpbird 258 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄)))
5453ex 413 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄))))
55 cvrat3.m . . . . . . . 8 = (meet‘𝐾)
561, 3, 55, 4cvrexch 36436 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → ((𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄))))
5736, 21, 40, 56syl3anc 1363 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄))))
5854, 57sylibrd 260 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄)))
5958adantr 481 . . . 4 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃𝑄) → ((¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄)))
601, 55latmcl 17650 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
6111, 21, 40, 60syl3anc 1363 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
621, 3, 4, 5cvrat2 36445 . . . . . . 7 ((𝐾 ∈ HL ∧ ((𝑋 (𝑃 𝑄)) ∈ 𝐵𝑃𝐴𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄))) → (𝑋 (𝑃 𝑄)) ∈ 𝐴)
63623expia 1113 . . . . . 6 ((𝐾 ∈ HL ∧ ((𝑋 (𝑃 𝑄)) ∈ 𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))
6436, 61, 12, 15, 63syl13anc 1364 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))
6564expdimp 453 . . . 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 431 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃𝑄 → (¬ 𝑄 𝑋 → (𝑃 (𝑋 𝑄) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))))
68673impd 1340 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 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  meetcmee 17543  Latclat 17643  ccvr 36278  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-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367
This theorem is referenced by:  cvrat4  36459  2atjm  36461  1cvrat  36492  2llnma1b  36802
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