Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvrat3 Structured version   Visualization version   GIF version

Theorem cvrat3 37905
Description: A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 31338 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 2736 . . . . . . . . . . . 12 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
5 cvrat3.a . . . . . . . . . . . 12 𝐴 = (Atoms‘𝐾)
61, 2, 3, 4, 5cvr1 37873 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑄𝐴) → (¬ 𝑄 𝑋𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
763adant3r2 1183 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ 𝑄 𝑋𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
87biimpa 477 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ¬ 𝑄 𝑋) → 𝑋( ⋖ ‘𝐾)(𝑋 𝑄))
98adantrr 715 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → 𝑋( ⋖ ‘𝐾)(𝑋 𝑄))
10 hllat 37825 . . . . . . . . . . . . . . . . . 18 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1110adantr 481 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ Lat)
12 simpr2 1195 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐴)
131, 5atbase 37751 . . . . . . . . . . . . . . . . . 18 (𝑃𝐴𝑃𝐵)
1412, 13syl 17 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐵)
15 simpr3 1196 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐴)
161, 5atbase 37751 . . . . . . . . . . . . . . . . . 18 (𝑄𝐴𝑄𝐵)
1715, 16syl 17 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐵)
181, 3latjcom 18336 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) = (𝑄 𝑃))
1911, 14, 17, 18syl3anc 1371 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
2019oveq2d 7373 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) = (𝑋 (𝑄 𝑃)))
21 simpr1 1194 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑋𝐵)
221, 3latjass 18372 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑄𝐵𝑃𝐵)) → ((𝑋 𝑄) 𝑃) = (𝑋 (𝑄 𝑃)))
2311, 21, 17, 14, 22syl13anc 1372 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋 𝑄) 𝑃) = (𝑋 (𝑄 𝑃)))
2420, 23eqtr4d 2779 . . . . . . . . . . . . . 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 18328 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → (𝑋 𝑄) ∈ 𝐵)
2711, 21, 17, 26syl3anc 1371 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 𝑄) ∈ 𝐵)
281, 2, 3latjlej2 18343 . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . 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 5127 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) ((𝑋 𝑄) (𝑋 𝑄)))
321, 3latjidm 18351 . . . . . . . . . . . . . 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 5131 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) (𝑋 𝑄))
36 simpl 483 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ HL)
372, 3, 5hlatlej2 37838 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄 (𝑃 𝑄))
3836, 12, 15, 37syl3anc 1371 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄 (𝑃 𝑄))
391, 3latjcl 18328 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
4011, 14, 17, 39syl3anc 1371 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) ∈ 𝐵)
411, 2, 3latjlej2 18343 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑄𝐵 ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵)) → (𝑄 (𝑃 𝑄) → (𝑋 𝑄) (𝑋 (𝑃 𝑄))))
4211, 17, 40, 21, 41syl13anc 1372 . . . . . . . . . . . . 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 18328 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
4611, 21, 40, 45syl3anc 1371 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
471, 2latasymb 18331 . . . . . . . . . . . . 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 1371 . . . . . . . . . . . 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 710 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) = (𝑋 𝑄))
5150breq2d 5117 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄)) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
5251adantrl 714 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → (𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄)) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
539, 52mpbird 256 . . . . . . 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 37883 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → ((𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄))))
5736, 21, 40, 56syl3anc 1371 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄))))
5854, 57sylibrd 258 . . . . 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 18329 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
6111, 21, 40, 60syl3anc 1371 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
621, 3, 4, 5cvrat2 37892 . . . . . . 7 ((𝐾 ∈ HL ∧ ((𝑋 (𝑃 𝑄)) ∈ 𝐵𝑃𝐴𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄))) → (𝑋 (𝑃 𝑄)) ∈ 𝐴)
63623expia 1121 . . . . . 6 ((𝐾 ∈ HL ∧ ((𝑋 (𝑃 𝑄)) ∈ 𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))
6436, 61, 12, 15, 63syl13anc 1372 . . . . 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 1348 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  meetcmee 18201  Latclat 18320  ccvr 37724  Atomscatm 37725  HLchlt 37812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813
This theorem is referenced by:  cvrat4  37906  2atjm  37908  1cvrat  37939  2llnma1b  38249
  Copyright terms: Public domain W3C validator