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Theorem cvrat3 39425
Description: A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 32425 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 2735 . . . . . . . . . . . 12 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
5 cvrat3.a . . . . . . . . . . . 12 𝐴 = (Atoms‘𝐾)
61, 2, 3, 4, 5cvr1 39393 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑄𝐴) → (¬ 𝑄 𝑋𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
763adant3r2 1182 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ 𝑄 𝑋𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
87biimpa 476 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ¬ 𝑄 𝑋) → 𝑋( ⋖ ‘𝐾)(𝑋 𝑄))
98adantrr 717 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → 𝑋( ⋖ ‘𝐾)(𝑋 𝑄))
10 hllat 39345 . . . . . . . . . . . . . . . . . 18 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1110adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ Lat)
12 simpr2 1194 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐴)
131, 5atbase 39271 . . . . . . . . . . . . . . . . . 18 (𝑃𝐴𝑃𝐵)
1412, 13syl 17 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐵)
15 simpr3 1195 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐴)
161, 5atbase 39271 . . . . . . . . . . . . . . . . . 18 (𝑄𝐴𝑄𝐵)
1715, 16syl 17 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐵)
181, 3latjcom 18505 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) = (𝑄 𝑃))
1911, 14, 17, 18syl3anc 1370 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
2019oveq2d 7447 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) = (𝑋 (𝑄 𝑃)))
21 simpr1 1193 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑋𝐵)
221, 3latjass 18541 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑄𝐵𝑃𝐵)) → ((𝑋 𝑄) 𝑃) = (𝑋 (𝑄 𝑃)))
2311, 21, 17, 14, 22syl13anc 1371 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋 𝑄) 𝑃) = (𝑋 (𝑄 𝑃)))
2420, 23eqtr4d 2778 . . . . . . . . . . . . . 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 18497 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → (𝑋 𝑄) ∈ 𝐵)
2711, 21, 17, 26syl3anc 1370 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 𝑄) ∈ 𝐵)
281, 2, 3latjlej2 18512 . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 (𝑋 𝑄) → ((𝑋 𝑄) 𝑃) ((𝑋 𝑄) (𝑋 𝑄))))
3029imp 406 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋 𝑄) 𝑃) ((𝑋 𝑄) (𝑋 𝑄)))
3125, 30eqbrtrd 5170 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) ((𝑋 𝑄) (𝑋 𝑄)))
321, 3latjidm 18520 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋 𝑄) (𝑋 𝑄)) = (𝑋 𝑄))
3531, 34breqtrd 5174 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) (𝑋 𝑄))
36 simpl 482 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ HL)
372, 3, 5hlatlej2 39358 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄 (𝑃 𝑄))
3836, 12, 15, 37syl3anc 1370 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄 (𝑃 𝑄))
391, 3latjcl 18497 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
4011, 14, 17, 39syl3anc 1370 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) ∈ 𝐵)
411, 2, 3latjlej2 18512 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑄𝐵 ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵)) → (𝑄 (𝑃 𝑄) → (𝑋 𝑄) (𝑋 (𝑃 𝑄))))
4211, 17, 40, 21, 41syl13anc 1371 . . . . . . . . . . . . 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 18497 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
4611, 21, 40, 45syl3anc 1370 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
471, 2latasymb 18500 . . . . . . . . . . . . 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 1370 . . . . . . . . . . . 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 712 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) = (𝑋 𝑄))
5150breq2d 5160 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 (𝑋 𝑄)) → (𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄)) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
5251adantrl 716 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → (𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄)) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 𝑄)))
539, 52mpbird 257 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄)))
5453ex 412 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄))))
55 cvrat3.m . . . . . . . 8 = (meet‘𝐾)
561, 3, 55, 4cvrexch 39403 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → ((𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄))))
5736, 21, 40, 56syl3anc 1370 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 (𝑃 𝑄))))
5854, 57sylibrd 259 . . . . 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 18498 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
6111, 21, 40, 60syl3anc 1370 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋 (𝑃 𝑄)) ∈ 𝐵)
621, 3, 4, 5cvrat2 39412 . . . . . . 7 ((𝐾 ∈ HL ∧ ((𝑋 (𝑃 𝑄)) ∈ 𝐵𝑃𝐴𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄))) → (𝑋 (𝑃 𝑄)) ∈ 𝐴)
63623expia 1120 . . . . . 6 ((𝐾 ∈ HL ∧ ((𝑋 (𝑃 𝑄)) ∈ 𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ (𝑋 (𝑃 𝑄))( ⋖ ‘𝐾)(𝑃 𝑄)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))
6436, 61, 12, 15, 63syl13anc 1371 . . . . 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 430 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃𝑄 → (¬ 𝑄 𝑋 → (𝑃 (𝑋 𝑄) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))))
68673impd 1347 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  Latclat 18489  ccvr 39244  Atomscatm 39245  HLchlt 39332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333
This theorem is referenced by:  cvrat4  39426  2atjm  39428  1cvrat  39459  2llnma1b  39769
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