Metamath Proof Explorer


Theorem cvrat3

Description: A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of PtakPulmannova p. 68. ( atcvat3i analog.) (Contributed by NM, 30-Nov-2011)

Ref Expression
Hypotheses cvrat3.b 𝐵 = ( Base ‘ 𝐾 )
cvrat3.l = ( le ‘ 𝐾 )
cvrat3.j = ( join ‘ 𝐾 )
cvrat3.m = ( meet ‘ 𝐾 )
cvrat3.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion cvrat3 ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( ( 𝑃𝑄 ∧ ¬ 𝑄 𝑋𝑃 ( 𝑋 𝑄 ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) ∈ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 cvrat3.b 𝐵 = ( Base ‘ 𝐾 )
2 cvrat3.l = ( le ‘ 𝐾 )
3 cvrat3.j = ( join ‘ 𝐾 )
4 cvrat3.m = ( meet ‘ 𝐾 )
5 cvrat3.a 𝐴 = ( Atoms ‘ 𝐾 )
6 eqid ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 )
7 1 2 3 6 5 cvr1 ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑄𝐴 ) → ( ¬ 𝑄 𝑋𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 𝑄 ) ) )
8 7 3adant3r2 ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( ¬ 𝑄 𝑋𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 𝑄 ) ) )
9 8 biimpa ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ ¬ 𝑄 𝑋 ) → 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 𝑄 ) )
10 9 adantrr ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ ( ¬ 𝑄 𝑋𝑃 ( 𝑋 𝑄 ) ) ) → 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 𝑄 ) )
11 hllat ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
12 11 adantr ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → 𝐾 ∈ Lat )
13 simpr2 ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → 𝑃𝐴 )
14 1 5 atbase ( 𝑃𝐴𝑃𝐵 )
15 13 14 syl ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → 𝑃𝐵 )
16 simpr3 ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → 𝑄𝐴 )
17 1 5 atbase ( 𝑄𝐴𝑄𝐵 )
18 16 17 syl ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → 𝑄𝐵 )
19 1 3 latjcom ( ( 𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵 ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
20 12 15 18 19 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
21 20 oveq2d ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) = ( 𝑋 ( 𝑄 𝑃 ) ) )
22 simpr1 ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → 𝑋𝐵 )
23 1 3 latjass ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑄𝐵𝑃𝐵 ) ) → ( ( 𝑋 𝑄 ) 𝑃 ) = ( 𝑋 ( 𝑄 𝑃 ) ) )
24 12 22 18 15 23 syl13anc ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( ( 𝑋 𝑄 ) 𝑃 ) = ( 𝑋 ( 𝑄 𝑃 ) ) )
25 21 24 eqtr4d ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) = ( ( 𝑋 𝑄 ) 𝑃 ) )
26 25 adantr ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ 𝑃 ( 𝑋 𝑄 ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) = ( ( 𝑋 𝑄 ) 𝑃 ) )
27 1 3 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵 ) → ( 𝑋 𝑄 ) ∈ 𝐵 )
28 12 22 18 27 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( 𝑋 𝑄 ) ∈ 𝐵 )
29 1 2 3 latjlej2 ( ( 𝐾 ∈ Lat ∧ ( 𝑃𝐵 ∧ ( 𝑋 𝑄 ) ∈ 𝐵 ∧ ( 𝑋 𝑄 ) ∈ 𝐵 ) ) → ( 𝑃 ( 𝑋 𝑄 ) → ( ( 𝑋 𝑄 ) 𝑃 ) ( ( 𝑋 𝑄 ) ( 𝑋 𝑄 ) ) ) )
30 12 15 28 28 29 syl13anc ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( 𝑃 ( 𝑋 𝑄 ) → ( ( 𝑋 𝑄 ) 𝑃 ) ( ( 𝑋 𝑄 ) ( 𝑋 𝑄 ) ) ) )
31 30 imp ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ 𝑃 ( 𝑋 𝑄 ) ) → ( ( 𝑋 𝑄 ) 𝑃 ) ( ( 𝑋 𝑄 ) ( 𝑋 𝑄 ) ) )
32 26 31 eqbrtrd ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ 𝑃 ( 𝑋 𝑄 ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) ( ( 𝑋 𝑄 ) ( 𝑋 𝑄 ) ) )
33 1 3 latjidm ( ( 𝐾 ∈ Lat ∧ ( 𝑋 𝑄 ) ∈ 𝐵 ) → ( ( 𝑋 𝑄 ) ( 𝑋 𝑄 ) ) = ( 𝑋 𝑄 ) )
34 12 28 33 syl2anc ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( ( 𝑋 𝑄 ) ( 𝑋 𝑄 ) ) = ( 𝑋 𝑄 ) )
35 34 adantr ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ 𝑃 ( 𝑋 𝑄 ) ) → ( ( 𝑋 𝑄 ) ( 𝑋 𝑄 ) ) = ( 𝑋 𝑄 ) )
36 32 35 breqtrd ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ 𝑃 ( 𝑋 𝑄 ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) ( 𝑋 𝑄 ) )
37 simpl ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → 𝐾 ∈ HL )
38 2 3 5 hlatlej2 ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → 𝑄 ( 𝑃 𝑄 ) )
39 37 13 16 38 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → 𝑄 ( 𝑃 𝑄 ) )
40 1 3 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵 ) → ( 𝑃 𝑄 ) ∈ 𝐵 )
41 12 15 18 40 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( 𝑃 𝑄 ) ∈ 𝐵 )
42 1 2 3 latjlej2 ( ( 𝐾 ∈ Lat ∧ ( 𝑄𝐵 ∧ ( 𝑃 𝑄 ) ∈ 𝐵𝑋𝐵 ) ) → ( 𝑄 ( 𝑃 𝑄 ) → ( 𝑋 𝑄 ) ( 𝑋 ( 𝑃 𝑄 ) ) ) )
43 12 18 41 22 42 syl13anc ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( 𝑄 ( 𝑃 𝑄 ) → ( 𝑋 𝑄 ) ( 𝑋 ( 𝑃 𝑄 ) ) ) )
44 39 43 mpd ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( 𝑋 𝑄 ) ( 𝑋 ( 𝑃 𝑄 ) ) )
45 44 adantr ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ 𝑃 ( 𝑋 𝑄 ) ) → ( 𝑋 𝑄 ) ( 𝑋 ( 𝑃 𝑄 ) ) )
46 1 3 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑃 𝑄 ) ∈ 𝐵 ) → ( 𝑋 ( 𝑃 𝑄 ) ) ∈ 𝐵 )
47 12 22 41 46 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) ∈ 𝐵 )
48 1 2 latasymb ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ( 𝑃 𝑄 ) ) ∈ 𝐵 ∧ ( 𝑋 𝑄 ) ∈ 𝐵 ) → ( ( ( 𝑋 ( 𝑃 𝑄 ) ) ( 𝑋 𝑄 ) ∧ ( 𝑋 𝑄 ) ( 𝑋 ( 𝑃 𝑄 ) ) ) ↔ ( 𝑋 ( 𝑃 𝑄 ) ) = ( 𝑋 𝑄 ) ) )
49 12 47 28 48 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( ( ( 𝑋 ( 𝑃 𝑄 ) ) ( 𝑋 𝑄 ) ∧ ( 𝑋 𝑄 ) ( 𝑋 ( 𝑃 𝑄 ) ) ) ↔ ( 𝑋 ( 𝑃 𝑄 ) ) = ( 𝑋 𝑄 ) ) )
50 49 adantr ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ 𝑃 ( 𝑋 𝑄 ) ) → ( ( ( 𝑋 ( 𝑃 𝑄 ) ) ( 𝑋 𝑄 ) ∧ ( 𝑋 𝑄 ) ( 𝑋 ( 𝑃 𝑄 ) ) ) ↔ ( 𝑋 ( 𝑃 𝑄 ) ) = ( 𝑋 𝑄 ) ) )
51 36 45 50 mpbi2and ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ 𝑃 ( 𝑋 𝑄 ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) = ( 𝑋 𝑄 ) )
52 51 breq2d ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ 𝑃 ( 𝑋 𝑄 ) ) → ( 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ( 𝑃 𝑄 ) ) ↔ 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 𝑄 ) ) )
53 52 adantrl ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ ( ¬ 𝑄 𝑋𝑃 ( 𝑋 𝑄 ) ) ) → ( 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ( 𝑃 𝑄 ) ) ↔ 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 𝑄 ) ) )
54 10 53 mpbird ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ ( ¬ 𝑄 𝑋𝑃 ( 𝑋 𝑄 ) ) ) → 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ( 𝑃 𝑄 ) ) )
55 54 ex ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( ( ¬ 𝑄 𝑋𝑃 ( 𝑋 𝑄 ) ) → 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ( 𝑃 𝑄 ) ) ) )
56 1 3 4 6 cvrexch ( ( 𝐾 ∈ HL ∧ 𝑋𝐵 ∧ ( 𝑃 𝑄 ) ∈ 𝐵 ) → ( ( 𝑋 ( 𝑃 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 𝑄 ) ↔ 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ( 𝑃 𝑄 ) ) ) )
57 37 22 41 56 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( ( 𝑋 ( 𝑃 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 𝑄 ) ↔ 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ( 𝑃 𝑄 ) ) ) )
58 55 57 sylibrd ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( ( ¬ 𝑄 𝑋𝑃 ( 𝑋 𝑄 ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 𝑄 ) ) )
59 58 adantr ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ 𝑃𝑄 ) → ( ( ¬ 𝑄 𝑋𝑃 ( 𝑋 𝑄 ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 𝑄 ) ) )
60 1 4 latmcl ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑃 𝑄 ) ∈ 𝐵 ) → ( 𝑋 ( 𝑃 𝑄 ) ) ∈ 𝐵 )
61 12 22 41 60 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) ∈ 𝐵 )
62 1 3 6 5 cvrat2 ( ( 𝐾 ∈ HL ∧ ( ( 𝑋 ( 𝑃 𝑄 ) ) ∈ 𝐵𝑃𝐴𝑄𝐴 ) ∧ ( 𝑃𝑄 ∧ ( 𝑋 ( 𝑃 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 𝑄 ) ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) ∈ 𝐴 )
63 62 3expia ( ( 𝐾 ∈ HL ∧ ( ( 𝑋 ( 𝑃 𝑄 ) ) ∈ 𝐵𝑃𝐴𝑄𝐴 ) ) → ( ( 𝑃𝑄 ∧ ( 𝑋 ( 𝑃 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 𝑄 ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) ∈ 𝐴 ) )
64 37 61 13 16 63 syl13anc ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( ( 𝑃𝑄 ∧ ( 𝑋 ( 𝑃 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 𝑄 ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) ∈ 𝐴 ) )
65 64 expdimp ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ 𝑃𝑄 ) → ( ( 𝑋 ( 𝑃 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 𝑄 ) → ( 𝑋 ( 𝑃 𝑄 ) ) ∈ 𝐴 ) )
66 59 65 syld ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) ∧ 𝑃𝑄 ) → ( ( ¬ 𝑄 𝑋𝑃 ( 𝑋 𝑄 ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) ∈ 𝐴 ) )
67 66 exp4b ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( 𝑃𝑄 → ( ¬ 𝑄 𝑋 → ( 𝑃 ( 𝑋 𝑄 ) → ( 𝑋 ( 𝑃 𝑄 ) ) ∈ 𝐴 ) ) ) )
68 67 3impd ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐴𝑄𝐴 ) ) → ( ( 𝑃𝑄 ∧ ¬ 𝑄 𝑋𝑃 ( 𝑋 𝑄 ) ) → ( 𝑋 ( 𝑃 𝑄 ) ) ∈ 𝐴 ) )