Metamath Proof Explorer


Theorem 1cvrat

Description: Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in Crawley p. 112. (Contributed by NM, 30-Apr-2012)

Ref Expression
Hypotheses 1cvrat.b 𝐵 = ( Base ‘ 𝐾 )
1cvrat.l = ( le ‘ 𝐾 )
1cvrat.j = ( join ‘ 𝐾 )
1cvrat.m = ( meet ‘ 𝐾 )
1cvrat.u 1 = ( 1. ‘ 𝐾 )
1cvrat.c 𝐶 = ( ⋖ ‘ 𝐾 )
1cvrat.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion 1cvrat ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → ( ( 𝑃 𝑄 ) 𝑋 ) ∈ 𝐴 )

Proof

Step Hyp Ref Expression
1 1cvrat.b 𝐵 = ( Base ‘ 𝐾 )
2 1cvrat.l = ( le ‘ 𝐾 )
3 1cvrat.j = ( join ‘ 𝐾 )
4 1cvrat.m = ( meet ‘ 𝐾 )
5 1cvrat.u 1 = ( 1. ‘ 𝐾 )
6 1cvrat.c 𝐶 = ( ⋖ ‘ 𝐾 )
7 1cvrat.a 𝐴 = ( Atoms ‘ 𝐾 )
8 hllat ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
9 8 3ad2ant1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → 𝐾 ∈ Lat )
10 simp21 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → 𝑃𝐴 )
11 1 7 atbase ( 𝑃𝐴𝑃𝐵 )
12 10 11 syl ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → 𝑃𝐵 )
13 simp22 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → 𝑄𝐴 )
14 1 7 atbase ( 𝑄𝐴𝑄𝐵 )
15 13 14 syl ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → 𝑄𝐵 )
16 1 3 latjcom ( ( 𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵 ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
17 9 12 15 16 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
18 17 oveq1d ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → ( ( 𝑃 𝑄 ) 𝑋 ) = ( ( 𝑄 𝑃 ) 𝑋 ) )
19 1 3 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑄𝐵𝑃𝐵 ) → ( 𝑄 𝑃 ) ∈ 𝐵 )
20 9 15 12 19 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → ( 𝑄 𝑃 ) ∈ 𝐵 )
21 simp23 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → 𝑋𝐵 )
22 1 4 latmcom ( ( 𝐾 ∈ Lat ∧ ( 𝑄 𝑃 ) ∈ 𝐵𝑋𝐵 ) → ( ( 𝑄 𝑃 ) 𝑋 ) = ( 𝑋 ( 𝑄 𝑃 ) ) )
23 9 20 21 22 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → ( ( 𝑄 𝑃 ) 𝑋 ) = ( 𝑋 ( 𝑄 𝑃 ) ) )
24 18 23 eqtrd ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → ( ( 𝑃 𝑄 ) 𝑋 ) = ( 𝑋 ( 𝑄 𝑃 ) ) )
25 simp1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → 𝐾 ∈ HL )
26 21 13 10 3jca ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → ( 𝑋𝐵𝑄𝐴𝑃𝐴 ) )
27 simp31 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → 𝑃𝑄 )
28 27 necomd ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → 𝑄𝑃 )
29 simp33 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → ¬ 𝑃 𝑋 )
30 hlop ( 𝐾 ∈ HL → 𝐾 ∈ OP )
31 30 3ad2ant1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → 𝐾 ∈ OP )
32 1 2 5 ople1 ( ( 𝐾 ∈ OP ∧ 𝑄𝐵 ) → 𝑄 1 )
33 31 15 32 syl2anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → 𝑄 1 )
34 simp32 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → 𝑋 𝐶 1 )
35 1 2 3 5 6 7 1cvrjat ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴 ) ∧ ( 𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → ( 𝑋 𝑃 ) = 1 )
36 25 21 10 34 29 35 syl32anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → ( 𝑋 𝑃 ) = 1 )
37 33 36 breqtrrd ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → 𝑄 ( 𝑋 𝑃 ) )
38 1 2 3 4 7 cvrat3 ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑄𝐴𝑃𝐴 ) ) → ( ( 𝑄𝑃 ∧ ¬ 𝑃 𝑋𝑄 ( 𝑋 𝑃 ) ) → ( 𝑋 ( 𝑄 𝑃 ) ) ∈ 𝐴 ) )
39 38 imp ( ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑄𝐴𝑃𝐴 ) ) ∧ ( 𝑄𝑃 ∧ ¬ 𝑃 𝑋𝑄 ( 𝑋 𝑃 ) ) ) → ( 𝑋 ( 𝑄 𝑃 ) ) ∈ 𝐴 )
40 25 26 28 29 37 39 syl23anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → ( 𝑋 ( 𝑄 𝑃 ) ) ∈ 𝐴 )
41 24 40 eqeltrd ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑋𝐵 ) ∧ ( 𝑃𝑄𝑋 𝐶 1 ∧ ¬ 𝑃 𝑋 ) ) → ( ( 𝑃 𝑄 ) 𝑋 ) ∈ 𝐴 )