Metamath Proof Explorer


Theorem atmod3i1

Description: Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 4-Jun-2012) (Revised by Mario Carneiro, 10-May-2013)

Ref Expression
Hypotheses atmod.b 𝐵 = ( Base ‘ 𝐾 )
atmod.l = ( le ‘ 𝐾 )
atmod.j = ( join ‘ 𝐾 )
atmod.m = ( meet ‘ 𝐾 )
atmod.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion atmod3i1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑃 𝑋 ) → ( 𝑃 ( 𝑋 𝑌 ) ) = ( 𝑋 ( 𝑃 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 atmod.b 𝐵 = ( Base ‘ 𝐾 )
2 atmod.l = ( le ‘ 𝐾 )
3 atmod.j = ( join ‘ 𝐾 )
4 atmod.m = ( meet ‘ 𝐾 )
5 atmod.a 𝐴 = ( Atoms ‘ 𝐾 )
6 simp1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑃 𝑋 ) → 𝐾 ∈ HL )
7 simp21 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑃 𝑋 ) → 𝑃𝐴 )
8 simp23 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑃 𝑋 ) → 𝑌𝐵 )
9 simp22 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑃 𝑋 ) → 𝑋𝐵 )
10 simp3 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑃 𝑋 ) → 𝑃 𝑋 )
11 1 2 3 4 5 atmod1i1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑌𝐵𝑋𝐵 ) ∧ 𝑃 𝑋 ) → ( 𝑃 ( 𝑌 𝑋 ) ) = ( ( 𝑃 𝑌 ) 𝑋 ) )
12 6 7 8 9 10 11 syl131anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑃 𝑋 ) → ( 𝑃 ( 𝑌 𝑋 ) ) = ( ( 𝑃 𝑌 ) 𝑋 ) )
13 hllat ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
14 13 3ad2ant1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑃 𝑋 ) → 𝐾 ∈ Lat )
15 1 4 latmcom ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 𝑌 ) = ( 𝑌 𝑋 ) )
16 14 9 8 15 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑃 𝑋 ) → ( 𝑋 𝑌 ) = ( 𝑌 𝑋 ) )
17 16 oveq2d ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑃 𝑋 ) → ( 𝑃 ( 𝑋 𝑌 ) ) = ( 𝑃 ( 𝑌 𝑋 ) ) )
18 1 5 atbase ( 𝑃𝐴𝑃𝐵 )
19 7 18 syl ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑃 𝑋 ) → 𝑃𝐵 )
20 1 3 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑃𝐵𝑌𝐵 ) → ( 𝑃 𝑌 ) ∈ 𝐵 )
21 14 19 8 20 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑃 𝑋 ) → ( 𝑃 𝑌 ) ∈ 𝐵 )
22 1 4 latmcom ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑃 𝑌 ) ∈ 𝐵 ) → ( 𝑋 ( 𝑃 𝑌 ) ) = ( ( 𝑃 𝑌 ) 𝑋 ) )
23 14 9 21 22 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑃 𝑋 ) → ( 𝑋 ( 𝑃 𝑌 ) ) = ( ( 𝑃 𝑌 ) 𝑋 ) )
24 12 17 23 3eqtr4d ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑃 𝑋 ) → ( 𝑃 ( 𝑋 𝑌 ) ) = ( 𝑋 ( 𝑃 𝑌 ) ) )