Metamath Proof Explorer

Theorem atmod4i1

Description: Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-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 atmod4i1 ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑌 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑃 ) = ( ( 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 hllat ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
7 6 3ad2ant1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑌 ) → 𝐾 ∈ Lat )
8 simp22 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑌 ) → 𝑋 ∈ 𝐵 )
9 simp23 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑌 ) → 𝑌 ∈ 𝐵 )
10 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
11 7 8 9 10 syl3anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑌 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
12 simp21 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑌 ) → 𝑃 ∈ 𝐴 )
13 1 5 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 )
14 12 13 syl ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑌 ) → 𝑃 ∈ 𝐵 )
15 1 3 latjcom ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑃 ) = ( 𝑃 ∨ ( 𝑋 ∧ 𝑌 ) ) )
16 7 11 14 15 syl3anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑌 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑃 ) = ( 𝑃 ∨ ( 𝑋 ∧ 𝑌 ) ) )
17 1 2 3 4 5 atmod1i1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑌 ) → ( 𝑃 ∨ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝑃 ∨ 𝑋 ) ∧ 𝑌 ) )
18 1 3 latjcom ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑃 ∨ 𝑋 ) = ( 𝑋 ∨ 𝑃 ) )
19 7 14 8 18 syl3anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑌 ) → ( 𝑃 ∨ 𝑋 ) = ( 𝑋 ∨ 𝑃 ) )
20 19 oveq1d ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑌 ) → ( ( 𝑃 ∨ 𝑋 ) ∧ 𝑌 ) = ( ( 𝑋 ∨ 𝑃 ) ∧ 𝑌 ) )
21 16 17 20 3eqtrd ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑌 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑃 ) = ( ( 𝑋 ∨ 𝑃 ) ∧ 𝑌 ) )