Metamath Proof Explorer

Theorem llnmod2i2

Description: Version of modular law pmod1i that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join P .\/ Q ). (Contributed by NM, 16-Sep-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 llnmod2i2 ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∨ 𝑌 ) = ( 𝑋 ∧ ( ( 𝑃 ∨ 𝑄 ) ∨ 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 simp11 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → 𝐾 ∈ HL )
7 6 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → 𝐾 ∈ Lat )
8 simp13 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → 𝑌 ∈ 𝐵 )
9 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → 𝑃 ∈ 𝐴 )
10 simp2r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → 𝑄 ∈ 𝐴 )
11 1 3 5 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 )
12 6 9 10 11 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 )
13 simp12 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → 𝑋 ∈ 𝐵 )
14 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ∈ 𝐵 )
15 7 12 13 14 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ∈ 𝐵 )
16 1 3 latjcom ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ∈ 𝐵 ) → ( 𝑌 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ) = ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ∨ 𝑌 ) )
17 7 8 15 16 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → ( 𝑌 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ) = ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ∨ 𝑌 ) )
18 1 3 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) → ( 𝑌 ∨ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐵 )
19 7 8 12 18 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → ( 𝑌 ∨ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐵 )
20 1 4 latmcom ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑌 ∨ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐵 ) → ( 𝑋 ∧ ( 𝑌 ∨ ( 𝑃 ∨ 𝑄 ) ) ) = ( ( 𝑌 ∨ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑋 ) )
21 7 13 19 20 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → ( 𝑋 ∧ ( 𝑌 ∨ ( 𝑃 ∨ 𝑄 ) ) ) = ( ( 𝑌 ∨ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑋 ) )
22 1 3 latjcom ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑌 ) = ( 𝑌 ∨ ( 𝑃 ∨ 𝑄 ) ) )
23 7 12 8 22 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑌 ) = ( 𝑌 ∨ ( 𝑃 ∨ 𝑄 ) ) )
24 23 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → ( 𝑋 ∧ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑌 ) ) = ( 𝑋 ∧ ( 𝑌 ∨ ( 𝑃 ∨ 𝑄 ) ) ) )
25 simp3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → 𝑌 ≤ 𝑋 )
26 1 2 3 4 5 llnmod1i2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → ( 𝑌 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ) = ( ( 𝑌 ∨ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑋 ) )
27 6 8 13 9 10 25 26 syl321anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → ( 𝑌 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ) = ( ( 𝑌 ∨ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑋 ) )
28 21 24 27 3eqtr4d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → ( 𝑋 ∧ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑌 ) ) = ( 𝑌 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ) )
29 1 4 latmcom ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) )
30 7 13 12 29 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) )
31 30 oveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∨ 𝑌 ) = ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ∨ 𝑌 ) )
32 17 28 31 3eqtr4rd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∨ 𝑌 ) = ( 𝑋 ∧ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑌 ) ) )