Metamath Proof Explorer

Theorem llnmod1i2

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 llnmod1i2 ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑋 ≤ 𝑌 ) → ( 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 simpl1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝐾 ∈ HL )
7 simpl2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑋 ∈ 𝐵 )
8 simprl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐴 )
9 simprr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑄 ∈ 𝐴 )
10 eqid ⊢ ( pmap ‘ 𝐾 ) = ( pmap ‘ 𝐾 )
11 eqid ⊢ ( +𝑃 ‘ 𝐾 ) = ( +𝑃 ‘ 𝐾 )
12 1 3 5 10 11 pmapjlln1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( pmap ‘ 𝐾 ) ‘ ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ) = ( ( ( pmap ‘ 𝐾 ) ‘ 𝑋 ) ( +𝑃 ‘ 𝐾 ) ( ( pmap ‘ 𝐾 ) ‘ ( 𝑃 ∨ 𝑄 ) ) ) )
13 6 7 8 9 12 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( pmap ‘ 𝐾 ) ‘ ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ) = ( ( ( pmap ‘ 𝐾 ) ‘ 𝑋 ) ( +𝑃 ‘ 𝐾 ) ( ( pmap ‘ 𝐾 ) ‘ ( 𝑃 ∨ 𝑄 ) ) ) )
14 6 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝐾 ∈ Lat )
15 1 5 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 )
16 8 15 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐵 )
17 1 5 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 )
18 9 17 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑄 ∈ 𝐵 )
19 1 3 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 )
20 14 16 18 19 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 )
21 simpl3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑌 ∈ 𝐵 )
22 1 2 3 4 10 11 hlmod1i ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ ( ( pmap ‘ 𝐾 ) ‘ ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ) = ( ( ( pmap ‘ 𝐾 ) ‘ 𝑋 ) ( +𝑃 ‘ 𝐾 ) ( ( pmap ‘ 𝐾 ) ‘ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑌 ) = ( 𝑋 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑌 ) ) ) )
23 6 7 20 21 22 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ ( ( pmap ‘ 𝐾 ) ‘ ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ) = ( ( ( pmap ‘ 𝐾 ) ‘ 𝑋 ) ( +𝑃 ‘ 𝐾 ) ( ( pmap ‘ 𝐾 ) ‘ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑌 ) = ( 𝑋 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑌 ) ) ) )
24 13 23 mpan2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 ≤ 𝑌 → ( ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑌 ) = ( 𝑋 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑌 ) ) ) )
25 24 3impia ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑋 ≤ 𝑌 ) → ( ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑌 ) = ( 𝑋 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑌 ) ) )
26 25 eqcomd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑌 ) ) = ( ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑌 ) )