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Metamath Proof Explorer

Theorem dihmeetlem5

Description: Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014)

Ref Expression
Hypotheses dihmeetlem5.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihmeetlem5.l ⊢ ≤ = ( le ‘ 𝐾 )
dihmeetlem5.j ⊢ ∨ = ( join ‘ 𝐾 )
dihmeetlem5.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihmeetlem5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion dihmeetlem5 ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) )

Proof

Step Hyp Ref Expression
1 dihmeetlem5.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihmeetlem5.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihmeetlem5.j ⊢ ∨ = ( join ‘ 𝐾 )
4 dihmeetlem5.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 dihmeetlem5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 simpl1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → 𝐾 ∈ HL )
7 simprl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → 𝑄 ∈ 𝐴 )
8 simpl2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → 𝑋 ∈ 𝐵 )
9 simpl3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → 𝑌 ∈ 𝐵 )
10 simprr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → 𝑄 ≤ 𝑋 )
11 1 2 3 4 5 atmod2i1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑄 ≤ 𝑋 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) = ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) )
12 6 7 8 9 10 11 syl131anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) = ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) )
13 12 eqcomd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) )