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 ) )