Theorem dihmeetlem8N

Description: Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change .\/ order of ( X ./\ Y ) .\/ p here and down? (Contributed by NM, 6-Apr-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dihmeetlem8.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dihmeetlem8.l	⊢ ≤ = ( le ‘ 𝐾 )
	dihmeetlem8.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihmeetlem8.j	⊢ ∨ = ( join ‘ 𝐾 )
	dihmeetlem8.m	⊢ ∧ = ( meet ‘ 𝐾 )
	dihmeetlem8.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	dihmeetlem8.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	dihmeetlem8.s	⊢ ⊕ = ( LSSum ‘ 𝑈 )
	dihmeetlem8.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dihmeetlem8N	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ) = ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		dihmeetlem8.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dihmeetlem8.l	⊢ ≤ = ( le ‘ 𝐾 )
3		dihmeetlem8.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		dihmeetlem8.j	⊢ ∨ = ( join ‘ 𝐾 )
5		dihmeetlem8.m	⊢ ∧ = ( meet ‘ 𝐾 )
6		dihmeetlem8.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
7		dihmeetlem8.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
8		dihmeetlem8.s	⊢ ⊕ = ( LSSum ‘ 𝑈 )
9		dihmeetlem8.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
10	1 2 3 4 5 6 7 8 9	dihjatc1	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ) = ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) )