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Metamath Proof Explorer

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 ) ) ) )