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Metamath Proof Explorer

Theorem trlval5

Description: The value of the trace of a lattice translation in terms of itself. (Contributed by NM, 19-Jul-2013)

Ref Expression
Hypotheses trlval3.l ⊢ ≤ = ( le ‘ 𝐾 )
trlval3.j ⊢ ∨ = ( join ‘ 𝐾 )
trlval3.m ⊢ ∧ = ( meet ‘ 𝐾 )
trlval3.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
trlval3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
trlval3.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
trlval3.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion trlval5 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ 𝑊 ) )

Proof

Step Hyp Ref Expression
1 trlval3.l ⊢ ≤ = ( le ‘ 𝐾 )
2 trlval3.j ⊢ ∨ = ( join ‘ 𝐾 )
3 trlval3.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 trlval3.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 trlval3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 trlval3.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 trlval3.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 1 2 3 4 5 6 7 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) )
9 1 2 4 5 6 7 trljat1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
10 9 oveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ 𝑊 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) )
11 8 10 eqtr4d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ 𝑊 ) )