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Metamath Proof Explorer

Theorem trlcoabs

Description: Absorption into a composition by joining with trace. (Contributed by NM, 22-Jul-2013)

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

Proof

Step Hyp Ref Expression
1 trlcoabs.l ⊢ ≤ = ( le ‘ 𝐾 )
2 trlcoabs.j ⊢ ∨ = ( join ‘ 𝐾 )
3 trlcoabs.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 trlcoabs.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 trlcoabs.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 trlcoabs.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
7 1 3 4 5 ltrncoval ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) )
8 7 3adant3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) )
9 8 oveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝑅 ‘ 𝐹 ) ) )
10 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
12 1 3 4 5 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
13 12 3adant2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
14 1 2 3 4 5 6 trljat3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝑅 ‘ 𝐹 ) ) )
15 10 11 13 14 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝑅 ‘ 𝐹 ) ) )
16 9 15 eqtr4d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) )