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 ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( ( 𝐹𝐺 ) ‘ 𝑃 ) ( 𝑅𝐹 ) ) = ( ( 𝐺𝑃 ) ( 𝑅𝐹 ) ) )

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 ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑃𝐴 ) → ( ( 𝐹𝐺 ) ‘ 𝑃 ) = ( 𝐹 ‘ ( 𝐺𝑃 ) ) )
8 7 3adant3r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝐹𝐺 ) ‘ 𝑃 ) = ( 𝐹 ‘ ( 𝐺𝑃 ) ) )
9 8 oveq1d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( ( 𝐹𝐺 ) ‘ 𝑃 ) ( 𝑅𝐹 ) ) = ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝑅𝐹 ) ) )
10 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
11 simp2l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝐹𝑇 )
12 1 3 4 5 ltrnel ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐺𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝐺𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺𝑃 ) 𝑊 ) )
13 12 3adant2l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝐺𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺𝑃 ) 𝑊 ) )
14 1 2 3 4 5 6 trljat3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( ( 𝐺𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺𝑃 ) 𝑊 ) ) → ( ( 𝐺𝑃 ) ( 𝑅𝐹 ) ) = ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝑅𝐹 ) ) )
15 10 11 13 14 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝐺𝑃 ) ( 𝑅𝐹 ) ) = ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝑅𝐹 ) ) )
16 9 15 eqtr4d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( ( 𝐹𝐺 ) ‘ 𝑃 ) ( 𝑅𝐹 ) ) = ( ( 𝐺𝑃 ) ( 𝑅𝐹 ) ) )