Theorem trlcocnv

Description: Swap the arguments of the trace of a composition with converse. (Contributed by NM, 1-Jul-2013)

	Ref	Expression
Hypotheses	trlcocnv.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	trlcocnv.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	trlcocnv.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion	trlcocnv	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) = ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) )

Proof

Step	Hyp	Ref	Expression
1		trlcocnv.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		trlcocnv.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		trlcocnv.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
4		simp1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
5	1 2	ltrncnv	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → ◡ 𝐺 ∈ 𝑇 )
6	5	3adant2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ◡ 𝐺 ∈ 𝑇 )
7	1 2	ltrnco	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ◡ 𝐺 ∈ 𝑇 ) → ( 𝐹 ∘ ◡ 𝐺 ) ∈ 𝑇 )
8	6 7	syld3an3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐹 ∘ ◡ 𝐺 ) ∈ 𝑇 )
9	1 2 3	trlcnv	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ∈ 𝑇 ) → ( 𝑅 ‘ ◡ ( 𝐹 ∘ ◡ 𝐺 ) ) = ( 𝑅 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) )
10	4 8 9	syl2anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑅 ‘ ◡ ( 𝐹 ∘ ◡ 𝐺 ) ) = ( 𝑅 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) )
11		cnvco	⊢ ◡ ( 𝐹 ∘ ◡ 𝐺 ) = ( ◡ ◡ 𝐺 ∘ ◡ 𝐹 )
12		cocnvcnv1	⊢ ( ◡ ◡ 𝐺 ∘ ◡ 𝐹 ) = ( 𝐺 ∘ ◡ 𝐹 )
13	11 12	eqtri	⊢ ◡ ( 𝐹 ∘ ◡ 𝐺 ) = ( 𝐺 ∘ ◡ 𝐹 )
14	13	fveq2i	⊢ ( 𝑅 ‘ ◡ ( 𝐹 ∘ ◡ 𝐺 ) ) = ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) )
15	10 14	eqtr3di	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) = ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) )