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		cnvco	⊢ ◡ ( 𝐹 ∘ ◡ 𝐺 ) = ( ◡ ◡ 𝐺 ∘ ◡ 𝐹 )
5		cocnvcnv1	⊢ ( ◡ ◡ 𝐺 ∘ ◡ 𝐹 ) = ( 𝐺 ∘ ◡ 𝐹 )
6	4 5	eqtri	⊢ ◡ ( 𝐹 ∘ ◡ 𝐺 ) = ( 𝐺 ∘ ◡ 𝐹 )
7	6	fveq2i	⊢ ( 𝑅 ‘ ◡ ( 𝐹 ∘ ◡ 𝐺 ) ) = ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) )
8		simp1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9	1 2	ltrncnv	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → ◡ 𝐺 ∈ 𝑇 )
10	9	3adant2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ◡ 𝐺 ∈ 𝑇 )
11	1 2	ltrnco	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ◡ 𝐺 ∈ 𝑇 ) → ( 𝐹 ∘ ◡ 𝐺 ) ∈ 𝑇 )
12	10 11	syld3an3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐹 ∘ ◡ 𝐺 ) ∈ 𝑇 )
13	1 2 3	trlcnv	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ∈ 𝑇 ) → ( 𝑅 ‘ ◡ ( 𝐹 ∘ ◡ 𝐺 ) ) = ( 𝑅 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) )
14	8 12 13	syl2anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑅 ‘ ◡ ( 𝐹 ∘ ◡ 𝐺 ) ) = ( 𝑅 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) )
15	7 14	syl5reqr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) = ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) )