Theorem cycpm3cl

Description: Closure of the 3-cycles in the permutations. (Contributed by Thierry Arnoux, 19-Sep-2023)

	Ref	Expression
Hypotheses	cycpm3.c	$$⊢C=\mathrm{toCyc}\left(D\right)$$
	cycpm3.s	$$⊢S=\mathrm{SymGrp}\left(D\right)$$
	cycpm3.d	$$⊢\phi \to D\in V$$
	cycpm3.i	$$⊢\phi \to I\in D$$
	cycpm3.j	$$⊢\phi \to J\in D$$
	cycpm3.k	$$⊢\phi \to K\in D$$
	cycpm3.1	$$⊢\phi \to I\ne J$$
	cycpm3.2	$$⊢\phi \to J\ne K$$
	cycpm3.3	$$⊢\phi \to K\ne I$$
Assertion	cycpm3cl	$$⊢\phi \to C\left(⟨“IJK”⟩\right)\in {\mathrm{Base}}_S$$

Proof

Step	Hyp	Ref	Expression
1		cycpm3.c	$$⊢C=\mathrm{toCyc}\left(D\right)$$
2		cycpm3.s	$$⊢S=\mathrm{SymGrp}\left(D\right)$$
3		cycpm3.d	$$⊢\phi \to D\in V$$
4		cycpm3.i	$$⊢\phi \to I\in D$$
5		cycpm3.j	$$⊢\phi \to J\in D$$
6		cycpm3.k	$$⊢\phi \to K\in D$$
7		cycpm3.1	$$⊢\phi \to I\ne J$$
8		cycpm3.2	$$⊢\phi \to J\ne K$$
9		cycpm3.3	$$⊢\phi \to K\ne I$$
10	4 5 6	s3cld	$$⊢\phi \to ⟨“IJK”⟩\in \mathrm{Word}D$$
11	4 5 6 7 8 9	s3f1	$$⊢\phi \to ⟨“IJK”⟩:\mathrm{dom}⟨“IJK”⟩\underset{1-1}{⟶}D$$
12	1 3 10 11 2	cycpmcl	$$⊢\phi \to C\left(⟨“IJK”⟩\right)\in {\mathrm{Base}}_S$$