Theorem cyc3fv3

Description: Function value of a 3-cycle at the third point. (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	cyc3fv3	$$⊢\phi \to C\left(⟨“IJK”⟩\right)\left(K\right)=I$$

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		3pos	$$⊢0<3$$
13		s3len	$$⊢\left|⟨“IJK”⟩\right|=3$$
14	12 13	breqtrri	$$⊢0<\left|⟨“IJK”⟩\right|$$
15	14	a1i	$$⊢\phi \to 0<\left|⟨“IJK”⟩\right|$$
16	13	oveq1i	$$⊢\left|⟨“IJK”⟩\right|-1=3-1$$
17		3m1e2	$$⊢3-1=2$$
18	16 17	eqtr2i	$$⊢2=\left|⟨“IJK”⟩\right|-1$$
19	18	a1i	$$⊢\phi \to 2=\left|⟨“IJK”⟩\right|-1$$
20	1 3 10 11 15 19	cycpmfv2	$$⊢\phi \to C\left(⟨“IJK”⟩\right)\left(⟨“IJK”⟩\left(2\right)\right)=⟨“IJK”⟩\left(0\right)$$
21		s3fv2	$$⊢K\in D\to ⟨“IJK”⟩\left(2\right)=K$$
22	6 21	syl	$$⊢\phi \to ⟨“IJK”⟩\left(2\right)=K$$
23	22	fveq2d	$$⊢\phi \to C\left(⟨“IJK”⟩\right)\left(⟨“IJK”⟩\left(2\right)\right)=C\left(⟨“IJK”⟩\right)\left(K\right)$$
24		s3fv0	$$⊢I\in D\to ⟨“IJK”⟩\left(0\right)=I$$
25	4 24	syl	$$⊢\phi \to ⟨“IJK”⟩\left(0\right)=I$$
26	20 23 25	3eqtr3d	$$⊢\phi \to C\left(⟨“IJK”⟩\right)\left(K\right)=I$$