Theorem cycsubggenodd

Description: Relationship between the order of a subgroup and the order of a generator of the subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	cycsubggenodd.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
	cycsubggenodd.2	⊢ · = ( .g ‘ 𝐺 )
	cycsubggenodd.3	⊢ 𝑂 = ( od ‘ 𝐺 )
	cycsubggenodd.4	⊢ ( 𝜑 → 𝐺 ∈ Grp )
	cycsubggenodd.5	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
	cycsubggenodd.6	⊢ ( 𝜑 → 𝐶 = ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝐴 ) ) )
Assertion	cycsubggenodd	⊢ ( 𝜑 → ( 𝑂 ‘ 𝐴 ) = if ( 𝐶 ∈ Fin , ( ♯ ‘ 𝐶 ) , 0 ) )

Proof

Step	Hyp	Ref	Expression
1		cycsubggenodd.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		cycsubggenodd.2	⊢ · = ( .g ‘ 𝐺 )
3		cycsubggenodd.3	⊢ 𝑂 = ( od ‘ 𝐺 )
4		cycsubggenodd.4	⊢ ( 𝜑 → 𝐺 ∈ Grp )
5		cycsubggenodd.5	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
6		cycsubggenodd.6	⊢ ( 𝜑 → 𝐶 = ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝐴 ) ) )
7		eqid	⊢ ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝐴 ) ) = ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝐴 ) )
8	1 3 2 7	dfod2	⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵 ) → ( 𝑂 ‘ 𝐴 ) = if ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝐴 ) ) ∈ Fin , ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝐴 ) ) ) , 0 ) )
9	4 5 8	syl2anc	⊢ ( 𝜑 → ( 𝑂 ‘ 𝐴 ) = if ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝐴 ) ) ∈ Fin , ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝐴 ) ) ) , 0 ) )
10	6	eqcomd	⊢ ( 𝜑 → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝐴 ) ) = 𝐶 )
11	10	eleq1d	⊢ ( 𝜑 → ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝐴 ) ) ∈ Fin ↔ 𝐶 ∈ Fin ) )
12	10	fveq2d	⊢ ( 𝜑 → ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝐴 ) ) ) = ( ♯ ‘ 𝐶 ) )
13	11 12	ifbieq1d	⊢ ( 𝜑 → if ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝐴 ) ) ∈ Fin , ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝐴 ) ) ) , 0 ) = if ( 𝐶 ∈ Fin , ( ♯ ‘ 𝐶 ) , 0 ) )
14	9 13	eqtrd	⊢ ( 𝜑 → ( 𝑂 ‘ 𝐴 ) = if ( 𝐶 ∈ Fin , ( ♯ ‘ 𝐶 ) , 0 ) )