Theorem cyggenod2

Description: In an infinite cyclic group, the generator must have infinite order, but this property no longer characterizes the generators. (Contributed by Mario Carneiro, 21-Apr-2016)

	Ref	Expression
Hypotheses	iscyg.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
	iscyg.2	⊢ · = ( .g ‘ 𝐺 )
	iscyg3.e	⊢ 𝐸 = { 𝑥 ∈ 𝐵 ∣ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 }
	cyggenod.o	⊢ 𝑂 = ( od ‘ 𝐺 )
Assertion	cyggenod2	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸 ) → ( 𝑂 ‘ 𝑋 ) = if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 ) )

Proof

Step	Hyp	Ref	Expression
1		iscyg.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		iscyg.2	⊢ · = ( .g ‘ 𝐺 )
3		iscyg3.e	⊢ 𝐸 = { 𝑥 ∈ 𝐵 ∣ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 }
4		cyggenod.o	⊢ 𝑂 = ( od ‘ 𝐺 )
5	1 2 3	iscyggen	⊢ ( 𝑋 ∈ 𝐸 ↔ ( 𝑋 ∈ 𝐵 ∧ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) = 𝐵 ) )
6	5	simplbi	⊢ ( 𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐵 )
7		eqid	⊢ ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) = ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) )
8	1 4 2 7	dfod2	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑂 ‘ 𝑋 ) = if ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ∈ Fin , ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ) , 0 ) )
9	6 8	sylan2	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸 ) → ( 𝑂 ‘ 𝑋 ) = if ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ∈ Fin , ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ) , 0 ) )
10	5	simprbi	⊢ ( 𝑋 ∈ 𝐸 → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) = 𝐵 )
11	10	adantl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸 ) → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) = 𝐵 )
12	11	eleq1d	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸 ) → ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ∈ Fin ↔ 𝐵 ∈ Fin ) )
13	11	fveq2d	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸 ) → ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ) = ( ♯ ‘ 𝐵 ) )
14	12 13	ifbieq1d	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸 ) → if ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ∈ Fin , ( ♯ ‘ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑋 ) ) ) , 0 ) = if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 ) )
15	9 14	eqtrd	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸 ) → ( 𝑂 ‘ 𝑋 ) = if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 ) )