Theorem cyggex

Description: The exponent of a finite cyclic group is the order of the group. (Contributed by Mario Carneiro, 24-Apr-2016)

	Ref	Expression
Hypotheses	cygctb.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
	cyggex.o	⊢ 𝐸 = ( gEx ‘ 𝐺 )
Assertion	cyggex	⊢ ( ( 𝐺 ∈ CycGrp ∧ 𝐵 ∈ Fin ) → 𝐸 = ( ♯ ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		cygctb.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		cyggex.o	⊢ 𝐸 = ( gEx ‘ 𝐺 )
3	1 2	cyggex2	⊢ ( 𝐺 ∈ CycGrp → 𝐸 = if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 ) )
4		iftrue	⊢ ( 𝐵 ∈ Fin → if ( 𝐵 ∈ Fin , ( ♯ ‘ 𝐵 ) , 0 ) = ( ♯ ‘ 𝐵 ) )
5	3 4	sylan9eq	⊢ ( ( 𝐺 ∈ CycGrp ∧ 𝐵 ∈ Fin ) → 𝐸 = ( ♯ ‘ 𝐵 ) )