Theorem gexod

Description: Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016)

	Ref	Expression
Hypotheses	gexod.1	⊢ 𝑋 = ( Base ‘ 𝐺 )
	gexod.2	⊢ 𝐸 = ( gEx ‘ 𝐺 )
	gexod.3	⊢ 𝑂 = ( od ‘ 𝐺 )
Assertion	gexod	⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑂 ‘ 𝐴 ) ∥ 𝐸 )

Proof

Step	Hyp	Ref	Expression
1		gexod.1	⊢ 𝑋 = ( Base ‘ 𝐺 )
2		gexod.2	⊢ 𝐸 = ( gEx ‘ 𝐺 )
3		gexod.3	⊢ 𝑂 = ( od ‘ 𝐺 )
4		eqid	⊢ ( .g ‘ 𝐺 ) = ( .g ‘ 𝐺 )
5		eqid	⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
6	1 2 4 5	gexid	⊢ ( 𝐴 ∈ 𝑋 → ( 𝐸 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) )
7	6	adantl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐸 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) )
8	1 2	gexcl	⊢ ( 𝐺 ∈ Grp → 𝐸 ∈ ℕ0 )
9	8	adantr	⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → 𝐸 ∈ ℕ0 )
10	9	nn0zd	⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → 𝐸 ∈ ℤ )
11	1 3 4 5	oddvds	⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ ℤ ) → ( ( 𝑂 ‘ 𝐴 ) ∥ 𝐸 ↔ ( 𝐸 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) ) )
12	10 11	mpd3an3	⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑂 ‘ 𝐴 ) ∥ 𝐸 ↔ ( 𝐸 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) ) )
13	7 12	mpbird	⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑂 ‘ 𝐴 ) ∥ 𝐸 )