Theorem odcl

Description: The order of a group element is always a nonnegative integer. (Contributed by Mario Carneiro, 14-Jan-2015) (Revised by Stefan O'Rear, 5-Sep-2015)

	Ref	Expression
Hypotheses	odcl.1	⊢ 𝑋 = ( Base ‘ 𝐺 )
	odcl.2	⊢ 𝑂 = ( od ‘ 𝐺 )
Assertion	odcl	⊢ ( 𝐴 ∈ 𝑋 → ( 𝑂 ‘ 𝐴 ) ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		odcl.1	⊢ 𝑋 = ( Base ‘ 𝐺 )
2		odcl.2	⊢ 𝑂 = ( od ‘ 𝐺 )
3		eqid	⊢ ( .g ‘ 𝐺 ) = ( .g ‘ 𝐺 )
4		eqid	⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
5		eqid	⊢ { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) } = { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) }
6	1 3 4 2 5	odlem1	⊢ ( 𝐴 ∈ 𝑋 → ( ( ( 𝑂 ‘ 𝐴 ) = 0 ∧ { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) } = ∅ ) ∨ ( 𝑂 ‘ 𝐴 ) ∈ { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) } ) )
7		simpl	⊢ ( ( ( 𝑂 ‘ 𝐴 ) = 0 ∧ { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) } = ∅ ) → ( 𝑂 ‘ 𝐴 ) = 0 )
8		elrabi	⊢ ( ( 𝑂 ‘ 𝐴 ) ∈ { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) } → ( 𝑂 ‘ 𝐴 ) ∈ ℕ )
9	7 8	orim12i	⊢ ( ( ( ( 𝑂 ‘ 𝐴 ) = 0 ∧ { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) } = ∅ ) ∨ ( 𝑂 ‘ 𝐴 ) ∈ { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) } ) → ( ( 𝑂 ‘ 𝐴 ) = 0 ∨ ( 𝑂 ‘ 𝐴 ) ∈ ℕ ) )
10	6 9	syl	⊢ ( 𝐴 ∈ 𝑋 → ( ( 𝑂 ‘ 𝐴 ) = 0 ∨ ( 𝑂 ‘ 𝐴 ) ∈ ℕ ) )
11	10	orcomd	⊢ ( 𝐴 ∈ 𝑋 → ( ( 𝑂 ‘ 𝐴 ) ∈ ℕ ∨ ( 𝑂 ‘ 𝐴 ) = 0 ) )
12		elnn0	⊢ ( ( 𝑂 ‘ 𝐴 ) ∈ ℕ0 ↔ ( ( 𝑂 ‘ 𝐴 ) ∈ ℕ ∨ ( 𝑂 ‘ 𝐴 ) = 0 ) )
13	11 12	sylibr	⊢ ( 𝐴 ∈ 𝑋 → ( 𝑂 ‘ 𝐴 ) ∈ ℕ0 )