Metamath Proof Explorer


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 ( 𝐴𝑋 → ( 𝑂𝐴 ) ∈ ℕ0 )

Proof

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