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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 ( 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 )