Metamath Proof Explorer


Theorem gexcl

Description: The exponent of a group is a nonnegative integer. (Contributed by Mario Carneiro, 23-Apr-2016)

Ref Expression
Hypotheses gexcl.1 𝑋 = ( Base ‘ 𝐺 )
gexcl.2 𝐸 = ( gEx ‘ 𝐺 )
Assertion gexcl ( 𝐺𝑉𝐸 ∈ ℕ0 )

Proof

Step Hyp Ref Expression
1 gexcl.1 𝑋 = ( Base ‘ 𝐺 )
2 gexcl.2 𝐸 = ( gEx ‘ 𝐺 )
3 eqid ( .g𝐺 ) = ( .g𝐺 )
4 eqid ( 0g𝐺 ) = ( 0g𝐺 )
5 eqid { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 ( .g𝐺 ) 𝑥 ) = ( 0g𝐺 ) } = { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 ( .g𝐺 ) 𝑥 ) = ( 0g𝐺 ) }
6 1 3 4 2 5 gexlem1 ( 𝐺𝑉 → ( ( 𝐸 = 0 ∧ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 ( .g𝐺 ) 𝑥 ) = ( 0g𝐺 ) } = ∅ ) ∨ 𝐸 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 ( .g𝐺 ) 𝑥 ) = ( 0g𝐺 ) } ) )
7 simpl ( ( 𝐸 = 0 ∧ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 ( .g𝐺 ) 𝑥 ) = ( 0g𝐺 ) } = ∅ ) → 𝐸 = 0 )
8 elrabi ( 𝐸 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 ( .g𝐺 ) 𝑥 ) = ( 0g𝐺 ) } → 𝐸 ∈ ℕ )
9 7 8 orim12i ( ( ( 𝐸 = 0 ∧ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 ( .g𝐺 ) 𝑥 ) = ( 0g𝐺 ) } = ∅ ) ∨ 𝐸 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥𝑋 ( 𝑦 ( .g𝐺 ) 𝑥 ) = ( 0g𝐺 ) } ) → ( 𝐸 = 0 ∨ 𝐸 ∈ ℕ ) )
10 6 9 syl ( 𝐺𝑉 → ( 𝐸 = 0 ∨ 𝐸 ∈ ℕ ) )
11 10 orcomd ( 𝐺𝑉 → ( 𝐸 ∈ ℕ ∨ 𝐸 = 0 ) )
12 elnn0 ( 𝐸 ∈ ℕ0 ↔ ( 𝐸 ∈ ℕ ∨ 𝐸 = 0 ) )
13 11 12 sylibr ( 𝐺𝑉𝐸 ∈ ℕ0 )