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	⊢ ( 𝐺 ∈ 𝑉 → 𝐸 ∈ ℕ0 )

Proof

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