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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 ( 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 )