Theorem subgmulgcl

Description: Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015)

	Ref	Expression
Hypothesis	subgmulgcl.t	⊢ · = ( .g ‘ 𝐺 )
Assertion	subgmulgcl	⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆 ) → ( 𝑁 · 𝑋 ) ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		subgmulgcl.t	⊢ · = ( .g ‘ 𝐺 )
2		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
3		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
4		subgrcl	⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp )
5	2	subgss	⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝑆 ⊆ ( Base ‘ 𝐺 ) )
6	3	subgcl	⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 )
7		eqid	⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
8	7	subg0cl	⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 0g ‘ 𝐺 ) ∈ 𝑆 )
9		eqid	⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
10	9	subginvcl	⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥 ∈ 𝑆 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑆 )
11	2 1 3 4 5 6 7 8 9 10	mulgsubcl	⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆 ) → ( 𝑁 · 𝑋 ) ∈ 𝑆 )