Theorem frgpgrp

Description: The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015)

	Ref	Expression
Hypothesis	frgpgrp.g	⊢ 𝐺 = ( freeGrp ‘ 𝐼 )
Assertion	frgpgrp	⊢ ( 𝐼 ∈ 𝑉 → 𝐺 ∈ Grp )

Proof

Step	Hyp	Ref	Expression
1		frgpgrp.g	⊢ 𝐺 = ( freeGrp ‘ 𝐼 )
2		eqid	⊢ ( ~FG ‘ 𝐼 ) = ( ~FG ‘ 𝐼 )
3	1 2	frgp0	⊢ ( 𝐼 ∈ 𝑉 → ( 𝐺 ∈ Grp ∧ [ ∅ ] ( ~FG ‘ 𝐼 ) = ( 0g ‘ 𝐺 ) ) )
4	3	simpld	⊢ ( 𝐼 ∈ 𝑉 → 𝐺 ∈ Grp )