Theorem tgrpgrp

Description: The translation group is a group. (Contributed by NM, 6-Jun-2013)

	Ref	Expression
Hypotheses	tgrpgrp.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	tgrpgrp.g	⊢ 𝐺 = ( ( TGrp ‘ 𝐾 ) ‘ 𝑊 )
Assertion	tgrpgrp	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐺 ∈ Grp )

Proof

Step	Hyp	Ref	Expression
1		tgrpgrp.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		tgrpgrp.g	⊢ 𝐺 = ( ( TGrp ‘ 𝐾 ) ‘ 𝑊 )
3		eqid	⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
5		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
6	1 3 2 4 5	tgrpgrplem	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐺 ∈ Grp )