Theorem gicrcl

Description: Isomorphism implies the right side is a group. (Contributed by Mario Carneiro, 6-May-2015)

	Ref	Expression
Assertion	gicrcl	⊢ ( 𝑅 ≃𝑔 𝑆 → 𝑆 ∈ Grp )

Proof

Step	Hyp	Ref	Expression
1		brgic	⊢ ( 𝑅 ≃𝑔 𝑆 ↔ ( 𝑅 GrpIso 𝑆 ) ≠ ∅ )
2		n0	⊢ ( ( 𝑅 GrpIso 𝑆 ) ≠ ∅ ↔ ∃ 𝑓 𝑓 ∈ ( 𝑅 GrpIso 𝑆 ) )
3	1 2	bitri	⊢ ( 𝑅 ≃𝑔 𝑆 ↔ ∃ 𝑓 𝑓 ∈ ( 𝑅 GrpIso 𝑆 ) )
4		gimghm	⊢ ( 𝑓 ∈ ( 𝑅 GrpIso 𝑆 ) → 𝑓 ∈ ( 𝑅 GrpHom 𝑆 ) )
5		ghmgrp2	⊢ ( 𝑓 ∈ ( 𝑅 GrpHom 𝑆 ) → 𝑆 ∈ Grp )
6	4 5	syl	⊢ ( 𝑓 ∈ ( 𝑅 GrpIso 𝑆 ) → 𝑆 ∈ Grp )
7	6	exlimiv	⊢ ( ∃ 𝑓 𝑓 ∈ ( 𝑅 GrpIso 𝑆 ) → 𝑆 ∈ Grp )
8	3 7	sylbi	⊢ ( 𝑅 ≃𝑔 𝑆 → 𝑆 ∈ Grp )