Theorem psgndmsubg

Description: The finitary permutations are a subgroup. (Contributed by Stefan O'Rear, 28-Aug-2015)

	Ref	Expression
Hypotheses	psgneldm.g	⊢ 𝐺 = ( SymGrp ‘ 𝐷 )
	psgneldm.n	⊢ 𝑁 = ( pmSgn ‘ 𝐷 )
Assertion	psgndmsubg	⊢ ( 𝐷 ∈ 𝑉 → dom 𝑁 ∈ ( SubGrp ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		psgneldm.g	⊢ 𝐺 = ( SymGrp ‘ 𝐷 )
2		psgneldm.n	⊢ 𝑁 = ( pmSgn ‘ 𝐷 )
3		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
4		eqid	⊢ { 𝑝 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } = { 𝑝 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin }
5	1 3 4 2	psgnfn	⊢ 𝑁 Fn { 𝑝 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin }
6		fndm	⊢ ( 𝑁 Fn { 𝑝 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } → dom 𝑁 = { 𝑝 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } )
7	5 6	ax-mp	⊢ dom 𝑁 = { 𝑝 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin }
8	1 3	symgfisg	⊢ ( 𝐷 ∈ 𝑉 → { 𝑝 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } ∈ ( SubGrp ‘ 𝐺 ) )
9	7 8	eqeltrid	⊢ ( 𝐷 ∈ 𝑉 → dom 𝑁 ∈ ( SubGrp ‘ 𝐺 ) )