Theorem zringmpg

Description: The multiplicative group of the ring of integers is the restriction of the multiplicative group of the complex numbers to the integers. (Contributed by AV, 15-Jun-2019)

	Ref	Expression
Assertion	zringmpg	⊢ ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) = ( mulGrp ‘ ℤring )

Proof

Step	Hyp	Ref	Expression
1		cndrng	⊢ ℂfld ∈ DivRing
2		zex	⊢ ℤ ∈ V
3		df-zring	⊢ ℤring = ( ℂfld ↾s ℤ )
4		eqid	⊢ ( mulGrp ‘ ℂfld ) = ( mulGrp ‘ ℂfld )
5	3 4	mgpress	⊢ ( ( ℂfld ∈ DivRing ∧ ℤ ∈ V ) → ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) = ( mulGrp ‘ ℤring ) )
6	1 2 5	mp2an	⊢ ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) = ( mulGrp ‘ ℤring )