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Metamath Proof Explorer

Theorem zring1

Description: The unity element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017) (Revised by AV, 9-Jun-2019)

Ref Expression
Assertion zring1 1 = ( 1r ‘ ℤring )

Proof

Step Hyp Ref Expression
1 zsubrg ⊢ ℤ ∈ ( SubRing ‘ ℂfld )
2 df-zring ⊢ ℤring = ( ℂfld ↾s ℤ )
3 cnfld1 ⊢ 1 = ( 1r ‘ ℂfld )
4 2 3 subrg1 ⊢ ( ℤ ∈ ( SubRing ‘ ℂfld ) → 1 = ( 1r ‘ ℤring ) )
5 1 4 ax-mp ⊢ 1 = ( 1r ‘ ℤring )