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Metamath Proof Explorer

Theorem re1r

Description: The unity element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017)

Ref Expression
Assertion re1r 1 = ( 1r ‘ ℝfld )

Proof

Step Hyp Ref Expression
1 resubdrg ⊢ ( ℝ ∈ ( SubRing ‘ ℂfld ) ∧ ℝfld ∈ DivRing )
2 1 simpli ⊢ ℝ ∈ ( SubRing ‘ ℂfld )
3 df-refld ⊢ ℝfld = ( ℂfld ↾s ℝ )
4 cnfld1 ⊢ 1 = ( 1r ‘ ℂfld )
5 3 4 subrg1 ⊢ ( ℝ ∈ ( SubRing ‘ ℂfld ) → 1 = ( 1r ‘ ℝfld ) )
6 2 5 ax-mp ⊢ 1 = ( 1r ‘ ℝfld )