Theorem re1r

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

	Ref	Expression
Assertion	re1r	$$⊢1=1_{{ℝ}_{\mathrm{fld}}}$$

Proof

Step	Hyp	Ref	Expression
1		resubdrg	$$⊢\left(ℝ\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\wedge {ℝ}_{\mathrm{fld}}\in \mathrm{DivRing}\right)$$
2	1	simpli	$$⊢ℝ\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)$$
3		df-refld	$$⊢{ℝ}_{\mathrm{fld}}={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}ℝ$$
4		cnfld1	$$⊢1=1_{{ℂ}_{\mathrm{fld}}}$$
5	3 4	subrg1	$$⊢ℝ\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\to 1=1_{{ℝ}_{\mathrm{fld}}}$$
6	2 5	ax-mp	$$⊢1=1_{{ℝ}_{\mathrm{fld}}}$$