Theorem reixi

Description: ixi without ax-mulcom . (Contributed by SN, 5-May-2024)

	Ref	Expression
Assertion	reixi	$$⊢\mathrm{i}\mathrm{i}=0{-}_{ℝ}1$$

Proof

Step	Hyp	Ref	Expression
1		ax-i2m1	$$⊢\mathrm{i}\mathrm{i}+1=0$$
2		1re	$$⊢1\in ℝ$$
3		renegid2	$$⊢1\in ℝ\to \left(0{-}_{ℝ}1\right)+1=0$$
4	2 3	ax-mp	$$⊢\left(0{-}_{ℝ}1\right)+1=0$$
5	1 4	eqtr4i	$$⊢\mathrm{i}\mathrm{i}+1=\left(0{-}_{ℝ}1\right)+1$$
6		ax-icn	$$⊢\mathrm{i}\in ℂ$$
7	6 6	mulcli	$$⊢\mathrm{i}\mathrm{i}\in ℂ$$
8	7	a1i	$$⊢\top \to \mathrm{i}\mathrm{i}\in ℂ$$
9		rernegcl	$$⊢1\in ℝ\to 0{-}_{ℝ}1\in ℝ$$
10	9	recnd	$$⊢1\in ℝ\to 0{-}_{ℝ}1\in ℂ$$
11	2 10	mp1i	$$⊢\top \to 0{-}_{ℝ}1\in ℂ$$
12		1cnd	$$⊢\top \to 1\in ℂ$$
13	8 11 12	sn-addcan2d	$$⊢\top \to \left(\mathrm{i}\mathrm{i}+1=\left(0{-}_{ℝ}1\right)+1\leftrightarrow \mathrm{i}\mathrm{i}=0{-}_{ℝ}1\right)$$
14	5 13	mpbii	$$⊢\top \to \mathrm{i}\mathrm{i}=0{-}_{ℝ}1$$
15	14	mptru	$$⊢\mathrm{i}\mathrm{i}=0{-}_{ℝ}1$$