Theorem reixi

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

	Ref	Expression
Assertion	reixi	⊢ ( i · i ) = ( 0 −ℝ 1 )

Proof

Step	Hyp	Ref	Expression
1		ax-i2m1	⊢ ( ( i · i ) + 1 ) = 0
2		1re	⊢ 1 ∈ ℝ
3		renegid2	⊢ ( 1 ∈ ℝ → ( ( 0 −ℝ 1 ) + 1 ) = 0 )
4	2 3	ax-mp	⊢ ( ( 0 −ℝ 1 ) + 1 ) = 0
5	1 4	eqtr4i	⊢ ( ( i · i ) + 1 ) = ( ( 0 −ℝ 1 ) + 1 )
6		ax-icn	⊢ i ∈ ℂ
7	6 6	mulcli	⊢ ( i · i ) ∈ ℂ
8	7	a1i	⊢ ( ⊤ → ( i · i ) ∈ ℂ )
9		rernegcl	⊢ ( 1 ∈ ℝ → ( 0 −ℝ 1 ) ∈ ℝ )
10	9	recnd	⊢ ( 1 ∈ ℝ → ( 0 −ℝ 1 ) ∈ ℂ )
11	2 10	mp1i	⊢ ( ⊤ → ( 0 −ℝ 1 ) ∈ ℂ )
12		1cnd	⊢ ( ⊤ → 1 ∈ ℂ )
13	8 11 12	sn-addcan2d	⊢ ( ⊤ → ( ( ( i · i ) + 1 ) = ( ( 0 −ℝ 1 ) + 1 ) ↔ ( i · i ) = ( 0 −ℝ 1 ) ) )
14	5 13	mpbii	⊢ ( ⊤ → ( i · i ) = ( 0 −ℝ 1 ) )
15	14	mptru	⊢ ( i · i ) = ( 0 −ℝ 1 )