Theorem sn-00idlem3

Description: Lemma for sn-00id . (Contributed by SN, 25-Dec-2023)

	Ref	Expression
Assertion	sn-00idlem3	$$⊢0{-}_{ℝ}0=1\to 0+0=0$$

Proof

Step	Hyp	Ref	Expression
1		oveq2	$$⊢0{-}_{ℝ}0=1\to 0\cdot \left(0{-}_{ℝ}0\right)=0\cdot 1$$
2	1	oveq1d	$$⊢0{-}_{ℝ}0=1\to 0\cdot \left(0{-}_{ℝ}0\right)+0=0\cdot 1+0$$
3		0re	$$⊢0\in ℝ$$
4		sn-00idlem1	$$⊢0\in ℝ\to 0\cdot \left(0{-}_{ℝ}0\right)=0{-}_{ℝ}0$$
5	4	adantr	$$⊢\left(0\in ℝ\wedge 0\in ℝ\right)\to 0\cdot \left(0{-}_{ℝ}0\right)=0{-}_{ℝ}0$$
6	5	oveq1d	$$⊢\left(0\in ℝ\wedge 0\in ℝ\right)\to 0\cdot \left(0{-}_{ℝ}0\right)+0=\left(0{-}_{ℝ}0\right)+0$$
7		resubidaddid1	$$⊢\left(0\in ℝ\wedge 0\in ℝ\right)\to \left(0{-}_{ℝ}0\right)+0=0$$
8	6 7	eqtrd	$$⊢\left(0\in ℝ\wedge 0\in ℝ\right)\to 0\cdot \left(0{-}_{ℝ}0\right)+0=0$$
9	3 3 8	mp2an	$$⊢0\cdot \left(0{-}_{ℝ}0\right)+0=0$$
10	9	a1i	$$⊢0{-}_{ℝ}0=1\to 0\cdot \left(0{-}_{ℝ}0\right)+0=0$$
11		ax-1rid	$$⊢0\in ℝ\to 0\cdot 1=0$$
12	3 11	mp1i	$$⊢0{-}_{ℝ}0=1\to 0\cdot 1=0$$
13	12	oveq1d	$$⊢0{-}_{ℝ}0=1\to 0\cdot 1+0=0+0$$
14	2 10 13	3eqtr3rd	$$⊢0{-}_{ℝ}0=1\to 0+0=0$$