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		0re	$$⊢0\in ℝ$$
3		sn-00idlem1	$$⊢0\in ℝ\to 0\cdot \left(0{-}_{ℝ}0\right)=0{-}_{ℝ}0$$
4	2 3	ax-mp	$$⊢0\cdot \left(0{-}_{ℝ}0\right)=0{-}_{ℝ}0$$
5		ax-1rid	$$⊢0\in ℝ\to 0\cdot 1=0$$
6	2 5	ax-mp	$$⊢0\cdot 1=0$$
7	1 4 6	3eqtr3g	$$⊢0{-}_{ℝ}0=1\to 0{-}_{ℝ}0=0$$
8	7	oveq1d	$$⊢0{-}_{ℝ}0=1\to \left(0{-}_{ℝ}0\right)+0=0+0$$
9		resubidaddlid	$$⊢\left(0\in ℝ\wedge 0\in ℝ\right)\to \left(0{-}_{ℝ}0\right)+0=0$$
10	2 2 9	mp2an	$$⊢\left(0{-}_{ℝ}0\right)+0=0$$
11	8 10	eqtr3di	$$⊢0{-}_{ℝ}0=1\to 0+0=0$$