Theorem sn-00idlem3

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

	Ref	Expression
Assertion	sn-00idlem3	⊢ ( ( 0 −ℝ 0 ) = 1 → ( 0 + 0 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		oveq2	⊢ ( ( 0 −ℝ 0 ) = 1 → ( 0 · ( 0 −ℝ 0 ) ) = ( 0 · 1 ) )
2		0re	⊢ 0 ∈ ℝ
3		sn-00idlem1	⊢ ( 0 ∈ ℝ → ( 0 · ( 0 −ℝ 0 ) ) = ( 0 −ℝ 0 ) )
4	2 3	ax-mp	⊢ ( 0 · ( 0 −ℝ 0 ) ) = ( 0 −ℝ 0 )
5		ax-1rid	⊢ ( 0 ∈ ℝ → ( 0 · 1 ) = 0 )
6	2 5	ax-mp	⊢ ( 0 · 1 ) = 0
7	1 4 6	3eqtr3g	⊢ ( ( 0 −ℝ 0 ) = 1 → ( 0 −ℝ 0 ) = 0 )
8	7	oveq1d	⊢ ( ( 0 −ℝ 0 ) = 1 → ( ( 0 −ℝ 0 ) + 0 ) = ( 0 + 0 ) )
9		resubidaddlid	⊢ ( ( 0 ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( 0 −ℝ 0 ) + 0 ) = 0 )
10	2 2 9	mp2an	⊢ ( ( 0 −ℝ 0 ) + 0 ) = 0
11	8 10	eqtr3di	⊢ ( ( 0 −ℝ 0 ) = 1 → ( 0 + 0 ) = 0 )