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Metamath Proof Explorer

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 )