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 1 oveq1d ⊢ ( ( 0 −ℝ 0 ) = 1 → ( ( 0 · ( 0 −ℝ 0 ) ) + 0 ) = ( ( 0 · 1 ) + 0 ) )
3 0re ⊢ 0 ∈ ℝ
4 sn-00idlem1 ⊢ ( 0 ∈ ℝ → ( 0 · ( 0 −ℝ 0 ) ) = ( 0 −ℝ 0 ) )
5 4 adantr ⊢ ( ( 0 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 0 · ( 0 −ℝ 0 ) ) = ( 0 −ℝ 0 ) )
6 5 oveq1d ⊢ ( ( 0 ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( 0 · ( 0 −ℝ 0 ) ) + 0 ) = ( ( 0 −ℝ 0 ) + 0 ) )
7 resubidaddid1 ⊢ ( ( 0 ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( 0 −ℝ 0 ) + 0 ) = 0 )
8 6 7 eqtrd ⊢ ( ( 0 ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( 0 · ( 0 −ℝ 0 ) ) + 0 ) = 0 )
9 3 3 8 mp2an ⊢ ( ( 0 · ( 0 −ℝ 0 ) ) + 0 ) = 0
10 9 a1i ⊢ ( ( 0 −ℝ 0 ) = 1 → ( ( 0 · ( 0 −ℝ 0 ) ) + 0 ) = 0 )
11 ax-1rid ⊢ ( 0 ∈ ℝ → ( 0 · 1 ) = 0 )
12 3 11 mp1i ⊢ ( ( 0 −ℝ 0 ) = 1 → ( 0 · 1 ) = 0 )
13 12 oveq1d ⊢ ( ( 0 −ℝ 0 ) = 1 → ( ( 0 · 1 ) + 0 ) = ( 0 + 0 ) )
14 2 10 13 3eqtr3rd ⊢ ( ( 0 −ℝ 0 ) = 1 → ( 0 + 0 ) = 0 )