Theorem renegadd

Description: Relationship between real negation and addition. (Contributed by Steven Nguyen, 7-Jan-2023)

	Ref	Expression
Assertion	renegadd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 −ℝ 𝐴 ) = 𝐵 ↔ ( 𝐴 + 𝐵 ) = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		elre0re	⊢ ( 𝐴 ∈ ℝ → 0 ∈ ℝ )
2		resubval	⊢ ( ( 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 −ℝ 𝐴 ) = ( ℩ 𝑥 ∈ ℝ ( 𝐴 + 𝑥 ) = 0 ) )
3	1 2	mpancom	⊢ ( 𝐴 ∈ ℝ → ( 0 −ℝ 𝐴 ) = ( ℩ 𝑥 ∈ ℝ ( 𝐴 + 𝑥 ) = 0 ) )
4	3	eqeq1d	⊢ ( 𝐴 ∈ ℝ → ( ( 0 −ℝ 𝐴 ) = 𝐵 ↔ ( ℩ 𝑥 ∈ ℝ ( 𝐴 + 𝑥 ) = 0 ) = 𝐵 ) )
5	4	adantr	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 −ℝ 𝐴 ) = 𝐵 ↔ ( ℩ 𝑥 ∈ ℝ ( 𝐴 + 𝑥 ) = 0 ) = 𝐵 ) )
6		renegeu	⊢ ( 𝐴 ∈ ℝ → ∃! 𝑥 ∈ ℝ ( 𝐴 + 𝑥 ) = 0 )
7		oveq2	⊢ ( 𝑥 = 𝐵 → ( 𝐴 + 𝑥 ) = ( 𝐴 + 𝐵 ) )
8	7	eqeq1d	⊢ ( 𝑥 = 𝐵 → ( ( 𝐴 + 𝑥 ) = 0 ↔ ( 𝐴 + 𝐵 ) = 0 ) )
9	8	riota2	⊢ ( ( 𝐵 ∈ ℝ ∧ ∃! 𝑥 ∈ ℝ ( 𝐴 + 𝑥 ) = 0 ) → ( ( 𝐴 + 𝐵 ) = 0 ↔ ( ℩ 𝑥 ∈ ℝ ( 𝐴 + 𝑥 ) = 0 ) = 𝐵 ) )
10	6 9	sylan2	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ( 𝐴 + 𝐵 ) = 0 ↔ ( ℩ 𝑥 ∈ ℝ ( 𝐴 + 𝑥 ) = 0 ) = 𝐵 ) )
11	10	ancoms	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 + 𝐵 ) = 0 ↔ ( ℩ 𝑥 ∈ ℝ ( 𝐴 + 𝑥 ) = 0 ) = 𝐵 ) )
12	5 11	bitr4d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 −ℝ 𝐴 ) = 𝐵 ↔ ( 𝐴 + 𝐵 ) = 0 ) )