Theorem resubadd

Description: Relation between real subtraction and addition. Based on subadd . (Contributed by Steven Nguyen, 7-Jan-2023)

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

Proof

Step	Hyp	Ref	Expression
1		resubval	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 −ℝ 𝐵 ) = ( ℩ 𝑥 ∈ ℝ ( 𝐵 + 𝑥 ) = 𝐴 ) )
2	1	eqeq1d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 −ℝ 𝐵 ) = 𝐶 ↔ ( ℩ 𝑥 ∈ ℝ ( 𝐵 + 𝑥 ) = 𝐴 ) = 𝐶 ) )
3	2	3adant3	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 −ℝ 𝐵 ) = 𝐶 ↔ ( ℩ 𝑥 ∈ ℝ ( 𝐵 + 𝑥 ) = 𝐴 ) = 𝐶 ) )
4		resubeu	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ∃! 𝑥 ∈ ℝ ( 𝐵 + 𝑥 ) = 𝐴 )
5		oveq2	⊢ ( 𝑥 = 𝐶 → ( 𝐵 + 𝑥 ) = ( 𝐵 + 𝐶 ) )
6	5	eqeq1d	⊢ ( 𝑥 = 𝐶 → ( ( 𝐵 + 𝑥 ) = 𝐴 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) )
7	6	riota2	⊢ ( ( 𝐶 ∈ ℝ ∧ ∃! 𝑥 ∈ ℝ ( 𝐵 + 𝑥 ) = 𝐴 ) → ( ( 𝐵 + 𝐶 ) = 𝐴 ↔ ( ℩ 𝑥 ∈ ℝ ( 𝐵 + 𝑥 ) = 𝐴 ) = 𝐶 ) )
8	4 7	sylan2	⊢ ( ( 𝐶 ∈ ℝ ∧ ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) ) → ( ( 𝐵 + 𝐶 ) = 𝐴 ↔ ( ℩ 𝑥 ∈ ℝ ( 𝐵 + 𝑥 ) = 𝐴 ) = 𝐶 ) )
9	8	3impb	⊢ ( ( 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ( 𝐵 + 𝐶 ) = 𝐴 ↔ ( ℩ 𝑥 ∈ ℝ ( 𝐵 + 𝑥 ) = 𝐴 ) = 𝐶 ) )
10	9	3com13	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐵 + 𝐶 ) = 𝐴 ↔ ( ℩ 𝑥 ∈ ℝ ( 𝐵 + 𝑥 ) = 𝐴 ) = 𝐶 ) )
11	3 10	bitr4d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 −ℝ 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) )