Theorem readdcan

Description: Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013) (Proof shortened by Mario Carneiro, 27-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		ltadd2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) ) )
2	1	notbid	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ¬ 𝐴 < 𝐵 ↔ ¬ ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) ) )
3		simp2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐵 ∈ ℝ )
4		simp1	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐴 ∈ ℝ )
5		simp3	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐶 ∈ ℝ )
6	3 4 5	ltadd2d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 < 𝐴 ↔ ( 𝐶 + 𝐵 ) < ( 𝐶 + 𝐴 ) ) )
7	6	notbid	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ¬ 𝐵 < 𝐴 ↔ ¬ ( 𝐶 + 𝐵 ) < ( 𝐶 + 𝐴 ) ) )
8	2 7	anbi12d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ↔ ( ¬ ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) ∧ ¬ ( 𝐶 + 𝐵 ) < ( 𝐶 + 𝐴 ) ) ) )
9	4 3	lttri3d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 = 𝐵 ↔ ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) )
10	5 4	readdcld	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐶 + 𝐴 ) ∈ ℝ )
11	5 3	readdcld	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐶 + 𝐵 ) ∈ ℝ )
12	10 11	lttri3d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐶 + 𝐴 ) = ( 𝐶 + 𝐵 ) ↔ ( ¬ ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) ∧ ¬ ( 𝐶 + 𝐵 ) < ( 𝐶 + 𝐴 ) ) ) )
13	8 9 12	3bitr4rd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐶 + 𝐴 ) = ( 𝐶 + 𝐵 ) ↔ 𝐴 = 𝐵 ) )