This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )