Metamath Proof Explorer


Theorem ltadd2

Description: Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion ltadd2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 axltadd ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐵 → ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) ) )
2 oveq2 ( 𝐴 = 𝐵 → ( 𝐶 + 𝐴 ) = ( 𝐶 + 𝐵 ) )
3 2 a1i ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 = 𝐵 → ( 𝐶 + 𝐴 ) = ( 𝐶 + 𝐵 ) ) )
4 axltadd ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 < 𝐴 → ( 𝐶 + 𝐵 ) < ( 𝐶 + 𝐴 ) ) )
5 4 3com12 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 < 𝐴 → ( 𝐶 + 𝐵 ) < ( 𝐶 + 𝐴 ) ) )
6 3 5 orim12d ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 = 𝐵𝐵 < 𝐴 ) → ( ( 𝐶 + 𝐴 ) = ( 𝐶 + 𝐵 ) ∨ ( 𝐶 + 𝐵 ) < ( 𝐶 + 𝐴 ) ) ) )
7 6 con3d ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ¬ ( ( 𝐶 + 𝐴 ) = ( 𝐶 + 𝐵 ) ∨ ( 𝐶 + 𝐵 ) < ( 𝐶 + 𝐴 ) ) → ¬ ( 𝐴 = 𝐵𝐵 < 𝐴 ) ) )
8 simp3 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐶 ∈ ℝ )
9 simp1 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐴 ∈ ℝ )
10 8 9 readdcld ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐶 + 𝐴 ) ∈ ℝ )
11 simp2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐵 ∈ ℝ )
12 8 11 readdcld ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐶 + 𝐵 ) ∈ ℝ )
13 axlttri ( ( ( 𝐶 + 𝐴 ) ∈ ℝ ∧ ( 𝐶 + 𝐵 ) ∈ ℝ ) → ( ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) ↔ ¬ ( ( 𝐶 + 𝐴 ) = ( 𝐶 + 𝐵 ) ∨ ( 𝐶 + 𝐵 ) < ( 𝐶 + 𝐴 ) ) ) )
14 10 12 13 syl2anc ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) ↔ ¬ ( ( 𝐶 + 𝐴 ) = ( 𝐶 + 𝐵 ) ∨ ( 𝐶 + 𝐵 ) < ( 𝐶 + 𝐴 ) ) ) )
15 axlttri ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ¬ ( 𝐴 = 𝐵𝐵 < 𝐴 ) ) )
16 9 11 15 syl2anc ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ¬ ( 𝐴 = 𝐵𝐵 < 𝐴 ) ) )
17 7 14 16 3imtr4d ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) → 𝐴 < 𝐵 ) )
18 1 17 impbid ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) ) )