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Metamath Proof Explorer

Theorem ltsubadd

Description: 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997) (Proof shortened by Mario Carneiro, 27-May-2016)

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

Proof

Step Hyp Ref Expression
1 simp1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐴 ∈ ℝ )
2 simp2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐵 ∈ ℝ )
3 1 2 resubcld ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 − 𝐵 ) ∈ ℝ )
4 simp3 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐶 ∈ ℝ )
5 ltadd1 ⊢ ( ( ( 𝐴 − 𝐵 ) ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) < 𝐶 ↔ ( ( 𝐴 − 𝐵 ) + 𝐵 ) < ( 𝐶 + 𝐵 ) ) )
6 3 4 2 5 syl3anc ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) < 𝐶 ↔ ( ( 𝐴 − 𝐵 ) + 𝐵 ) < ( 𝐶 + 𝐵 ) ) )
7 1 recnd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐴 ∈ ℂ )
8 2 recnd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐵 ∈ ℂ )
9 7 8 npcand ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) + 𝐵 ) = 𝐴 )
10 9 breq1d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( ( 𝐴 − 𝐵 ) + 𝐵 ) < ( 𝐶 + 𝐵 ) ↔ 𝐴 < ( 𝐶 + 𝐵 ) ) )
11 6 10 bitrd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) < 𝐶 ↔ 𝐴 < ( 𝐶 + 𝐵 ) ) )