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Metamath Proof Explorer

Theorem ltsubadd2

Description: 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997)

Ref Expression
Assertion ltsubadd2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) < 𝐶 ↔ 𝐴 < ( 𝐵 + 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 ltsubadd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) < 𝐶 ↔ 𝐴 < ( 𝐶 + 𝐵 ) ) )
2 simp2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐵 ∈ ℝ )
3 2 recnd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐵 ∈ ℂ )
4 simp3 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐶 ∈ ℝ )
5 4 recnd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐶 ∈ ℂ )
6 3 5 addcomd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 + 𝐶 ) = ( 𝐶 + 𝐵 ) )
7 6 breq2d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < ( 𝐵 + 𝐶 ) ↔ 𝐴 < ( 𝐶 + 𝐵 ) ) )
8 1 7 bitr4d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) < 𝐶 ↔ 𝐴 < ( 𝐵 + 𝐶 ) ) )