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Metamath Proof Explorer

Theorem lesub1

Description: Subtraction from both sides of 'less than or equal to'. (Contributed by NM, 13-May-2004) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion lesub1 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 − 𝐶 ) ≤ ( 𝐵 − 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 simp1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐴 ∈ ℝ )
2 simp3 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐶 ∈ ℝ )
3 simp2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐵 ∈ ℝ )
4 3 2 resubcld ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 − 𝐶 ) ∈ ℝ )
5 lesubadd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ ( 𝐵 − 𝐶 ) ∈ ℝ ) → ( ( 𝐴 − 𝐶 ) ≤ ( 𝐵 − 𝐶 ) ↔ 𝐴 ≤ ( ( 𝐵 − 𝐶 ) + 𝐶 ) ) )
6 1 2 4 5 syl3anc ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐶 ) ≤ ( 𝐵 − 𝐶 ) ↔ 𝐴 ≤ ( ( 𝐵 − 𝐶 ) + 𝐶 ) ) )
7 3 recnd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐵 ∈ ℂ )
8 2 recnd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐶 ∈ ℂ )
9 7 8 npcand ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐵 − 𝐶 ) + 𝐶 ) = 𝐵 )
10 9 breq2d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ≤ ( ( 𝐵 − 𝐶 ) + 𝐶 ) ↔ 𝐴 ≤ 𝐵 ) )
11 6 10 bitr2d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 − 𝐶 ) ≤ ( 𝐵 − 𝐶 ) ) )