Theorem lem1

Description: A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Assertion	lem1	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − 1 ) ≤ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ltm1	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − 1 ) < 𝐴 )
2		peano2rem	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − 1 ) ∈ ℝ )
3		ltle	⊢ ( ( ( 𝐴 − 1 ) ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ( 𝐴 − 1 ) < 𝐴 → ( 𝐴 − 1 ) ≤ 𝐴 ) )
4	2 3	mpancom	⊢ ( 𝐴 ∈ ℝ → ( ( 𝐴 − 1 ) < 𝐴 → ( 𝐴 − 1 ) ≤ 𝐴 ) )
5	1 4	mpd	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − 1 ) ≤ 𝐴 )