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Metamath Proof Explorer

Theorem xleadd2d

Description: Addition of extended reals preserves the "less than or equal to" relation, in the right slot. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	xleadd2d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ^* )
	xleadd2d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ^* )
	xleadd2d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ^* )
	xleadd2d.4	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
Assertion	xleadd2d	⊢ ( 𝜑 → ( 𝐶 +_𝑒 𝐴 ) ≤ ( 𝐶 +_𝑒 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		xleadd2d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ^* )
2		xleadd2d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ^* )
3		xleadd2d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ^* )
4		xleadd2d.4	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
5		xleadd2a	⊢ ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐶 +_𝑒 𝐴 ) ≤ ( 𝐶 +_𝑒 𝐵 ) )
6	1 2 3 4 5	syl31anc	⊢ ( 𝜑 → ( 𝐶 +_𝑒 𝐴 ) ≤ ( 𝐶 +_𝑒 𝐵 ) )