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Metamath Proof Explorer

Theorem xleadd1d

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

	Ref	Expression
Hypotheses	xleadd1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ^* )
	xleadd1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ^* )
	xleadd1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ^* )
	xleadd1d.4	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
Assertion	xleadd1d	⊢ ( 𝜑 → ( 𝐴 +_𝑒 𝐶 ) ≤ ( 𝐵 +_𝑒 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		xleadd1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ^* )
2		xleadd1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ^* )
3		xleadd1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ^* )
4		xleadd1d.4	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
5		xleadd1a	⊢ ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 +_𝑒 𝐶 ) ≤ ( 𝐵 +_𝑒 𝐶 ) )
6	1 2 3 4 5	syl31anc	⊢ ( 𝜑 → ( 𝐴 +_𝑒 𝐶 ) ≤ ( 𝐵 +_𝑒 𝐶 ) )