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Metamath Proof Explorer

Theorem ltaddrp2d

Description: Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses rpgecld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
rpgecld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
Assertion ltaddrp2d ( 𝜑 → 𝐴 < ( 𝐵 + 𝐴 ) )

Proof

Step Hyp Ref Expression
1 rpgecld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 rpgecld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3 1 2 ltaddrpd ⊢ ( 𝜑 → 𝐴 < ( 𝐴 + 𝐵 ) )
4 1 recnd ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
5 2 rpcnd ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
6 4 5 addcomd ⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
7 3 6 breqtrd ⊢ ( 𝜑 → 𝐴 < ( 𝐵 + 𝐴 ) )