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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