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Metamath Proof Explorer

Theorem xraddge02

Description: A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 28-Dec-2016)

Ref Expression
Assertion xraddge02 ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 0 ≤ 𝐵 → 𝐴 ≤ ( 𝐴 +𝑒 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 xrleid ⊢ ( 𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴 )
2 1 adantr ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → 𝐴 ≤ 𝐴 )
3 simpl ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → 𝐴 ∈ ℝ* )
4 0xr ⊢ 0 ∈ ℝ*
5 3 4 jctir ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ∈ ℝ* ∧ 0 ∈ ℝ* ) )
6 xle2add ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 0 ∈ ℝ* ) ∧ ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ) → ( ( 𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( 𝐴 +𝑒 0 ) ≤ ( 𝐴 +𝑒 𝐵 ) ) )
7 5 6 mpancom ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( 𝐴 +𝑒 0 ) ≤ ( 𝐴 +𝑒 𝐵 ) ) )
8 2 7 mpand ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 0 ≤ 𝐵 → ( 𝐴 +𝑒 0 ) ≤ ( 𝐴 +𝑒 𝐵 ) ) )
9 xaddrid ⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 +𝑒 0 ) = 𝐴 )
10 9 breq1d ⊢ ( 𝐴 ∈ ℝ* → ( ( 𝐴 +𝑒 0 ) ≤ ( 𝐴 +𝑒 𝐵 ) ↔ 𝐴 ≤ ( 𝐴 +𝑒 𝐵 ) ) )
11 10 adantr ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 +𝑒 0 ) ≤ ( 𝐴 +𝑒 𝐵 ) ↔ 𝐴 ≤ ( 𝐴 +𝑒 𝐵 ) ) )
12 8 11 sylibd ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 0 ≤ 𝐵 → 𝐴 ≤ ( 𝐴 +𝑒 𝐵 ) ) )