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Metamath Proof Explorer

Theorem lemulge12

Description: Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011)

Ref Expression
Assertion lemulge12 ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 1 ≤ 𝐵 ) ) → 𝐴 ≤ ( 𝐵 · 𝐴 ) )

Proof

Step Hyp Ref Expression
1 lemulge11 ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 1 ≤ 𝐵 ) ) → 𝐴 ≤ ( 𝐴 · 𝐵 ) )
2 recn ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
3 recn ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
4 mulcom ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
5 2 3 4 syl2an ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
6 5 breq2d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ ( 𝐴 · 𝐵 ) ↔ 𝐴 ≤ ( 𝐵 · 𝐴 ) ) )
7 6 adantr ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 1 ≤ 𝐵 ) ) → ( 𝐴 ≤ ( 𝐴 · 𝐵 ) ↔ 𝐴 ≤ ( 𝐵 · 𝐴 ) ) )
8 1 7 mpbid ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 1 ≤ 𝐵 ) ) → 𝐴 ≤ ( 𝐵 · 𝐴 ) )