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Metamath Proof Explorer

Theorem lemuldiv2

Description: 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006)

Ref Expression
Assertion lemuldiv2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐶 · 𝐴 ) ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐵 / 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 recn ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
2 recn ⊢ ( 𝐶 ∈ ℝ → 𝐶 ∈ ℂ )
3 mulcom ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
4 1 2 3 syl2an ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
5 4 adantrr ⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
6 5 3adant2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
7 6 breq1d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 · 𝐶 ) ≤ 𝐵 ↔ ( 𝐶 · 𝐴 ) ≤ 𝐵 ) )
8 lemuldiv ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 · 𝐶 ) ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐵 / 𝐶 ) ) )
9 7 8 bitr3d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐶 · 𝐴 ) ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐵 / 𝐶 ) ) )