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Metamath Proof Explorer

Theorem ltdiv23neg

Description: Swap denominator with other side of 'less than', when both are negative. (Contributed by Glauco Siliprandi, 26-Jun-2021)

Ref Expression
Hypotheses ltdiv23neg.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltdiv23neg.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
ltdiv23neg.3 ⊢ ( 𝜑 → 𝐵 < 0 )
ltdiv23neg.4 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
ltdiv23neg.5 ⊢ ( 𝜑 → 𝐶 < 0 )
Assertion ltdiv23neg ( 𝜑 → ( ( 𝐴 / 𝐵 ) < 𝐶 ↔ ( 𝐴 / 𝐶 ) < 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ltdiv23neg.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 ltdiv23neg.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 ltdiv23neg.3 ⊢ ( 𝜑 → 𝐵 < 0 )
4 ltdiv23neg.4 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
5 ltdiv23neg.5 ⊢ ( 𝜑 → 𝐶 < 0 )
6 2 3 ltned ⊢ ( 𝜑 → 𝐵 ≠ 0 )
7 1 2 6 redivcld ⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ∈ ℝ )
8 7 4 2 3 ltmulneg ⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) < 𝐶 ↔ ( 𝐶 · 𝐵 ) < ( ( 𝐴 / 𝐵 ) · 𝐵 ) ) )
9 recn ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
10 1 9 syl ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
11 recn ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
12 2 11 syl ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
13 10 12 6 divcan1d ⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) · 𝐵 ) = 𝐴 )
14 13 breq2d ⊢ ( 𝜑 → ( ( 𝐶 · 𝐵 ) < ( ( 𝐴 / 𝐵 ) · 𝐵 ) ↔ ( 𝐶 · 𝐵 ) < 𝐴 ) )
15 remulcl ⊢ ( ( 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐶 · 𝐵 ) ∈ ℝ )
16 4 2 15 syl2anc ⊢ ( 𝜑 → ( 𝐶 · 𝐵 ) ∈ ℝ )
17 4 5 ltned ⊢ ( 𝜑 → 𝐶 ≠ 0 )
18 4 17 rereccld ⊢ ( 𝜑 → ( 1 / 𝐶 ) ∈ ℝ )
19 4 5 reclt0d ⊢ ( 𝜑 → ( 1 / 𝐶 ) < 0 )
20 16 1 18 19 ltmulneg ⊢ ( 𝜑 → ( ( 𝐶 · 𝐵 ) < 𝐴 ↔ ( 𝐴 · ( 1 / 𝐶 ) ) < ( ( 𝐶 · 𝐵 ) · ( 1 / 𝐶 ) ) ) )
21 recn ⊢ ( 𝐶 ∈ ℝ → 𝐶 ∈ ℂ )
22 4 21 syl ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
23 10 22 17 divrecd ⊢ ( 𝜑 → ( 𝐴 / 𝐶 ) = ( 𝐴 · ( 1 / 𝐶 ) ) )
24 23 eqcomd ⊢ ( 𝜑 → ( 𝐴 · ( 1 / 𝐶 ) ) = ( 𝐴 / 𝐶 ) )
25 22 12 mulcld ⊢ ( 𝜑 → ( 𝐶 · 𝐵 ) ∈ ℂ )
26 25 22 17 divrecd ⊢ ( 𝜑 → ( ( 𝐶 · 𝐵 ) / 𝐶 ) = ( ( 𝐶 · 𝐵 ) · ( 1 / 𝐶 ) ) )
27 divcan3 ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( ( 𝐶 · 𝐵 ) / 𝐶 ) = 𝐵 )
28 27 3expb ⊢ ( ( 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐶 · 𝐵 ) / 𝐶 ) = 𝐵 )
29 12 22 17 28 syl12anc ⊢ ( 𝜑 → ( ( 𝐶 · 𝐵 ) / 𝐶 ) = 𝐵 )
30 26 29 eqtr3d ⊢ ( 𝜑 → ( ( 𝐶 · 𝐵 ) · ( 1 / 𝐶 ) ) = 𝐵 )
31 24 30 breq12d ⊢ ( 𝜑 → ( ( 𝐴 · ( 1 / 𝐶 ) ) < ( ( 𝐶 · 𝐵 ) · ( 1 / 𝐶 ) ) ↔ ( 𝐴 / 𝐶 ) < 𝐵 ) )
32 20 31 bitrd ⊢ ( 𝜑 → ( ( 𝐶 · 𝐵 ) < 𝐴 ↔ ( 𝐴 / 𝐶 ) < 𝐵 ) )
33 8 14 32 3bitrd ⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) < 𝐶 ↔ ( 𝐴 / 𝐶 ) < 𝐵 ) )