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Metamath Proof Explorer

Theorem ltrec

Description: The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion ltrec ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 𝐴 < 𝐵 ↔ ( 1 / 𝐵 ) < ( 1 / 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 1red ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 1 ∈ ℝ )
2 simprl ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 𝐵 ∈ ℝ )
3 simpll ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 𝐴 ∈ ℝ )
4 simplr ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 0 < 𝐴 )
5 ltmuldiv ⊢ ( ( 1 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ) → ( ( 1 · 𝐴 ) < 𝐵 ↔ 1 < ( 𝐵 / 𝐴 ) ) )
6 1 2 3 4 5 syl112anc ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( ( 1 · 𝐴 ) < 𝐵 ↔ 1 < ( 𝐵 / 𝐴 ) ) )
7 3 recnd ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 𝐴 ∈ ℂ )
8 7 mullidd ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 1 · 𝐴 ) = 𝐴 )
9 8 breq1d ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( ( 1 · 𝐴 ) < 𝐵 ↔ 𝐴 < 𝐵 ) )
10 2 recnd ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 𝐵 ∈ ℂ )
11 4 gt0ne0d ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 𝐴 ≠ 0 )
12 10 7 11 divrecd ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 𝐵 / 𝐴 ) = ( 𝐵 · ( 1 / 𝐴 ) ) )
13 12 breq2d ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 1 < ( 𝐵 / 𝐴 ) ↔ 1 < ( 𝐵 · ( 1 / 𝐴 ) ) ) )
14 6 9 13 3bitr3d ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 𝐴 < 𝐵 ↔ 1 < ( 𝐵 · ( 1 / 𝐴 ) ) ) )
15 3 11 rereccld ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 1 / 𝐴 ) ∈ ℝ )
16 simprr ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 0 < 𝐵 )
17 ltdivmul ⊢ ( ( 1 ∈ ℝ ∧ ( 1 / 𝐴 ) ∈ ℝ ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( ( 1 / 𝐵 ) < ( 1 / 𝐴 ) ↔ 1 < ( 𝐵 · ( 1 / 𝐴 ) ) ) )
18 1 15 2 16 17 syl112anc ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( ( 1 / 𝐵 ) < ( 1 / 𝐴 ) ↔ 1 < ( 𝐵 · ( 1 / 𝐴 ) ) ) )
19 14 18 bitr4d ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 𝐴 < 𝐵 ↔ ( 1 / 𝐵 ) < ( 1 / 𝐴 ) ) )