Metamath Proof Explorer


Theorem rec11

Description: Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion rec11 ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 1cnd ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → 1 ∈ ℂ )
2 reccl ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 1 / 𝐵 ) ∈ ℂ )
3 2 adantl ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 1 / 𝐵 ) ∈ ℂ )
4 simpl ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) )
5 divmul ( ( 1 ∈ ℂ ∧ ( 1 / 𝐵 ) ∈ ℂ ∧ ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ) → ( ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ↔ ( 𝐴 · ( 1 / 𝐵 ) ) = 1 ) )
6 1 3 4 5 syl3anc ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ↔ ( 𝐴 · ( 1 / 𝐵 ) ) = 1 ) )
7 simpll ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → 𝐴 ∈ ℂ )
8 simprl ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → 𝐵 ∈ ℂ )
9 simprr ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → 𝐵 ≠ 0 )
10 divrec ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 / 𝐵 ) = ( 𝐴 · ( 1 / 𝐵 ) ) )
11 7 8 9 10 syl3anc ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 𝐴 / 𝐵 ) = ( 𝐴 · ( 1 / 𝐵 ) ) )
12 11 eqeq1d ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) = 1 ↔ ( 𝐴 · ( 1 / 𝐵 ) ) = 1 ) )
13 diveq1 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = 1 ↔ 𝐴 = 𝐵 ) )
14 7 8 9 13 syl3anc ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) = 1 ↔ 𝐴 = 𝐵 ) )
15 6 12 14 3bitr2d ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ↔ 𝐴 = 𝐵 ) )