Theorem cj11

Description: Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005) (Proof shortened by Eric Schmidt, 2-Jul-2009)

	Ref	Expression
Assertion	cj11	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ∗ ‘ 𝐴 ) = ( ∗ ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( ( ∗ ‘ 𝐴 ) = ( ∗ ‘ 𝐵 ) → ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) = ( ∗ ‘ ( ∗ ‘ 𝐵 ) ) )
2		cjcj	⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) = 𝐴 )
3		cjcj	⊢ ( 𝐵 ∈ ℂ → ( ∗ ‘ ( ∗ ‘ 𝐵 ) ) = 𝐵 )
4	2 3	eqeqan12d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) = ( ∗ ‘ ( ∗ ‘ 𝐵 ) ) ↔ 𝐴 = 𝐵 ) )
5	1 4	syl5ib	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ∗ ‘ 𝐴 ) = ( ∗ ‘ 𝐵 ) → 𝐴 = 𝐵 ) )
6		fveq2	⊢ ( 𝐴 = 𝐵 → ( ∗ ‘ 𝐴 ) = ( ∗ ‘ 𝐵 ) )
7	5 6	impbid1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ∗ ‘ 𝐴 ) = ( ∗ ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )