Theorem sqeqor

Description: The squares of two complex numbers are equal iff one number equals the other or its negative. Lemma 15-4.7 of Gleason p. 311 and its converse. (Contributed by Paul Chapman, 15-Mar-2008)

	Ref	Expression
Assertion	sqeqor	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = - 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		oveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( 𝐴 ↑ 2 ) = ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) )
2	1	eqeq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) = ( 𝐵 ↑ 2 ) ) )
3		eqeq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( 𝐴 = 𝐵 ↔ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = 𝐵 ) )
4		eqeq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( 𝐴 = - 𝐵 ↔ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = - 𝐵 ) )
5	3 4	orbi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( ( 𝐴 = 𝐵 ∨ 𝐴 = - 𝐵 ) ↔ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = 𝐵 ∨ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = - 𝐵 ) ) )
6	2 5	bibi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( ( ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = - 𝐵 ) ) ↔ ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = 𝐵 ∨ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = - 𝐵 ) ) ) )
7		oveq1	⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( 𝐵 ↑ 2 ) = ( if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ↑ 2 ) )
8	7	eqeq2d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) = ( if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ↑ 2 ) ) )
9		eqeq2	⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = 𝐵 ↔ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) )
10		negeq	⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → - 𝐵 = - if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) )
11	10	eqeq2d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = - 𝐵 ↔ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = - if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) )
12	9 11	orbi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = 𝐵 ∨ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = - 𝐵 ) ↔ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ∨ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = - if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ) )
13	8 12	bibi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = 𝐵 ∨ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = - 𝐵 ) ) ↔ ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) = ( if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ↑ 2 ) ↔ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ∨ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = - if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ) ) )
14		0cn	⊢ 0 ∈ ℂ
15	14	elimel	⊢ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ∈ ℂ
16	14	elimel	⊢ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ∈ ℂ
17	15 16	sqeqori	⊢ ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) = ( if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ↑ 2 ) ↔ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ∨ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) = - if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) )
18	6 13 17	dedth2h	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = - 𝐵 ) ) )