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Metamath Proof Explorer

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 ) ) )