Metamath Proof Explorer


Theorem abs2sqle

Description: The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007)

Ref Expression
Assertion abs2sqle ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( abs ‘ 𝐴 ) ≤ ( abs ‘ 𝐵 ) ↔ ( ( abs ‘ 𝐴 ) ↑ 2 ) ≤ ( ( abs ‘ 𝐵 ) ↑ 2 ) ) )

Proof

Step Hyp Ref Expression
1 fveq2 ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( abs ‘ 𝐴 ) = ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) )
2 1 breq1d ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( ( abs ‘ 𝐴 ) ≤ ( abs ‘ 𝐵 ) ↔ ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ≤ ( abs ‘ 𝐵 ) ) )
3 1 oveq1d ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ↑ 2 ) )
4 3 breq1d ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( ( ( abs ‘ 𝐴 ) ↑ 2 ) ≤ ( ( abs ‘ 𝐵 ) ↑ 2 ) ↔ ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ↑ 2 ) ≤ ( ( abs ‘ 𝐵 ) ↑ 2 ) ) )
5 2 4 bibi12d ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( ( ( abs ‘ 𝐴 ) ≤ ( abs ‘ 𝐵 ) ↔ ( ( abs ‘ 𝐴 ) ↑ 2 ) ≤ ( ( abs ‘ 𝐵 ) ↑ 2 ) ) ↔ ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ≤ ( abs ‘ 𝐵 ) ↔ ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ↑ 2 ) ≤ ( ( abs ‘ 𝐵 ) ↑ 2 ) ) ) )
6 fveq2 ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( abs ‘ 𝐵 ) = ( abs ‘ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) )
7 6 breq2d ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ≤ ( abs ‘ 𝐵 ) ↔ ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ≤ ( abs ‘ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ) )
8 oveq1 ( ( abs ‘ 𝐵 ) = ( abs ‘ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) → ( ( abs ‘ 𝐵 ) ↑ 2 ) = ( ( abs ‘ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ↑ 2 ) )
9 8 breq2d ( ( abs ‘ 𝐵 ) = ( abs ‘ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) → ( ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ↑ 2 ) ≤ ( ( abs ‘ 𝐵 ) ↑ 2 ) ↔ ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ↑ 2 ) ≤ ( ( abs ‘ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ↑ 2 ) ) )
10 6 9 syl ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ↑ 2 ) ≤ ( ( abs ‘ 𝐵 ) ↑ 2 ) ↔ ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ↑ 2 ) ≤ ( ( abs ‘ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ↑ 2 ) ) )
11 7 10 bibi12d ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ≤ ( abs ‘ 𝐵 ) ↔ ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ↑ 2 ) ≤ ( ( abs ‘ 𝐵 ) ↑ 2 ) ) ↔ ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ≤ ( abs ‘ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ↔ ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ↑ 2 ) ≤ ( ( abs ‘ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ↑ 2 ) ) ) )
12 0cn 0 ∈ ℂ
13 12 elimel if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ∈ ℂ
14 12 elimel if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ∈ ℂ
15 13 14 abs2sqlei ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ≤ ( abs ‘ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ↔ ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) ↑ 2 ) ≤ ( ( abs ‘ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ↑ 2 ) )
16 5 11 15 dedth2h ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( abs ‘ 𝐴 ) ≤ ( abs ‘ 𝐵 ) ↔ ( ( abs ‘ 𝐴 ) ↑ 2 ) ≤ ( ( abs ‘ 𝐵 ) ↑ 2 ) ) )