Metamath Proof Explorer

Theorem binom2

Description: The square of a binomial. (Contributed by FL, 10-Dec-2006)

Ref Expression
Assertion binom2 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) ) )

Proof

Step Hyp Ref Expression
1 oveq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( 𝐴 + 𝐵 ) = ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) + 𝐵 ) )
2 1 oveq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( ( 𝐴 + 𝐵 ) ↑ 2 ) = ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) + 𝐵 ) ↑ 2 ) )
3 oveq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( 𝐴 ↑ 2 ) = ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) )
4 oveq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( 𝐴 · 𝐵 ) = ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · 𝐵 ) )
5 4 oveq2d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( 2 · ( 𝐴 · 𝐵 ) ) = ( 2 · ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · 𝐵 ) ) )
6 3 5 oveq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 𝐵 ) ) ) = ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) + ( 2 · ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · 𝐵 ) ) ) )
7 6 oveq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) ) = ( ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) + ( 2 · ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) ) )
8 2 7 eqeq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( ( ( 𝐴 + 𝐵 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) ) ↔ ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) + 𝐵 ) ↑ 2 ) = ( ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) + ( 2 · ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) ) ) )
9 oveq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) + 𝐵 ) = ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) + if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) )
10 9 oveq1d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) + 𝐵 ) ↑ 2 ) = ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) + if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ↑ 2 ) )
11 oveq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · 𝐵 ) = ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) )
12 11 oveq2d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( 2 · ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · 𝐵 ) ) = ( 2 · ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ) )
13 12 oveq2d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) + ( 2 · ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · 𝐵 ) ) ) = ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) + ( 2 · ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ) ) )
14 oveq1 ⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( 𝐵 ↑ 2 ) = ( if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ↑ 2 ) )
15 13 14 oveq12d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) + ( 2 · ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) ) = ( ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) + ( 2 · ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ) ) + ( if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ↑ 2 ) ) )
16 10 15 eqeq12d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) → ( ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) + 𝐵 ) ↑ 2 ) = ( ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) + ( 2 · ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) ) ↔ ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) + if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ↑ 2 ) = ( ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) + ( 2 · ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ) ) + ( if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ↑ 2 ) ) ) )
17 0cn ⊢ 0 ∈ ℂ
18 17 elimel ⊢ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ∈ ℂ
19 17 elimel ⊢ if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ∈ ℂ
20 18 19 binom2i ⊢ ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) + if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ↑ 2 ) = ( ( ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ↑ 2 ) + ( 2 · ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) · if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ) ) ) + ( if ( 𝐵 ∈ ℂ , 𝐵 , 0 ) ↑ 2 ) )
21 8 16 20 dedth2h ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) ) )