Metamath Proof Explorer


Theorem muladdd

Description: Product of two sums. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses mulm1d.1 ( 𝜑𝐴 ∈ ℂ )
mulnegd.2 ( 𝜑𝐵 ∈ ℂ )
subdid.3 ( 𝜑𝐶 ∈ ℂ )
muladdd.4 ( 𝜑𝐷 ∈ ℂ )
Assertion muladdd ( 𝜑 → ( ( 𝐴 + 𝐵 ) · ( 𝐶 + 𝐷 ) ) = ( ( ( 𝐴 · 𝐶 ) + ( 𝐷 · 𝐵 ) ) + ( ( 𝐴 · 𝐷 ) + ( 𝐶 · 𝐵 ) ) ) )

Proof

Step Hyp Ref Expression
1 mulm1d.1 ( 𝜑𝐴 ∈ ℂ )
2 mulnegd.2 ( 𝜑𝐵 ∈ ℂ )
3 subdid.3 ( 𝜑𝐶 ∈ ℂ )
4 muladdd.4 ( 𝜑𝐷 ∈ ℂ )
5 muladd ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) · ( 𝐶 + 𝐷 ) ) = ( ( ( 𝐴 · 𝐶 ) + ( 𝐷 · 𝐵 ) ) + ( ( 𝐴 · 𝐷 ) + ( 𝐶 · 𝐵 ) ) ) )
6 1 2 3 4 5 syl22anc ( 𝜑 → ( ( 𝐴 + 𝐵 ) · ( 𝐶 + 𝐷 ) ) = ( ( ( 𝐴 · 𝐶 ) + ( 𝐷 · 𝐵 ) ) + ( ( 𝐴 · 𝐷 ) + ( 𝐶 · 𝐵 ) ) ) )