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Metamath Proof Explorer

Theorem addmulsub

Description: The product of a sum and a difference. (Contributed by AV, 5-Mar-2023)

Ref Expression
Assertion addmulsub ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) · ( 𝐶 − 𝐷 ) ) = ( ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) − ( ( 𝐴 · 𝐷 ) + ( 𝐵 · 𝐷 ) ) ) )

Proof

Step Hyp Ref Expression
1 simpll ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → 𝐴 ∈ ℂ )
2 simplr ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → 𝐵 ∈ ℂ )
3 1 2 addcld ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( 𝐴 + 𝐵 ) ∈ ℂ )
4 simprl ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → 𝐶 ∈ ℂ )
5 simprr ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → 𝐷 ∈ ℂ )
6 3 4 5 subdid ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) · ( 𝐶 − 𝐷 ) ) = ( ( ( 𝐴 + 𝐵 ) · 𝐶 ) − ( ( 𝐴 + 𝐵 ) · 𝐷 ) ) )
7 1 2 4 adddird ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) )
8 1 2 5 adddird ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) · 𝐷 ) = ( ( 𝐴 · 𝐷 ) + ( 𝐵 · 𝐷 ) ) )
9 7 8 oveq12d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( ( 𝐴 + 𝐵 ) · 𝐶 ) − ( ( 𝐴 + 𝐵 ) · 𝐷 ) ) = ( ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) − ( ( 𝐴 · 𝐷 ) + ( 𝐵 · 𝐷 ) ) ) )
10 6 9 eqtrd ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) · ( 𝐶 − 𝐷 ) ) = ( ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) − ( ( 𝐴 · 𝐷 ) + ( 𝐵 · 𝐷 ) ) ) )