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Metamath Proof Explorer

Theorem muladd11

Description: A simple product of sums expansion. (Contributed by NM, 21-Feb-2005)

Ref Expression
Assertion muladd11 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 1 + 𝐴 ) · ( 1 + 𝐵 ) ) = ( ( 1 + 𝐴 ) + ( 𝐵 + ( 𝐴 · 𝐵 ) ) ) )

Proof

Step Hyp Ref Expression
1 ax-1cn ⊢ 1 ∈ ℂ
2 addcl ⊢ ( ( 1 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 1 + 𝐴 ) ∈ ℂ )
3 1 2 mpan ⊢ ( 𝐴 ∈ ℂ → ( 1 + 𝐴 ) ∈ ℂ )
4 adddi ⊢ ( ( ( 1 + 𝐴 ) ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 1 + 𝐴 ) · ( 1 + 𝐵 ) ) = ( ( ( 1 + 𝐴 ) · 1 ) + ( ( 1 + 𝐴 ) · 𝐵 ) ) )
5 1 4 mp3an2 ⊢ ( ( ( 1 + 𝐴 ) ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 1 + 𝐴 ) · ( 1 + 𝐵 ) ) = ( ( ( 1 + 𝐴 ) · 1 ) + ( ( 1 + 𝐴 ) · 𝐵 ) ) )
6 3 5 sylan ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 1 + 𝐴 ) · ( 1 + 𝐵 ) ) = ( ( ( 1 + 𝐴 ) · 1 ) + ( ( 1 + 𝐴 ) · 𝐵 ) ) )
7 3 mulridd ⊢ ( 𝐴 ∈ ℂ → ( ( 1 + 𝐴 ) · 1 ) = ( 1 + 𝐴 ) )
8 7 adantr ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 1 + 𝐴 ) · 1 ) = ( 1 + 𝐴 ) )
9 adddir ⊢ ( ( 1 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 1 + 𝐴 ) · 𝐵 ) = ( ( 1 · 𝐵 ) + ( 𝐴 · 𝐵 ) ) )
10 1 9 mp3an1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 1 + 𝐴 ) · 𝐵 ) = ( ( 1 · 𝐵 ) + ( 𝐴 · 𝐵 ) ) )
11 mullid ⊢ ( 𝐵 ∈ ℂ → ( 1 · 𝐵 ) = 𝐵 )
12 11 adantl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 1 · 𝐵 ) = 𝐵 )
13 12 oveq1d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 1 · 𝐵 ) + ( 𝐴 · 𝐵 ) ) = ( 𝐵 + ( 𝐴 · 𝐵 ) ) )
14 10 13 eqtrd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 1 + 𝐴 ) · 𝐵 ) = ( 𝐵 + ( 𝐴 · 𝐵 ) ) )
15 8 14 oveq12d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ( 1 + 𝐴 ) · 1 ) + ( ( 1 + 𝐴 ) · 𝐵 ) ) = ( ( 1 + 𝐴 ) + ( 𝐵 + ( 𝐴 · 𝐵 ) ) ) )
16 6 15 eqtrd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 1 + 𝐴 ) · ( 1 + 𝐵 ) ) = ( ( 1 + 𝐴 ) + ( 𝐵 + ( 𝐴 · 𝐵 ) ) ) )