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Metamath Proof Explorer

Theorem muldivdir

Description: Distribution of division over addition with a multiplication. (Contributed by AV, 1-Jul-2021)

Ref Expression
Assertion muldivdir ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( ( 𝐶 · 𝐴 ) + 𝐵 ) / 𝐶 ) = ( 𝐴 + ( 𝐵 / 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 simp3l ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → 𝐶 ∈ ℂ )
2 simp1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → 𝐴 ∈ ℂ )
3 1 2 mulcld ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐶 · 𝐴 ) ∈ ℂ )
4 divdir ⊢ ( ( ( 𝐶 · 𝐴 ) ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( ( 𝐶 · 𝐴 ) + 𝐵 ) / 𝐶 ) = ( ( ( 𝐶 · 𝐴 ) / 𝐶 ) + ( 𝐵 / 𝐶 ) ) )
5 3 4 syld3an1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( ( 𝐶 · 𝐴 ) + 𝐵 ) / 𝐶 ) = ( ( ( 𝐶 · 𝐴 ) / 𝐶 ) + ( 𝐵 / 𝐶 ) ) )
6 3anass ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ↔ ( 𝐴 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) )
7 6 biimpri ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) )
8 7 3adant2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) )
9 divcan3 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( ( 𝐶 · 𝐴 ) / 𝐶 ) = 𝐴 )
10 8 9 syl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐶 · 𝐴 ) / 𝐶 ) = 𝐴 )
11 10 oveq1d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( ( 𝐶 · 𝐴 ) / 𝐶 ) + ( 𝐵 / 𝐶 ) ) = ( 𝐴 + ( 𝐵 / 𝐶 ) ) )
12 5 11 eqtrd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( ( 𝐶 · 𝐴 ) + 𝐵 ) / 𝐶 ) = ( 𝐴 + ( 𝐵 / 𝐶 ) ) )