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Metamath Proof Explorer

Theorem divmulass

Description: An associative law for division and multiplication. (Contributed by AV, 10-Jul-2021)

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

Proof

Step Hyp Ref Expression
1 simpl1 ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) → 𝐴 ∈ ℂ )
2 simpl2 ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) → 𝐵 ∈ ℂ )
3 simpr ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) → ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) )
4 divass ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) → ( ( 𝐴 · 𝐵 ) / 𝐷 ) = ( 𝐴 · ( 𝐵 / 𝐷 ) ) )
5 1 2 3 4 syl3anc ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) → ( ( 𝐴 · 𝐵 ) / 𝐷 ) = ( 𝐴 · ( 𝐵 / 𝐷 ) ) )
6 5 eqcomd ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) → ( 𝐴 · ( 𝐵 / 𝐷 ) ) = ( ( 𝐴 · 𝐵 ) / 𝐷 ) )
7 6 oveq1d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) → ( ( 𝐴 · ( 𝐵 / 𝐷 ) ) · 𝐶 ) = ( ( ( 𝐴 · 𝐵 ) / 𝐷 ) · 𝐶 ) )
8 mulcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 𝐵 ) ∈ ℂ )
9 8 3adant3 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · 𝐵 ) ∈ ℂ )
10 9 adantr ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) → ( 𝐴 · 𝐵 ) ∈ ℂ )
11 simpl3 ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) → 𝐶 ∈ ℂ )
12 div32 ⊢ ( ( ( 𝐴 · 𝐵 ) ∈ ℂ ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( ( 𝐴 · 𝐵 ) / 𝐷 ) · 𝐶 ) = ( ( 𝐴 · 𝐵 ) · ( 𝐶 / 𝐷 ) ) )
13 10 3 11 12 syl3anc ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) → ( ( ( 𝐴 · 𝐵 ) / 𝐷 ) · 𝐶 ) = ( ( 𝐴 · 𝐵 ) · ( 𝐶 / 𝐷 ) ) )
14 7 13 eqtrd ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) → ( ( 𝐴 · ( 𝐵 / 𝐷 ) ) · 𝐶 ) = ( ( 𝐴 · 𝐵 ) · ( 𝐶 / 𝐷 ) ) )