Theorem mulassi 8341

Description: Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)

Hypotheses

Ref	Expression
axi.1	⊢ 𝐴 ∈ ℂ
axi.2	⊢ 𝐵 ∈ ℂ
axi.3	⊢ 𝐶 ∈ ℂ

Assertion

Ref	Expression
mulassi	⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))

Proof of Theorem mulassi

Step	Hyp	Ref	Expression
1		axi.1	. 2 ⊢ 𝐴 ∈ ℂ
2		axi.2	. 2 ⊢ 𝐵 ∈ ℂ
3		axi.3	. 2 ⊢ 𝐶 ∈ ℂ
4		mulass 8316	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
5	1, 2, 3, 4	mp3an 1378	1 ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))

This proof depends on syntax axioms:   = wceq 1402   ∈ wcel 2209  (class class class)co 6088  ℂcc 8183   · cmul 8190

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-mulass 8288

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  8th4div3  9535  numma  9835  decbin0  9931  sq4e2t8  11094  3dec  11173  ef01bndlem  12547  3dvdsdec  12656  3dvds2dec  12657  dec5dvds  13219  karatsuba  13238  sincos4thpi  16084  sincos6thpi  16086  log2ublem2  16239  log2ublem3  16240  log2ublog2  16241  bclbnd  16324  bposlem8  16335  2lgsoddprmlem3d  16451