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Theorem mulassd 7048
Description: Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
addcld.1 (𝜑𝐴 ∈ ℂ)
addcld.2 (𝜑𝐵 ∈ ℂ)
addassd.3 (𝜑𝐶 ∈ ℂ)
Assertion
Ref Expression
mulassd (𝜑 → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Proof of Theorem mulassd
StepHypRef Expression
1 addcld.1 . 2 (𝜑𝐴 ∈ ℂ)
2 addcld.2 . 2 (𝜑𝐵 ∈ ℂ)
3 addassd.3 . 2 (𝜑𝐶 ∈ ℂ)
4 mulass 7010 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
51, 2, 3, 4syl3anc 1135 1 (𝜑 → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393  (class class class)co 5512  cc 6885   · cmul 6892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-mulass 6985
This theorem depends on definitions:  df-bi 110  df-3an 887
This theorem is referenced by:  ltmul1  7581  recexap  7632  mulap0  7633  mulcanapd  7640  receuap  7648  divdivdivap  7687  divmuleqap  7691  conjmulap  7703  apmul1  7762  qapne  8572  expadd  9271  binom3  9340  crre  9431  remullem  9445  resqrexlemcalc1  9586  resqrexlemnm  9590  amgm2  9688
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