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Theorem mul12 10081
Description: Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
Assertion
Ref Expression
mul12 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))

Proof of Theorem mul12
StepHypRef Expression
1 mulcom 9901 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
21oveq1d 6564 . . 3 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐵 · 𝐴) · 𝐶))
323adant3 1074 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐵 · 𝐴) · 𝐶))
4 mulass 9903 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
5 mulass 9903 . . 3 ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐵 · 𝐴) · 𝐶) = (𝐵 · (𝐴 · 𝐶)))
653com12 1261 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐵 · 𝐴) · 𝐶) = (𝐵 · (𝐴 · 𝐶)))
73, 4, 63eqtr3d 2652 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  (class class class)co 6549  cc 9813   · cmul 9820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-mulcom 9879  ax-mulass 9881
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552
This theorem is referenced by:  mul02  10093  mul12i  10110  mul12d  10124  mulre  13709  sqreulem  13947  fsumcube  14630  demoivre  14769  demoivreALT  14770  dvdscmul  14846  dvdscmulr  14848  dvdstr  14856  ablfacrp  18288  nmoleub2lem3  22723  sinperlem  24036  coskpi  24076  sineq0  24077  efif1olem4  24095  rpvmasum2  25001  expgrowthi  37554
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