Theorem mul12i 10834

Description: Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Hypotheses

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

Assertion

Ref	Expression
mul12i	⊢ (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))

Proof of Theorem mul12i

Step	Hyp	Ref	Expression
1		mul.1	. 2 ⊢ 𝐴 ∈ ℂ
2		mul.2	. 2 ⊢ 𝐵 ∈ ℂ
3		mul.3	. 2 ⊢ 𝐶 ∈ ℂ
4		mul12 10804	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
5	1, 2, 3, 4	mp3an 1457	1 ⊢ (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))

Syntax hints:   = wceq 1533   ∈ wcel 2110  (class class class)co 7155  ℂcc 10534   · cmul 10541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-mulcom 10600  ax-mulass 10602

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-iota 6313  df-fv 6362  df-ov 7158

This theorem is referenced by:  decmul10add  12166  faclbnd4lem1  13652  bpoly3  15411  decsplit  16418  root1eq1  25335  cxpeq  25337  1cubrlem  25418  efiatan2  25494  2efiatan  25495  tanatan  25496  log2ublem2  25524  log2ublem3  25525  bposlem8  25866  ax5seglem7  26720  ip1ilem  28602  ipasslem10  28615  polid2i  28933  3exp4mod41  43780