Theorem mul12i 10110

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 10081	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
5	1, 2, 3, 4	mp3an 1416	1 ⊢ (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))

Syntax hints:   = wceq 1475   ∈ wcel 1977  (class class class)co 6549  ℂcc 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:  decmul10add  11469  decmul10addOLD  11470  faclbnd4lem1  12942  bpoly3  14628  decsplit  15625  decsplitOLD  15629  root1eq1  24296  cxpeq  24298  1cubrlem  24368  efiatan2  24444  2efiatan  24445  tanatan  24446  log2ublem2  24474  log2ublem3  24475  bposlem8  24816  ax5seglem7  25615  ip1ilem  27065  ipasslem10  27078  polid2i  27398  3exp4mod41  40071