Theorem mul32i 11316

Description: Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)

Hypotheses

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

Assertion

Ref	Expression
mul32i	⊢ ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)

Proof of Theorem mul32i

Step	Hyp	Ref	Expression
1		mul.1	. 2 ⊢ 𝐴 ∈ ℂ
2		mul.2	. 2 ⊢ 𝐵 ∈ ℂ
3		mul.3	. 2 ⊢ 𝐶 ∈ ℂ
4		mul32 11286	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))
5	1, 2, 3, 4	mp3an 1463	1 ⊢ ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)

Syntax hints:   = wceq 1541   ∈ wcel 2113  (class class class)co 7352  ℂcc 11011   · cmul 11018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-mulcom 11077  ax-mulass 11079

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6442  df-fv 6494  df-ov 7355

This theorem is referenced by:  8th4div3  12348  faclbnd4lem1  14202  bpoly4  15968  dec5nprm  16980  dec2nprm  16981  karatsuba  16997  quart1lem  26793  log2ublem2  26885  log2ub  26887  normlem3  31094  bcseqi  31102  dpmul100  32884  dpmul1000  32886