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Theorem mul32i 10438
 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
StepHypRef Expression
1 mul.1 . 2 𝐴 ∈ ℂ
2 mul.2 . 2 𝐵 ∈ ℂ
3 mul.3 . 2 𝐶 ∈ ℂ
4 mul32 10409 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))
51, 2, 3, 4mp3an 1572 1 ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631   ∈ wcel 2145  (class class class)co 6796  ℂcc 10140   · cmul 10147 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-mulcom 10206  ax-mulass 10208 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038  df-ov 6799 This theorem is referenced by:  8th4div3  11459  faclbnd4lem1  13284  bpoly4  14996  dec5nprm  15977  dec2nprm  15978  karatsuba  15999  karatsubaOLD  16000  quart1lem  24803  log2ublem2  24895  log2ub  24897  normlem3  28309  bcseqi  28317  dpmul100  29945  dpmul1000  29947
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