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Theorem mul32i 11458
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 11428 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))
51, 2, 3, 4mp3an 1462 1 ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  (class class class)co 7432  cc 11154   · cmul 11161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-mulcom 11220  ax-mulass 11222
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435
This theorem is referenced by:  8th4div3  12488  faclbnd4lem1  14333  bpoly4  16096  dec5nprm  17105  dec2nprm  17106  karatsuba  17122  quart1lem  26899  log2ublem2  26991  log2ub  26993  normlem3  31132  bcseqi  31140  dpmul100  32880  dpmul1000  32882
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