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Theorem mul32d 7226
Description: Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
muld.1 (𝜑𝐴 ∈ ℂ)
addcomd.2 (𝜑𝐵 ∈ ℂ)
mul12d.3 (𝜑𝐶 ∈ ℂ)
Assertion
Ref Expression
mul32d (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))

Proof of Theorem mul32d
StepHypRef Expression
1 muld.1 . 2 (𝜑𝐴 ∈ ℂ)
2 addcomd.2 . 2 (𝜑𝐵 ∈ ℂ)
3 mul12d.3 . 2 (𝜑𝐶 ∈ ℂ)
4 mul32 7203 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))
51, 2, 3, 4syl3anc 1146 1 (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  (class class class)co 5539  cc 6944   · cmul 6951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-mulcom 7042  ax-mulass 7044
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-iota 4894  df-fv 4937  df-ov 5542
This theorem is referenced by:  conjmulap  7779  modqmul1  9326  binom3  9533  bernneq  9536  bcm1k  9627  bcp1n  9628  resqrexlemcalc1  9840  resqrexlemnm  9844  dvds2ln  10139
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