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Theorem mul32i 8118
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 8101 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))
51, 2, 3, 4mp3an 1347 1 ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1363  wcel 2158  (class class class)co 5888  cc 7823   · cmul 7830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-mulcom 7926  ax-mulass 7928
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891
This theorem is referenced by:  8th4div3  9152
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