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Theorem mul12d 6942
 Description: Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
muld.1 (φA ℂ)
mul12d.3 (φ𝐶 ℂ)
Assertion
Ref Expression
mul12d (φ → (A · (B · 𝐶)) = (B · (A · 𝐶)))

Proof of Theorem mul12d
StepHypRef Expression
1 muld.1 . 2 (φA ℂ)
2 addcomd.2 . 2 (φB ℂ)
3 mul12d.3 . 2 (φ𝐶 ℂ)
4 mul12 6919 . 2 ((A B 𝐶 ℂ) → (A · (B · 𝐶)) = (B · (A · 𝐶)))
51, 2, 3, 4syl3anc 1134 1 (φ → (A · (B · 𝐶)) = (B · (A · 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  (class class class)co 5455  ℂcc 6689   · cmul 6696 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-mulcom 6764  ax-mulass 6766 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458 This theorem is referenced by:  mulreim  7368  divrecap  7429  remullem  9079
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