Theorem mul32i 8326

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

Step	Hyp	Ref	Expression
1		mul.1	. 2 ⊢ 𝐴 ∈ ℂ
2		mul.2	. 2 ⊢ 𝐵 ∈ ℂ
3		mul.3	. 2 ⊢ 𝐶 ∈ ℂ
4		mul32 8309	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))
5	1, 2, 3, 4	mp3an 1373	1 ⊢ ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)

Syntax hints:   = wceq 1397   ∈ wcel 2202  (class class class)co 6018  ℂcc 8030   · cmul 8037

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-mulcom 8133  ax-mulass 8135

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6021

This theorem is referenced by:  8th4div3  9363  dec5nprm  12992  dec2nprm  12993  karatsuba  13008