Theorem mul13d 46501

Description: Commutative/associative law that swaps the first and the third factor in a triple product. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
mul13d.1	⊢ (𝜑 → 𝐴 ∈ ℂ)
mul13d.2	⊢ (𝜑 → 𝐵 ∈ ℂ)
mul13d.3	⊢ (𝜑 → 𝐶 ∈ ℂ)

Assertion

Ref	Expression
mul13d	⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐶 · (𝐵 · 𝐴)))

Proof of Theorem mul13d

Step	Hyp	Ref	Expression
1		mul13d.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ)
2		mul13d.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ)
3		mul13d.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
4	1, 2, 3	mul12d 11546	. 2 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
5	2, 1, 3	mulassd 11359	. 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐵 · (𝐴 · 𝐶)))
6	2, 1	mulcld 11356	. . 3 ⊢ (𝜑 → (𝐵 · 𝐴) ∈ ℂ)
7	6, 3	mulcomd 11357	. 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐶 · (𝐵 · 𝐴)))
8	4, 5, 7	3eqtr2d 2804	1 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐶 · (𝐵 · 𝐴)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  (class class class)co 7443  ℂcc 11225   · cmul 11232

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-mulcl 11289  ax-mulcom 11291  ax-mulass 11293

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6512  df-fv 6566  df-ov 7446

This theorem is used by:  dirkertrigeqlem3  47315  fourierdlem83  47404