Theorem mul13d 45230

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 11468	. 2 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
5	2, 1, 3	mulassd 11282	. 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐵 · (𝐴 · 𝐶)))
6	2, 1	mulcld 11279	. . 3 ⊢ (𝜑 → (𝐵 · 𝐴) ∈ ℂ)
7	6, 3	mulcomd 11280	. 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐶 · (𝐵 · 𝐴)))
8	4, 5, 7	3eqtr2d 2781	1 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐶 · (𝐵 · 𝐴)))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  (class class class)co 7431  ℂcc 11151   · cmul 11158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-mulcl 11215  ax-mulcom 11217  ax-mulass 11219

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434

This theorem is referenced by:  dirkertrigeqlem3  46056  fourierdlem83  46145