Theorem mul13d 45524

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 11342	. 2 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
5	2, 1, 3	mulassd 11155	. 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐵 · (𝐴 · 𝐶)))
6	2, 1	mulcld 11152	. . 3 ⊢ (𝜑 → (𝐵 · 𝐴) ∈ ℂ)
7	6, 3	mulcomd 11153	. 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐶 · (𝐵 · 𝐴)))
8	4, 5, 7	3eqtr2d 2777	1 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐶 · (𝐵 · 𝐴)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  (class class class)co 7358  ℂcc 11024   · cmul 11031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-mulcl 11088  ax-mulcom 11090  ax-mulass 11092

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361

This theorem is referenced by:  dirkertrigeqlem3  46340  fourierdlem83  46429