Theorem mul13d 41565

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 10849	. 2 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
5	2, 1, 3	mulassd 10664	. 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐵 · (𝐴 · 𝐶)))
6	2, 1	mulcld 10661	. . 3 ⊢ (𝜑 → (𝐵 · 𝐴) ∈ ℂ)
7	6, 3	mulcomd 10662	. 2 ⊢ (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐶 · (𝐵 · 𝐴)))
8	4, 5, 7	3eqtr2d 2862	1 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐶 · (𝐵 · 𝐴)))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  (class class class)co 7156  ℂcc 10535   · cmul 10542

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-mulcl 10599  ax-mulcom 10601  ax-mulass 10603

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-iota 6314  df-fv 6363  df-ov 7159

This theorem is referenced by:  dirkertrigeqlem3  42405  fourierdlem83  42494