Theorem mul31d 11392

Description: Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)

Hypotheses

Ref	Expression
muld.1	⊢ (𝜑 → 𝐴 ∈ ℂ)
addcomd.2	⊢ (𝜑 → 𝐵 ∈ ℂ)
addcand.3	⊢ (𝜑 → 𝐶 ∈ ℂ)

Assertion

Ref	Expression
mul31d	⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))

Proof of Theorem mul31d

Step	Hyp	Ref	Expression
1		muld.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℂ)
2		addcomd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℂ)
3		addcand.3	. 2 ⊢ (𝜑 → 𝐶 ∈ ℂ)
4		mul31 11348	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))
5	1, 2, 3, 4	syl3anc 1373	1 ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  (class class class)co 7390  ℂcc 11073   · cmul 11080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-mulcl 11137  ax-mulcom 11139  ax-mulass 11141

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393

This theorem is referenced by:  lawcoslem1  26732