Theorem int-mulassocd 43672

Description: MultiplicationAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)

Hypotheses

Ref	Expression
int-mulassocd.1	⊢ (𝜑 → 𝐵 ∈ ℝ)
int-mulassocd.2	⊢ (𝜑 → 𝐶 ∈ ℝ)
int-mulassocd.3	⊢ (𝜑 → 𝐷 ∈ ℝ)
int-mulassocd.4	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
int-mulassocd	⊢ (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷))

Proof of Theorem int-mulassocd

Step	Hyp	Ref	Expression
1		int-mulassocd.1	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
2	1	recnd 11272	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ)
3		int-mulassocd.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4	3	recnd 11272	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5		int-mulassocd.3	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ)
6	5	recnd 11272	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ)
7	2, 4, 6	mulassd 11267	. 2 ⊢ (𝜑 → ((𝐵 · 𝐶) · 𝐷) = (𝐵 · (𝐶 · 𝐷)))
8		int-mulassocd.4	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵)
9	8	eqcomd 2731	. . . 4 ⊢ (𝜑 → 𝐵 = 𝐴)
10	9	oveq1d 7431	. . 3 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐴 · 𝐶))
11	10	oveq1d 7431	. 2 ⊢ (𝜑 → ((𝐵 · 𝐶) · 𝐷) = ((𝐴 · 𝐶) · 𝐷))
12	7, 11	eqtr3d 2767	1 ⊢ (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  (class class class)co 7416  ℝcr 11137   · cmul 11143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-resscn 11195  ax-mulass 11204

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-iota 6495  df-fv 6551  df-ov 7419

This theorem is referenced by: (None)