Theorem int-mulassocd 42800

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 11229	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ)
3		int-mulassocd.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4	3	recnd 11229	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5		int-mulassocd.3	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ)
6	5	recnd 11229	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ)
7	2, 4, 6	mulassd 11224	. 2 ⊢ (𝜑 → ((𝐵 · 𝐶) · 𝐷) = (𝐵 · (𝐶 · 𝐷)))
8		int-mulassocd.4	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵)
9	8	eqcomd 2739	. . . 4 ⊢ (𝜑 → 𝐵 = 𝐴)
10	9	oveq1d 7411	. . 3 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐴 · 𝐶))
11	10	oveq1d 7411	. 2 ⊢ (𝜑 → ((𝐵 · 𝐶) · 𝐷) = ((𝐴 · 𝐶) · 𝐷))
12	7, 11	eqtr3d 2775	1 ⊢ (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  (class class class)co 7396  ℝcr 11096   · cmul 11102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-resscn 11154  ax-mulass 11163

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-iota 6487  df-fv 6543  df-ov 7399

This theorem is referenced by: (None)