Theorem int-mulassocd 44139

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 11318	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ)
3		int-mulassocd.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4	3	recnd 11318	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5		int-mulassocd.3	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ)
6	5	recnd 11318	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ)
7	2, 4, 6	mulassd 11313	. 2 ⊢ (𝜑 → ((𝐵 · 𝐶) · 𝐷) = (𝐵 · (𝐶 · 𝐷)))
8		int-mulassocd.4	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵)
9	8	eqcomd 2746	. . . 4 ⊢ (𝜑 → 𝐵 = 𝐴)
10	9	oveq1d 7463	. . 3 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐴 · 𝐶))
11	10	oveq1d 7463	. 2 ⊢ (𝜑 → ((𝐵 · 𝐶) · 𝐷) = ((𝐴 · 𝐶) · 𝐷))
12	7, 11	eqtr3d 2782	1 ⊢ (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  (class class class)co 7448  ℝcr 11183   · cmul 11189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-resscn 11241  ax-mulass 11250

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451

This theorem is referenced by: (None)