Theorem int-mulassocd 44753

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 11210	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ)
3		int-mulassocd.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4	3	recnd 11210	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5		int-mulassocd.3	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ)
6	5	recnd 11210	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ)
7	2, 4, 6	mulassd 11205	. 2 ⊢ (𝜑 → ((𝐵 · 𝐶) · 𝐷) = (𝐵 · (𝐶 · 𝐷)))
8		int-mulassocd.4	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵)
9	8	eqcomd 2768	. . . 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 2799	1 ⊢ (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷))

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142  (class class class)co 7396  ℝcr 11072   · cmul 11078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-resscn 11130  ax-mulass 11139

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6477  df-fv 6529  df-ov 7399

This theorem is referenced by: (None)