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Theorem int-mulassocd 40408
Description: MultiplicationAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
Hypotheses
Ref Expression
int-mulassocd.1 (𝜑𝐵 ∈ ℝ)
int-mulassocd.2 (𝜑𝐶 ∈ ℝ)
int-mulassocd.3 (𝜑𝐷 ∈ ℝ)
int-mulassocd.4 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
int-mulassocd (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷))

Proof of Theorem int-mulassocd
StepHypRef Expression
1 int-mulassocd.1 . . . 4 (𝜑𝐵 ∈ ℝ)
21recnd 10657 . . 3 (𝜑𝐵 ∈ ℂ)
3 int-mulassocd.2 . . . 4 (𝜑𝐶 ∈ ℝ)
43recnd 10657 . . 3 (𝜑𝐶 ∈ ℂ)
5 int-mulassocd.3 . . . 4 (𝜑𝐷 ∈ ℝ)
65recnd 10657 . . 3 (𝜑𝐷 ∈ ℂ)
72, 4, 6mulassd 10652 . 2 (𝜑 → ((𝐵 · 𝐶) · 𝐷) = (𝐵 · (𝐶 · 𝐷)))
8 int-mulassocd.4 . . . . 5 (𝜑𝐴 = 𝐵)
98eqcomd 2824 . . . 4 (𝜑𝐵 = 𝐴)
109oveq1d 7160 . . 3 (𝜑 → (𝐵 · 𝐶) = (𝐴 · 𝐶))
1110oveq1d 7160 . 2 (𝜑 → ((𝐵 · 𝐶) · 𝐷) = ((𝐴 · 𝐶) · 𝐷))
127, 11eqtr3d 2855 1 (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  (class class class)co 7145  cr 10524   · cmul 10530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-resscn 10582  ax-mulass 10591
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148
This theorem is referenced by: (None)
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