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Theorem int-mulassocd 42800
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 11229 . . 3 (𝜑𝐵 ∈ ℂ)
3 int-mulassocd.2 . . . 4 (𝜑𝐶 ∈ ℝ)
43recnd 11229 . . 3 (𝜑𝐶 ∈ ℂ)
5 int-mulassocd.3 . . . 4 (𝜑𝐷 ∈ ℝ)
65recnd 11229 . . 3 (𝜑𝐷 ∈ ℂ)
72, 4, 6mulassd 11224 . 2 (𝜑 → ((𝐵 · 𝐶) · 𝐷) = (𝐵 · (𝐶 · 𝐷)))
8 int-mulassocd.4 . . . . 5 (𝜑𝐴 = 𝐵)
98eqcomd 2739 . . . 4 (𝜑𝐵 = 𝐴)
109oveq1d 7411 . . 3 (𝜑 → (𝐵 · 𝐶) = (𝐴 · 𝐶))
1110oveq1d 7411 . 2 (𝜑 → ((𝐵 · 𝐶) · 𝐷) = ((𝐴 · 𝐶) · 𝐷))
127, 11eqtr3d 2775 1 (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  (class class class)co 7396  cr 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)
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