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Theorem int-mulassocd 40661
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 10643 . . 3 (𝜑𝐵 ∈ ℂ)
3 int-mulassocd.2 . . . 4 (𝜑𝐶 ∈ ℝ)
43recnd 10643 . . 3 (𝜑𝐶 ∈ ℂ)
5 int-mulassocd.3 . . . 4 (𝜑𝐷 ∈ ℝ)
65recnd 10643 . . 3 (𝜑𝐷 ∈ ℂ)
72, 4, 6mulassd 10638 . 2 (𝜑 → ((𝐵 · 𝐶) · 𝐷) = (𝐵 · (𝐶 · 𝐷)))
8 int-mulassocd.4 . . . . 5 (𝜑𝐴 = 𝐵)
98eqcomd 2826 . . . 4 (𝜑𝐵 = 𝐴)
109oveq1d 7144 . . 3 (𝜑 → (𝐵 · 𝐶) = (𝐴 · 𝐶))
1110oveq1d 7144 . 2 (𝜑 → ((𝐵 · 𝐶) · 𝐷) = ((𝐴 · 𝐶) · 𝐷))
127, 11eqtr3d 2857 1 (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  (class class class)co 7129  cr 10510   · cmul 10516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-resscn 10568  ax-mulass 10577
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3472  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-iota 6286  df-fv 6335  df-ov 7132
This theorem is referenced by: (None)
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