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Theorem mulcli 7032
Description: Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
Hypotheses
Ref Expression
axi.1 𝐴 ∈ ℂ
axi.2 𝐵 ∈ ℂ
Assertion
Ref Expression
mulcli (𝐴 · 𝐵) ∈ ℂ

Proof of Theorem mulcli
StepHypRef Expression
1 axi.1 . 2 𝐴 ∈ ℂ
2 axi.2 . 2 𝐵 ∈ ℂ
3 mulcl 7008 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
41, 2, 3mp2an 402 1 (𝐴 · 𝐵) ∈ ℂ
Colors of variables: wff set class
Syntax hints:  wcel 1393  (class class class)co 5512  cc 6887   · cmul 6894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia3 101  ax-mulcl 6982
This theorem is referenced by:  ixi  7574  2mulicn  8147  numma  8398  nummac  8399  decbin2  8471  irec  9352  binom2i  9360  rei  9499  imi  9500
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