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Theorem mulcli 7090
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 7066 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
41, 2, 3mp2an 410 1 (𝐴 · 𝐵) ∈ ℂ
Colors of variables: wff set class
Syntax hints:  wcel 1409  (class class class)co 5540  cc 6945   · cmul 6952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia3 105  ax-mulcl 7040
This theorem is referenced by:  ixi  7648  2mulicn  8204  numma  8470  nummac  8471  9t11e99  8556  decbin2  8567  irec  9518  binom2i  9527  3dec  9586  rei  9727  imi  9728  3dvdsdec  10176  3dvds2dec  10177  odd2np1  10184
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