Metamath Proof Explorer


Axiom ax-mulcl

Description: Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by Theorem axmulcl . Proofs should normally use mulcl instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994)

Ref Expression
Assertion ax-mulcl
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA
 |-  A
1 cc
 |-  CC
2 0 1 wcel
 |-  A e. CC
3 cB
 |-  B
4 3 1 wcel
 |-  B e. CC
5 2 4 wa
 |-  ( A e. CC /\ B e. CC )
6 cmul
 |-  x.
7 0 3 6 co
 |-  ( A x. B )
8 7 1 wcel
 |-  ( A x. B ) e. CC
9 5 8 wi
 |-  ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )