Description: Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by Theorem axmulrcl . Proofs should normally use remulcl instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax-mulrcl | |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | cA | |- A |
|
| 1 | cr | |- RR |
|
| 2 | 0 1 | wcel | |- A e. RR |
| 3 | cB | |- B |
|
| 4 | 3 1 | wcel | |- B e. RR |
| 5 | 2 4 | wa | |- ( A e. RR /\ B e. RR ) |
| 6 | cmul | |- x. |
|
| 7 | 0 3 6 | co | |- ( A x. B ) |
| 8 | 7 1 | wcel | |- ( A x. B ) e. RR |
| 9 | 5 8 | wi | |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR ) |