Description: Product of two negatives. Theorem I.12 of Apostol p. 18. (Contributed by NM, 30-Jul-2004) (Proof shortened by Andrew Salmon, 19-Nov-2011)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | mul2neg | |- ( ( A e. CC /\ B e. CC ) -> ( -u A x. -u B ) = ( A x. B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | negcl | |- ( B e. CC -> -u B e. CC ) |
|
| 2 | mulneg12 | |- ( ( A e. CC /\ -u B e. CC ) -> ( -u A x. -u B ) = ( A x. -u -u B ) ) |
|
| 3 | 1 2 | sylan2 | |- ( ( A e. CC /\ B e. CC ) -> ( -u A x. -u B ) = ( A x. -u -u B ) ) |
| 4 | negneg | |- ( B e. CC -> -u -u B = B ) |
|
| 5 | 4 | adantl | |- ( ( A e. CC /\ B e. CC ) -> -u -u B = B ) |
| 6 | 5 | oveq2d | |- ( ( A e. CC /\ B e. CC ) -> ( A x. -u -u B ) = ( A x. B ) ) |
| 7 | 3 6 | eqtrd | |- ( ( A e. CC /\ B e. CC ) -> ( -u A x. -u B ) = ( A x. B ) ) |