# Metamath Proof Explorer

## Theorem rpmtmip

Description: "Minus times minus is plus", see also nnmtmip , holds for positive reals, too (formalized to "The product of two negative reals is a positive real"). "The reason for this" in this case is that ( -u A x. -u B ) = ( A x. B ) for all complex numbers A and B because of mul2neg , A and B are complex numbers because of rpcn , and ( A x. B ) e. RR+ because of rpmulcl . Note that the opposites -u A and -u B of the positive reals A and B are negative reals. (Contributed by AV, 23-Dec-2022)

Ref Expression
Assertion rpmtmip
`|- ( ( A e. RR+ /\ B e. RR+ ) -> ( -u A x. -u B ) e. RR+ )`

### Proof

Step Hyp Ref Expression
1 rpcn
` |-  ( A e. RR+ -> A e. CC )`
2 rpcn
` |-  ( B e. RR+ -> B e. CC )`
3 mul2neg
` |-  ( ( A e. CC /\ B e. CC ) -> ( -u A x. -u B ) = ( A x. B ) )`
4 1 2 3 syl2an
` |-  ( ( A e. RR+ /\ B e. RR+ ) -> ( -u A x. -u B ) = ( A x. B ) )`
5 rpmulcl
` |-  ( ( A e. RR+ /\ B e. RR+ ) -> ( A x. B ) e. RR+ )`
6 4 5 eqeltrd
` |-  ( ( A e. RR+ /\ B e. RR+ ) -> ( -u A x. -u B ) e. RR+ )`