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 + B + A B +

Proof

Step Hyp Ref Expression
1 rpcn A + A
2 rpcn B + B
3 mul2neg A B A B = A B
4 1 2 3 syl2an A + B + A B = A B
5 rpmulcl A + B + A B +
6 4 5 eqeltrd A + B + A B +