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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 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) → ( - 𝐴 · - 𝐵 ) ∈ ℝ+ )

Proof

Step Hyp Ref Expression
1 rpcn ⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ )
2 rpcn ⊢ ( 𝐵 ∈ ℝ+ → 𝐵 ∈ ℂ )
3 mul2neg ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 · - 𝐵 ) = ( 𝐴 · 𝐵 ) )
4 1 2 3 syl2an ⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) → ( - 𝐴 · - 𝐵 ) = ( 𝐴 · 𝐵 ) )
5 rpmulcl ⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 · 𝐵 ) ∈ ℝ+ )
6 4 5 eqeltrd ⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) → ( - 𝐴 · - 𝐵 ) ∈ ℝ+ )