I am reading Dummit and Foote Section 9.6 Polynomials in Several Variables Over a Field and Grobner Bases.

I have a problem understanding a step in the proof of Proposition 24, Page 322 of D&F

Proposition 24 reads as follows:

Proposition 24.Fix a monomial ordering on $\displaystyle R= F[x_1, ... , x_n] $ and let I be a non-zero ideal in R

(1) If $\displaystyle g_1, ... , g_m $ are any elements of I such that $\displaystyle LT(I) = (LT(g_1), ... ... LT(g_m) )$

then $\displaystyle \{ g_1, ... , g_m \} $ is a Grobner Basis for I

(2) The ideal I has a Grobner Basis

The proof of Proposition 24 begins as follows:

Proof:Suppose $\displaystyle g_1, ... , g_m \in I $ with $\displaystyle LT(I) = (LT(g_1), ... ... LT(g_m) )$ .

We need to see that $\displaystyle g_1, ... , g_m $ generate the ideal I.

If $\displaystyle f \in I $ use general polynomial division to write $\displaystyle f = \sum q_i g_i + r $ where no non-zero term in the remainder r is divisible by any $\displaystyle LT(g_i) $

Since $\displaystyle f \in I $, also $\displaystyle r \in I $ ... ... etc etc

My question: How do we know $\displaystyle f \in I \Longrightarrow r \in I $?

Would appreciate some help

Peter