Modulo Operation/Examples/0.11 mod 0.1/Proof 2

Theorem

$0 \cdotp 11 \bmod 0 \cdotp 1 = 0 \cdotp 01$

Proof

From Modulo Operation: $1 \cdotp 1 \bmod 1$:

$1 \cdotp 1 \bmod 1 = 0 \cdotp 1$

From Product Distributes over Modulo Operation:

$z \left({x \bmod y}\right) = \left({z x}\right) \bmod \left({z y}\right)$

and so:

\(\ds 0 \cdotp 11 \bmod 0 \cdotp 1\)

\(=\)

\(\ds 0 \cdotp 1 \left({1 \cdotp 1 \bmod 1}\right)\)

\(\ds \)

\(=\)

\(\ds 0 \cdotp 1 \times 0 \cdotp 1\)

\(\ds \)

\(=\)

\(\ds 0 \cdotp 01\)

$\blacksquare$