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Why not think about it like the following?
NOM$/Y = REAL$/Y * (1+P$)/(1+PY)
You agree with that?  The nominal rate is always real * inflation.  Notice that $ on top and Yen on bottom throughout.
What’s the real rate?
REAL$/Y = NOM$/Y / (1+P$)/(1+PY)
………… = NOM$/Y * (1+Py)/(1+P$)

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