A PA28 FLT INSTRUCTOR AND STUDENT; WHILE ON A GPS INSTRUMENT APCH; DSNDED PREMATURELY BELOW THE MINIMUM PUBLISHED ALT.

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I WAS ON MY LAST FLT OF THE DAY; AN IFR TRAINING FLT. THE FLT WAS AN IFR XCOUNTRY IN NIGHT; VMC WITH INTERMITTENT MODERATE PRECIPITATION. WE WERE SHOOTING OUR 3RD INST APCH WHEN I LOOKED UP AT THE ALTIMETER AND REALIZED MY STUDENT HAD BEGUN A DSCNT WHICH PUT US 500 FT BELOW THE MINIMUM PUBLISHED ALT FOR THAT PORTION OF THE APCH. I IMMEDIATELY TOOK THE CTLS AND BEGAN A CLB BACK TO OUR MINIMUM ALT. IN HINDSIGHT; THERE WERE SEVERAL FACTORS THAT CONTRIBUTED TO THIS ERROR. UPON COMPLETION OF OUR APCH; WE BOTH DISCUSSED WHAT HAD LED TO THE ERROR. ALTHOUGH NEITHER OF US HAD MENTIONED IT AT THE TIME; WE BOTH FELT FATIGUED PRIOR TO THE FLT AND HAD CONSIDERED CALLING THE FLT OFF. UNFORTUNATELY; NEITHER OF US HAD MENTIONED IT OR APPLIED THE 'I'M SAFE' CHKLIST TO OURSELVES. WE BOTH AGREED THE COMPLEXITIES OF NIGHT FLT HAD ADDED TO OUR FATIGUE AND WORKLOAD. IN ADDITION; A LACK OF CRM PROCS (ALT CALLOUT THAT WE WERE LEAVING AN ALT) PREVENTED COM AND EARLY DETECTION OF THE ERROR. ON FUTURE FLTS; I WILL DEFINITELY PUT MORE EMPHASIS ON THE 'I'M SAFE' CHKLIST FOR BOTH MYSELF AND STUDENTS AS WELL AS MORE EMPHASIS ON CRM PROCS.