Metamath Proof Explorer

Theorem pell14qrreccl

Description: Positive Pell solutions are closed under reciprocal. (Contributed by Stefan O'Rear, 18-Sep-2014)

Ref Expression
Assertion pell14qrreccl 
|- ( ( D e. ( NN \ []NN ) /\ A e. ( Pell14QR ` D ) ) -> ( 1 / A ) e. ( Pell14QR ` D ) )

Proof

Step Hyp Ref Expression
1 pell1234qrreccl 
 |-  ( ( D e. ( NN \ []NN ) /\ A e. ( Pell1234QR ` D ) ) -> ( 1 / A ) e. ( Pell1234QR ` D ) )
2 1 adantrr 
 |-  ( ( D e. ( NN \ []NN ) /\ ( A e. ( Pell1234QR ` D ) /\ 0 < A ) ) -> ( 1 / A ) e. ( Pell1234QR ` D ) )
3 pell1234qrre 
 |-  ( ( D e. ( NN \ []NN ) /\ A e. ( Pell1234QR ` D ) ) -> A e. RR )
4 3 adantrr 
 |-  ( ( D e. ( NN \ []NN ) /\ ( A e. ( Pell1234QR ` D ) /\ 0 < A ) ) -> A e. RR )
5 simprr 
 |-  ( ( D e. ( NN \ []NN ) /\ ( A e. ( Pell1234QR ` D ) /\ 0 < A ) ) -> 0 < A )
6 4 5 recgt0d 
 |-  ( ( D e. ( NN \ []NN ) /\ ( A e. ( Pell1234QR ` D ) /\ 0 < A ) ) -> 0 < ( 1 / A ) )
7 2 6 jca 
 |-  ( ( D e. ( NN \ []NN ) /\ ( A e. ( Pell1234QR ` D ) /\ 0 < A ) ) -> ( ( 1 / A ) e. ( Pell1234QR ` D ) /\ 0 < ( 1 / A ) ) )
8 7 ex 
 |-  ( D e. ( NN \ []NN ) -> ( ( A e. ( Pell1234QR ` D ) /\ 0 < A ) -> ( ( 1 / A ) e. ( Pell1234QR ` D ) /\ 0 < ( 1 / A ) ) ) )
9 elpell14qr2 
 |-  ( D e. ( NN \ []NN ) -> ( A e. ( Pell14QR ` D ) <-> ( A e. ( Pell1234QR ` D ) /\ 0 < A ) ) )
10 elpell14qr2 
 |-  ( D e. ( NN \ []NN ) -> ( ( 1 / A ) e. ( Pell14QR ` D ) <-> ( ( 1 / A ) e. ( Pell1234QR ` D ) /\ 0 < ( 1 / A ) ) ) )
11 8 9 10 3imtr4d 
 |-  ( D e. ( NN \ []NN ) -> ( A e. ( Pell14QR ` D ) -> ( 1 / A ) e. ( Pell14QR ` D ) ) )
12 11 imp 
 |-  ( ( D e. ( NN \ []NN ) /\ A e. ( Pell14QR ` D ) ) -> ( 1 / A ) e. ( Pell14QR ` D ) )