Metamath Proof Explorer

Theorem pell14qrgap

Description: Positive Pell solutions are bounded away from 1. (Contributed by Stefan O'Rear, 18-Sep-2014)

Ref Expression
Assertion pell14qrgap 
|- ( ( D e. ( NN \ []NN ) /\ A e. ( Pell14QR ` D ) /\ 1 < A ) -> ( ( sqrt ` ( D + 1 ) ) + ( sqrt ` D ) ) <_ A )

Proof

Step Hyp Ref Expression
1 simp2 
 |-  ( ( D e. ( NN \ []NN ) /\ A e. ( Pell14QR ` D ) /\ 1 < A ) -> A e. ( Pell14QR ` D ) )
2 1re 
 |-  1 e. RR
3 pell14qrre 
 |-  ( ( D e. ( NN \ []NN ) /\ A e. ( Pell14QR ` D ) ) -> A e. RR )
4 ltle 
 |-  ( ( 1 e. RR /\ A e. RR ) -> ( 1 < A -> 1 <_ A ) )
5 2 3 4 sylancr 
 |-  ( ( D e. ( NN \ []NN ) /\ A e. ( Pell14QR ` D ) ) -> ( 1 < A -> 1 <_ A ) )
6 5 3impia 
 |-  ( ( D e. ( NN \ []NN ) /\ A e. ( Pell14QR ` D ) /\ 1 < A ) -> 1 <_ A )
7 elpell1qr2 
 |-  ( D e. ( NN \ []NN ) -> ( A e. ( Pell1QR ` D ) <-> ( A e. ( Pell14QR ` D ) /\ 1 <_ A ) ) )
8 7 3ad2ant1 
 |-  ( ( D e. ( NN \ []NN ) /\ A e. ( Pell14QR ` D ) /\ 1 < A ) -> ( A e. ( Pell1QR ` D ) <-> ( A e. ( Pell14QR ` D ) /\ 1 <_ A ) ) )
9 1 6 8 mpbir2and 
 |-  ( ( D e. ( NN \ []NN ) /\ A e. ( Pell14QR ` D ) /\ 1 < A ) -> A e. ( Pell1QR ` D ) )
10 pell1qrgap 
 |-  ( ( D e. ( NN \ []NN ) /\ A e. ( Pell1QR ` D ) /\ 1 < A ) -> ( ( sqrt ` ( D + 1 ) ) + ( sqrt ` D ) ) <_ A )
11 9 10 syld3an2 
 |-  ( ( D e. ( NN \ []NN ) /\ A e. ( Pell14QR ` D ) /\ 1 < A ) -> ( ( sqrt ` ( D + 1 ) ) + ( sqrt ` D ) ) <_ A )