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Theorem pellfundlb 37448
Description: A nontrivial first quadrant solution is at least as large as the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Proof shortened by AV, 15-Sep-2020.)
Assertion
Ref Expression
pellfundlb |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ 1 < A ) -> (PellFund ` D ) <_ A )
Proof of Theorem pellfundlb
Dummy variables a b c d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pellfundval 37444 . . 3 |- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) = inf ( { a e. (Pell14QR ` D ) | 1 < a } , RR , < ) )
213ad2ant1 1082 . 2 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ 1 < A ) -> (PellFund ` D ) = inf ( { a e. (Pell14QR ` D ) | 1 < a } , RR , < ) )
3 ssrab2 3687 . . . . 5 |- { a e. (Pell14QR ` D ) | 1 < a } C_ (Pell14QR ` D )
4 pell14qrre 37421 . . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ d e. (Pell14QR ` D ) ) -> d e. RR )
54ex 450 . . . . . 6 |- ( D e. ( NN \ ◻NN ) -> ( d e. (Pell14QR ` D ) -> d e. RR ) )
65ssrdv 3609 . . . . 5 |- ( D e. ( NN \ ◻NN ) -> (Pell14QR ` D ) C_ RR )
73, 6syl5ss 3614 . . . 4 |- ( D e. ( NN \ ◻NN ) -> { a e. (Pell14QR ` D ) | 1 < a } C_ RR )
873ad2ant1 1082 . . 3 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ 1 < A ) -> { a e. (Pell14QR ` D ) | 1 < a } C_ RR )
9 1re 10039 . . . 4 |- 1 e. RR
10 breq2 4657 . . . . . . . 8 |- ( a = c -> ( 1 < a <-> 1 < c ) )
1110elrab 3363 . . . . . . 7 |- ( c e. { a e. (Pell14QR ` D ) | 1 < a } <-> ( c e. (Pell14QR ` D ) /\ 1 < c ) )
12 pell14qrre 37421 . . . . . . . . 9 |- ( ( D e. ( NN \ ◻NN ) /\ c e. (Pell14QR ` D ) ) -> c e. RR )
13 ltle 10126 . . . . . . . . 9 |- ( ( 1 e. RR /\ c e. RR ) -> ( 1 < c -> 1 <_ c ) )
149, 12, 13sylancr 695 . . . . . . . 8 |- ( ( D e. ( NN \ ◻NN ) /\ c e. (Pell14QR ` D ) ) -> ( 1 < c -> 1 <_ c ) )
1514expimpd 629 . . . . . . 7 |- ( D e. ( NN \ ◻NN ) -> ( ( c e. (Pell14QR ` D ) /\ 1 < c ) -> 1 <_ c ) )
1611, 15syl5bi 232 . . . . . 6 |- ( D e. ( NN \ ◻NN ) -> ( c e. { a e. (Pell14QR ` D ) | 1 < a } -> 1 <_ c ) )
1716ralrimiv 2965 . . . . 5 |- ( D e. ( NN \ ◻NN ) -> A. c e. { a e. (Pell14QR ` D ) | 1 < a } 1 <_ c )
18173ad2ant1 1082 . . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ 1 < A ) -> A. c e. { a e. (Pell14QR ` D ) | 1 < a } 1 <_ c )
19 breq1 4656 . . . . . 6 |- ( b = 1 -> ( b <_ c <-> 1 <_ c ) )
2019ralbidv 2986 . . . . 5 |- ( b = 1 -> ( A. c e. { a e. (Pell14QR ` D ) | 1 < a } b <_ c <-> A. c e. { a e. (Pell14QR ` D ) | 1 < a } 1 <_ c ) )
2120rspcev 3309 . . . 4 |- ( ( 1 e. RR /\ A. c e. { a e. (Pell14QR ` D ) | 1 < a } 1 <_ c ) -> E. b e. RR A. c e. { a e. (Pell14QR ` D ) | 1 < a } b <_ c )
229, 18, 21sylancr 695 . . 3 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ 1 < A ) -> E. b e. RR A. c e. { a e. (Pell14QR ` D ) | 1 < a } b <_ c )
23 simp2 1062 . . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ 1 < A ) -> A e. (Pell14QR ` D ) )
24 simp3 1063 . . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ 1 < A ) -> 1 < A )
25 breq2 4657 . . . . 5 |- ( a = A -> ( 1 < a <-> 1 < A ) )
2625elrab 3363 . . . 4 |- ( A e. { a e. (Pell14QR ` D ) | 1 < a } <-> ( A e. (Pell14QR ` D ) /\ 1 < A ) )
2723, 24, 26sylanbrc 698 . . 3 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ 1 < A ) -> A e. { a e. (Pell14QR ` D ) | 1 < a } )
28 infrelb 11008 . . 3 |- ( ( { a e. (Pell14QR ` D ) | 1 < a } C_ RR /\ E. b e. RR A. c e. { a e. (Pell14QR ` D ) | 1 < a } b <_ c /\ A e. { a e. (Pell14QR ` D ) | 1 < a } ) -> inf ( { a e. (Pell14QR ` D ) | 1 < a } , RR , < ) <_ A )
298, 22, 27, 28syl3anc 1326 . 2 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ 1 < A ) -> inf ( { a e. (Pell14QR ` D ) | 1 < a } , RR , < ) <_ A )
302, 29eqbrtrd 4675 1 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ 1 < A ) -> (PellFund ` D ) <_ A )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   \ cdif 3571   C_ wss 3574   class class class wbr 4653   ` cfv 5888  infcinf 8347   RRcr 9935   1c1 9937   < clt 10074   <_ cle 10075   NNcn 11020  ◻NNcsquarenn 37400  Pell14QRcpell14qr 37403  PellFundcpellfund 37404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-pell14qr 37407  df-pell1234qr 37408  df-pellfund 37409
This theorem is referenced by:  pellfundglb  37449  pellfund14gap  37451  rmspecfund  37474
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