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Theorem pell14qrss1234 35702
Description: A positive Pell solution is a general Pell solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pell14qrss1234  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  (Pell1234QR `
 D ) )

Proof of Theorem pell14qrss1234
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0z 10960 . . . . . . 7  |-  ( b  e.  NN0  ->  b  e.  ZZ )
21a1i 11 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( b  e.  NN0  ->  b  e.  ZZ ) )
32anim1d 568 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( ( b  e. 
NN0  /\  E. c  e.  ZZ  ( a  =  ( b  +  ( ( sqr `  D
)  x.  c ) )  /\  ( ( b ^ 2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1 ) )  -> 
( b  e.  ZZ  /\ 
E. c  e.  ZZ  ( a  =  ( b  +  ( ( sqr `  D )  x.  c ) )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  1 ) ) ) )
43reximdv2 2858 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( E. b  e. 
NN0  E. c  e.  ZZ  ( a  =  ( b  +  ( ( sqr `  D )  x.  c ) )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  1 )  ->  E. b  e.  ZZ  E. c  e.  ZZ  ( a  =  ( b  +  ( ( sqr `  D
)  x.  c ) )  /\  ( ( b ^ 2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1 ) ) )
54anim2d 569 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( ( a  e.  RR  /\  E. b  e.  NN0  E. c  e.  ZZ  ( a  =  ( b  +  ( ( sqr `  D
)  x.  c ) )  /\  ( ( b ^ 2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1 ) )  -> 
( a  e.  RR  /\ 
E. b  e.  ZZ  E. c  e.  ZZ  (
a  =  ( b  +  ( ( sqr `  D )  x.  c
) )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  1 ) ) ) )
6 elpell14qr 35695 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( a  e.  (Pell14QR `  D )  <->  ( a  e.  RR  /\  E. b  e.  NN0  E. c  e.  ZZ  ( a  =  ( b  +  ( ( sqr `  D
)  x.  c ) )  /\  ( ( b ^ 2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1 ) ) ) )
7 elpell1234qr 35697 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( a  e.  (Pell1234QR `  D )  <->  ( a  e.  RR  /\  E. b  e.  ZZ  E. c  e.  ZZ  ( a  =  ( b  +  ( ( sqr `  D
)  x.  c ) )  /\  ( ( b ^ 2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1 ) ) ) )
85, 6, 73imtr4d 272 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( a  e.  (Pell14QR `  D )  ->  a  e.  (Pell1234QR `  D ) ) )
98ssrdv 3438 1  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  (Pell1234QR `
 D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   E.wrex 2738    \ cdif 3401    C_ wss 3404   ` cfv 5582  (class class class)co 6290   RRcr 9538   1c1 9540    + caddc 9542    x. cmul 9544    - cmin 9860   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   ^cexp 12272   sqrcsqrt 13296  ◻NNcsquarenn 35680  Pell1234QRcpell1234qr 35682  Pell14QRcpell14qr 35683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-pell14qr 35688  df-pell1234qr 35689
This theorem is referenced by:  pell14qrre  35703  pell14qrne0  35704  elpell14qr2  35708
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