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Theorem pell14qrmulcl 29353
Description: Positive Pell solutions are closed under multiplication. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
pell14qrmulcl  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  B  e.  (Pell14QR `  D )
)  ->  ( A  x.  B )  e.  (Pell14QR `  D ) )

Proof of Theorem pell14qrmulcl
StepHypRef Expression
1 simpl 457 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  D  e.  ( NN  \NN ) )
2 simprll 761 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  A  e.  (Pell1234QR `  D ) )
3 simprrl 763 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  B  e.  (Pell1234QR `  D ) )
4 pell1234qrmulcl 29345 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D )  /\  B  e.  (Pell1234QR `  D ) )  ->  ( A  x.  B )  e.  (Pell1234QR `  D ) )
51, 2, 3, 4syl3anc 1219 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  ( A  x.  B )  e.  (Pell1234QR `  D ) )
6 pell1234qrre 29342 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  e.  RR )
72, 6syldan 470 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  A  e.  RR )
8 pell1234qrre 29342 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  B  e.  (Pell1234QR `  D ) )  ->  B  e.  RR )
93, 8syldan 470 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  B  e.  RR )
10 simprlr 762 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  0  <  A
)
11 simprrr 764 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  0  <  B
)
127, 9, 10, 11mulgt0d 9638 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  0  <  ( A  x.  B )
)
135, 12jca 532 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  ( ( A  x.  B )  e.  (Pell1234QR `  D )  /\  0  <  ( A  x.  B ) ) )
1413ex 434 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) )  -> 
( ( A  x.  B )  e.  (Pell1234QR `  D )  /\  0  <  ( A  x.  B
) ) ) )
15 elpell14qr2 29352 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell14QR `  D )  <->  ( A  e.  (Pell1234QR `  D )  /\  0  <  A ) ) )
16 elpell14qr2 29352 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( B  e.  (Pell14QR `  D )  <->  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )
1715, 16anbi12d 710 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( ( A  e.  (Pell14QR `  D )  /\  B  e.  (Pell14QR `  D ) )  <->  ( ( A  e.  (Pell1234QR `  D
)  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) ) )
18 elpell14qr2 29352 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( ( A  x.  B )  e.  (Pell14QR `  D )  <->  ( ( A  x.  B )  e.  (Pell1234QR `  D )  /\  0  <  ( A  x.  B ) ) ) )
1914, 17, 183imtr4d 268 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( ( A  e.  (Pell14QR `  D )  /\  B  e.  (Pell14QR `  D ) )  -> 
( A  x.  B
)  e.  (Pell14QR `  D
) ) )
20193impib 1186 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  B  e.  (Pell14QR `  D )
)  ->  ( A  x.  B )  e.  (Pell14QR `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1758    \ cdif 3434   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   RRcr 9393   0cc0 9394    x. cmul 9399    < clt 9530   NNcn 10434  ◻NNcsquarenn 29326  Pell1234QRcpell1234qr 29328  Pell14QRcpell14qr 29329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-sup 7803  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-n0 10692  df-z 10759  df-uz 10974  df-rp 11104  df-seq 11925  df-exp 11984  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-pell14qr 29333  df-pell1234qr 29334
This theorem is referenced by:  pell14qrdivcl  29355  pell14qrexpclnn0  29356  pellfund14  29388
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