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Theorem pell14qrmulcl 31004
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 455 . . . . . 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 30996 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D )  /\  B  e.  (Pell1234QR `  D ) )  ->  ( A  x.  B )  e.  (Pell1234QR `  D ) )
51, 2, 3, 4syl3anc 1226 . . . . 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 30993 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  e.  RR )
72, 6syldan 468 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  A  e.  RR )
8 pell1234qrre 30993 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  B  e.  (Pell1234QR `  D ) )  ->  B  e.  RR )
93, 8syldan 468 . . . . . 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 9670 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  0  <  ( A  x.  B )
)
135, 12jca 530 . . . 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 432 . . 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 31003 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell14QR `  D )  <->  ( A  e.  (Pell1234QR `  D )  /\  0  <  A ) ) )
16 elpell14qr2 31003 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( B  e.  (Pell14QR `  D )  <->  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )
1715, 16anbi12d 708 . . 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 31003 . . 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 1192 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 367    /\ w3a 971    e. wcel 1836    \ cdif 3403   class class class wbr 4384   ` cfv 5513  (class class class)co 6218   RRcr 9424   0cc0 9425    x. cmul 9430    < clt 9561   NNcn 10474  ◻NNcsquarenn 30977  Pell1234QRcpell1234qr 30979  Pell14QRcpell14qr 30980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502  ax-pre-sup 9503
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-2nd 6722  df-recs 6982  df-rdg 7016  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-sup 7838  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-div 10146  df-nn 10475  df-2 10533  df-3 10534  df-n0 10735  df-z 10804  df-uz 11024  df-rp 11162  df-seq 12034  df-exp 12093  df-cj 12957  df-re 12958  df-im 12959  df-sqrt 13093  df-abs 13094  df-pell14qr 30984  df-pell1234qr 30985
This theorem is referenced by:  pell14qrdivcl  31006  pell14qrexpclnn0  31007  pellfund14  31039
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