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Theorem pell14qrmulcl 26948
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 443 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  D  e.  ( NN  \NN ) )
2 simprll 738 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  A  e.  (Pell1234QR `  D ) )
3 simprrl 740 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  B  e.  (Pell1234QR `  D ) )
4 pell1234qrmulcl 26940 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D )  /\  B  e.  (Pell1234QR `  D ) )  ->  ( A  x.  B )  e.  (Pell1234QR `  D ) )
51, 2, 3, 4syl3anc 1182 . . . . 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 26937 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  e.  RR )
72, 6syldan 456 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  A  e.  RR )
8 pell1234qrre 26937 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  B  e.  (Pell1234QR `  D ) )  ->  B  e.  RR )
93, 8syldan 456 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  B  e.  RR )
10 simprlr 739 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  0  <  A
)
11 simprrr 741 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  0  <  B
)
127, 9, 10, 11mulgt0d 8971 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  ( ( A  e.  (Pell1234QR `  D )  /\  0  <  A )  /\  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )  ->  0  <  ( A  x.  B )
)
135, 12jca 518 . . . 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 423 . . 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 26947 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell14QR `  D )  <->  ( A  e.  (Pell1234QR `  D )  /\  0  <  A ) ) )
16 elpell14qr2 26947 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( B  e.  (Pell14QR `  D )  <->  ( B  e.  (Pell1234QR `  D )  /\  0  <  B ) ) )
1715, 16anbi12d 691 . . 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 26947 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( ( A  x.  B )  e.  (Pell14QR `  D )  <->  ( ( A  x.  B )  e.  (Pell1234QR `  D )  /\  0  <  ( A  x.  B ) ) ) )
1914, 17, 183imtr4d 259 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( ( A  e.  (Pell14QR `  D )  /\  B  e.  (Pell14QR `  D ) )  -> 
( A  x.  B
)  e.  (Pell14QR `  D
) ) )
20193impib 1149 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  B  e.  (Pell14QR `  D )
)  ->  ( A  x.  B )  e.  (Pell14QR `  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1684    \ cdif 3149   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737    x. cmul 8742    < clt 8867   NNcn 9746  ◻NNcsquarenn 26921  Pell1234QRcpell1234qr 26923  Pell14QRcpell14qr 26924
This theorem is referenced by:  pell14qrdivcl  26950  pell14qrexpclnn0  26951  pellfund14  26983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-pell14qr 26928  df-pell1234qr 26929
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