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Theorem pell14qrmulcl 39453
Description: Positive Pell solutions are closed under multiplication. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
pell14qrmulcl ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → (𝐴 · 𝐵) ∈ (Pell14QR‘𝐷))

Proof of Theorem pell14qrmulcl
StepHypRef Expression
1 simpl 485 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ ((𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴) ∧ (𝐵 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐵))) → 𝐷 ∈ (ℕ ∖ ◻NN))
2 simprll 777 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ ((𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴) ∧ (𝐵 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐵))) → 𝐴 ∈ (Pell1234QR‘𝐷))
3 simprrl 779 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ ((𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴) ∧ (𝐵 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐵))) → 𝐵 ∈ (Pell1234QR‘𝐷))
4 pell1234qrmulcl 39445 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷) ∧ 𝐵 ∈ (Pell1234QR‘𝐷)) → (𝐴 · 𝐵) ∈ (Pell1234QR‘𝐷))
51, 2, 3, 4syl3anc 1367 . . . . 5 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ ((𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴) ∧ (𝐵 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐵))) → (𝐴 · 𝐵) ∈ (Pell1234QR‘𝐷))
6 pell1234qrre 39442 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → 𝐴 ∈ ℝ)
72, 6syldan 593 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ ((𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴) ∧ (𝐵 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐵))) → 𝐴 ∈ ℝ)
8 pell1234qrre 39442 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐵 ∈ (Pell1234QR‘𝐷)) → 𝐵 ∈ ℝ)
93, 8syldan 593 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ ((𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴) ∧ (𝐵 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐵))) → 𝐵 ∈ ℝ)
10 simprlr 778 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ ((𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴) ∧ (𝐵 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐵))) → 0 < 𝐴)
11 simprrr 780 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ ((𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴) ∧ (𝐵 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐵))) → 0 < 𝐵)
127, 9, 10, 11mulgt0d 10789 . . . . 5 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ ((𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴) ∧ (𝐵 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐵))) → 0 < (𝐴 · 𝐵))
135, 12jca 514 . . . 4 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ ((𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴) ∧ (𝐵 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐵))) → ((𝐴 · 𝐵) ∈ (Pell1234QR‘𝐷) ∧ 0 < (𝐴 · 𝐵)))
1413ex 415 . . 3 (𝐷 ∈ (ℕ ∖ ◻NN) → (((𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴) ∧ (𝐵 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐵)) → ((𝐴 · 𝐵) ∈ (Pell1234QR‘𝐷) ∧ 0 < (𝐴 · 𝐵))))
15 elpell14qr2 39452 . . . 4 (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ (𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴)))
16 elpell14qr2 39452 . . . 4 (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐵 ∈ (Pell14QR‘𝐷) ↔ (𝐵 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐵)))
1715, 16anbi12d 632 . . 3 (𝐷 ∈ (ℕ ∖ ◻NN) → ((𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) ↔ ((𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴) ∧ (𝐵 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐵))))
18 elpell14qr2 39452 . . 3 (𝐷 ∈ (ℕ ∖ ◻NN) → ((𝐴 · 𝐵) ∈ (Pell14QR‘𝐷) ↔ ((𝐴 · 𝐵) ∈ (Pell1234QR‘𝐷) ∧ 0 < (𝐴 · 𝐵))))
1914, 17, 183imtr4d 296 . 2 (𝐷 ∈ (ℕ ∖ ◻NN) → ((𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → (𝐴 · 𝐵) ∈ (Pell14QR‘𝐷)))
20193impib 1112 1 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → (𝐴 · 𝐵) ∈ (Pell14QR‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083  wcel 2110  cdif 3933   class class class wbr 5059  cfv 6350  (class class class)co 7150  cr 10530  0cc0 10531   · cmul 10536   < clt 10669  cn 11632  NNcsquarenn 39426  Pell1234QRcpell1234qr 39428  Pell14QRcpell14qr 39429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-sup 8900  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12384  df-seq 13364  df-exp 13424  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-pell14qr 39433  df-pell1234qr 39434
This theorem is referenced by:  pell14qrdivcl  39455  pell14qrexpclnn0  39456  pellfund14  39488
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