Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pellfund14b | Structured version Visualization version GIF version |
Description: The positive Pell solutions are precisely the integer powers of the fundamental solution. To get the general solution set (which we will not be using), throw in a copy of Z/2Z. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
Ref | Expression |
---|---|
pellfund14b | ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pellfund14 40375 | . 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥)) | |
2 | simpll 767 | . . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → 𝐷 ∈ (ℕ ∖ ◻NN)) | |
3 | pell1qrss14 40345 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷)) | |
4 | pellfundex 40363 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell1QR‘𝐷)) | |
5 | 3, 4 | sseldd 3892 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷)) |
6 | 5 | ad2antrr 726 | . . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷)) |
7 | simplr 769 | . . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → 𝑥 ∈ ℤ) | |
8 | pell14qrexpcl 40344 | . . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ (PellFund‘𝐷) ∈ (Pell14QR‘𝐷) ∧ 𝑥 ∈ ℤ) → ((PellFund‘𝐷)↑𝑥) ∈ (Pell14QR‘𝐷)) | |
9 | 2, 6, 7, 8 | syl3anc 1373 | . . . 4 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → ((PellFund‘𝐷)↑𝑥) ∈ (Pell14QR‘𝐷)) |
10 | eleq1 2821 | . . . . 5 ⊢ (𝐴 = ((PellFund‘𝐷)↑𝑥) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ ((PellFund‘𝐷)↑𝑥) ∈ (Pell14QR‘𝐷))) | |
11 | 10 | adantl 485 | . . . 4 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ ((PellFund‘𝐷)↑𝑥) ∈ (Pell14QR‘𝐷))) |
12 | 9, 11 | mpbird 260 | . . 3 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → 𝐴 ∈ (Pell14QR‘𝐷)) |
13 | 12 | r19.29an 3200 | . 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → 𝐴 ∈ (Pell14QR‘𝐷)) |
14 | 1, 13 | impbida 801 | 1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∃wrex 3055 ∖ cdif 3854 ‘cfv 6369 (class class class)co 7202 ℕcn 11813 ℤcz 12159 ↑cexp 13618 ◻NNcsquarenn 40313 Pell1QRcpell1qr 40314 Pell14QRcpell14qr 40316 PellFundcpellfund 40317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-oadd 8195 df-omul 8196 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-fi 9016 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-acn 9541 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-xnn0 12146 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ioc 12923 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-mod 13426 df-seq 13558 df-exp 13619 df-fac 13823 df-bc 13852 df-hash 13880 df-shft 14613 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-limsup 15015 df-clim 15032 df-rlim 15033 df-sum 15233 df-ef 15610 df-sin 15612 df-cos 15613 df-pi 15615 df-dvds 15797 df-gcd 16035 df-numer 16272 df-denom 16273 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-rest 16899 df-topn 16900 df-0g 16918 df-gsum 16919 df-topgen 16920 df-pt 16921 df-prds 16924 df-xrs 16979 df-qtop 16984 df-imas 16985 df-xps 16987 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-mulg 18461 df-cntz 18683 df-cmn 19144 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-fbas 20332 df-fg 20333 df-cnfld 20336 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cld 21888 df-ntr 21889 df-cls 21890 df-nei 21967 df-lp 22005 df-perf 22006 df-cn 22096 df-cnp 22097 df-haus 22184 df-tx 22431 df-hmeo 22624 df-fil 22715 df-fm 22807 df-flim 22808 df-flf 22809 df-xms 23190 df-ms 23191 df-tms 23192 df-cncf 23747 df-limc 24735 df-dv 24736 df-log 25417 df-squarenn 40318 df-pell1qr 40319 df-pell14qr 40320 df-pell1234qr 40321 df-pellfund 40322 |
This theorem is referenced by: rmxyelqirr 40387 rmxycomplete 40394 |
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