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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 38307 | . 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥)) | |
2 | simpll 785 | . . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → 𝐷 ∈ (ℕ ∖ ◻NN)) | |
3 | pell1qrss14 38277 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷)) | |
4 | pellfundex 38295 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell1QR‘𝐷)) | |
5 | 3, 4 | sseldd 3829 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷)) |
6 | 5 | ad2antrr 719 | . . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷)) |
7 | simplr 787 | . . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → 𝑥 ∈ ℤ) | |
8 | pell14qrexpcl 38276 | . . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ (PellFund‘𝐷) ∈ (Pell14QR‘𝐷) ∧ 𝑥 ∈ ℤ) → ((PellFund‘𝐷)↑𝑥) ∈ (Pell14QR‘𝐷)) | |
9 | 2, 6, 7, 8 | syl3anc 1496 | . . . 4 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → ((PellFund‘𝐷)↑𝑥) ∈ (Pell14QR‘𝐷)) |
10 | eleq1 2895 | . . . . 5 ⊢ (𝐴 = ((PellFund‘𝐷)↑𝑥) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ ((PellFund‘𝐷)↑𝑥) ∈ (Pell14QR‘𝐷))) | |
11 | 10 | adantl 475 | . . . 4 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ ((PellFund‘𝐷)↑𝑥) ∈ (Pell14QR‘𝐷))) |
12 | 9, 11 | mpbird 249 | . . 3 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → 𝐴 ∈ (Pell14QR‘𝐷)) |
13 | 12 | r19.29an 3288 | . 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → 𝐴 ∈ (Pell14QR‘𝐷)) |
14 | 1, 13 | impbida 837 | 1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∃wrex 3119 ∖ cdif 3796 ‘cfv 6124 (class class class)co 6906 ℕcn 11351 ℤcz 11705 ↑cexp 13155 ◻NNcsquarenn 38245 Pell1QRcpell1qr 38246 Pell14QRcpell14qr 38248 PellFundcpellfund 38249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-inf2 8816 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 ax-addf 10332 ax-mulf 10333 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-se 5303 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-of 7158 df-om 7328 df-1st 7429 df-2nd 7430 df-supp 7561 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-2o 7828 df-oadd 7831 df-omul 7832 df-er 8010 df-map 8125 df-pm 8126 df-ixp 8177 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-fsupp 8546 df-fi 8587 df-sup 8618 df-inf 8619 df-oi 8685 df-card 9079 df-acn 9082 df-cda 9306 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-xnn0 11692 df-z 11706 df-dec 11823 df-uz 11970 df-q 12073 df-rp 12114 df-xneg 12233 df-xadd 12234 df-xmul 12235 df-ioo 12468 df-ioc 12469 df-ico 12470 df-icc 12471 df-fz 12621 df-fzo 12762 df-fl 12889 df-mod 12965 df-seq 13097 df-exp 13156 df-fac 13355 df-bc 13384 df-hash 13412 df-shft 14185 df-cj 14217 df-re 14218 df-im 14219 df-sqrt 14353 df-abs 14354 df-limsup 14580 df-clim 14597 df-rlim 14598 df-sum 14795 df-ef 15171 df-sin 15173 df-cos 15174 df-pi 15176 df-dvds 15359 df-gcd 15591 df-numer 15815 df-denom 15816 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-starv 16321 df-sca 16322 df-vsca 16323 df-ip 16324 df-tset 16325 df-ple 16326 df-ds 16328 df-unif 16329 df-hom 16330 df-cco 16331 df-rest 16437 df-topn 16438 df-0g 16456 df-gsum 16457 df-topgen 16458 df-pt 16459 df-prds 16462 df-xrs 16516 df-qtop 16521 df-imas 16522 df-xps 16524 df-mre 16600 df-mrc 16601 df-acs 16603 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-submnd 17690 df-mulg 17896 df-cntz 18101 df-cmn 18549 df-psmet 20099 df-xmet 20100 df-met 20101 df-bl 20102 df-mopn 20103 df-fbas 20104 df-fg 20105 df-cnfld 20108 df-top 21070 df-topon 21087 df-topsp 21109 df-bases 21122 df-cld 21195 df-ntr 21196 df-cls 21197 df-nei 21274 df-lp 21312 df-perf 21313 df-cn 21403 df-cnp 21404 df-haus 21491 df-tx 21737 df-hmeo 21930 df-fil 22021 df-fm 22113 df-flim 22114 df-flf 22115 df-xms 22496 df-ms 22497 df-tms 22498 df-cncf 23052 df-limc 24030 df-dv 24031 df-log 24703 df-squarenn 38250 df-pell1qr 38251 df-pell14qr 38252 df-pell1234qr 38253 df-pellfund 38254 |
This theorem is referenced by: rmxyelqirr 38319 rmxycomplete 38326 |
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