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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 41123 | . 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥)) | |
2 | simpll 766 | . . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → 𝐷 ∈ (ℕ ∖ ◻NN)) | |
3 | pell1qrss14 41093 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷)) | |
4 | pellfundex 41111 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell1QR‘𝐷)) | |
5 | 3, 4 | sseldd 3944 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷)) |
6 | 5 | ad2antrr 725 | . . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷)) |
7 | simplr 768 | . . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → 𝑥 ∈ ℤ) | |
8 | pell14qrexpcl 41092 | . . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ (PellFund‘𝐷) ∈ (Pell14QR‘𝐷) ∧ 𝑥 ∈ ℤ) → ((PellFund‘𝐷)↑𝑥) ∈ (Pell14QR‘𝐷)) | |
9 | 2, 6, 7, 8 | syl3anc 1372 | . . . 4 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → ((PellFund‘𝐷)↑𝑥) ∈ (Pell14QR‘𝐷)) |
10 | eleq1 2826 | . . . . 5 ⊢ (𝐴 = ((PellFund‘𝐷)↑𝑥) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ ((PellFund‘𝐷)↑𝑥) ∈ (Pell14QR‘𝐷))) | |
11 | 10 | adantl 483 | . . . 4 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ ((PellFund‘𝐷)↑𝑥) ∈ (Pell14QR‘𝐷))) |
12 | 9, 11 | mpbird 257 | . . 3 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ ℤ) ∧ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → 𝐴 ∈ (Pell14QR‘𝐷)) |
13 | 12 | r19.29an 3154 | . 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥)) → 𝐴 ∈ (Pell14QR‘𝐷)) |
14 | 1, 13 | impbida 800 | 1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3072 ∖ cdif 3906 ‘cfv 6492 (class class class)co 7350 ℕcn 12087 ℤcz 12433 ↑cexp 13897 ◻NNcsquarenn 41061 Pell1QRcpell1qr 41062 Pell14QRcpell14qr 41064 PellFundcpellfund 41065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-inf2 9511 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 ax-addf 11064 ax-mulf 11065 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-of 7608 df-om 7794 df-1st 7912 df-2nd 7913 df-supp 8061 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-omul 8385 df-er 8582 df-map 8701 df-pm 8702 df-ixp 8770 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-fsupp 9240 df-fi 9281 df-sup 9312 df-inf 9313 df-oi 9380 df-card 9809 df-acn 9812 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-xnn0 12420 df-z 12434 df-dec 12553 df-uz 12698 df-q 12804 df-rp 12846 df-xneg 12963 df-xadd 12964 df-xmul 12965 df-ioo 13198 df-ioc 13199 df-ico 13200 df-icc 13201 df-fz 13355 df-fzo 13498 df-fl 13627 df-mod 13705 df-seq 13837 df-exp 13898 df-fac 14103 df-bc 14132 df-hash 14160 df-shft 14887 df-cj 14919 df-re 14920 df-im 14921 df-sqrt 15055 df-abs 15056 df-limsup 15289 df-clim 15306 df-rlim 15307 df-sum 15507 df-ef 15886 df-sin 15888 df-cos 15889 df-pi 15891 df-dvds 16073 df-gcd 16311 df-numer 16546 df-denom 16547 df-struct 16955 df-sets 16972 df-slot 16990 df-ndx 17002 df-base 17020 df-ress 17049 df-plusg 17082 df-mulr 17083 df-starv 17084 df-sca 17085 df-vsca 17086 df-ip 17087 df-tset 17088 df-ple 17089 df-ds 17091 df-unif 17092 df-hom 17093 df-cco 17094 df-rest 17240 df-topn 17241 df-0g 17259 df-gsum 17260 df-topgen 17261 df-pt 17262 df-prds 17265 df-xrs 17320 df-qtop 17325 df-imas 17326 df-xps 17328 df-mre 17402 df-mrc 17403 df-acs 17405 df-mgm 18433 df-sgrp 18482 df-mnd 18493 df-submnd 18538 df-mulg 18808 df-cntz 19032 df-cmn 19499 df-psmet 20717 df-xmet 20718 df-met 20719 df-bl 20720 df-mopn 20721 df-fbas 20722 df-fg 20723 df-cnfld 20726 df-top 22171 df-topon 22188 df-topsp 22210 df-bases 22224 df-cld 22298 df-ntr 22299 df-cls 22300 df-nei 22377 df-lp 22415 df-perf 22416 df-cn 22506 df-cnp 22507 df-haus 22594 df-tx 22841 df-hmeo 23034 df-fil 23125 df-fm 23217 df-flim 23218 df-flf 23219 df-xms 23601 df-ms 23602 df-tms 23603 df-cncf 24169 df-limc 25158 df-dv 25159 df-log 25840 df-squarenn 41066 df-pell1qr 41067 df-pell14qr 41068 df-pell1234qr 41069 df-pellfund 41070 |
This theorem is referenced by: rmxyelqirr 41135 rmxyelqirrOLD 41136 rmxycomplete 41143 |
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