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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rmxynorm | Structured version Visualization version GIF version |
Description: The X and Y sequences define a solution to the corresponding Pell equation. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
Ref | Expression |
---|---|
rmxynorm | ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (((𝐴 Xrm 𝑁)↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 479 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
2 | eqidd 2779 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 Xrm 𝑁) = (𝐴 Xrm 𝑁)) | |
3 | eqidd 2779 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝐴 Yrm 𝑁) = (𝐴 Yrm 𝑁)) | |
4 | 2, 3 | anim12i 606 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm 𝑁) = (𝐴 Xrm 𝑁) ∧ (𝐴 Yrm 𝑁) = (𝐴 Yrm 𝑁))) |
5 | oveq2 6932 | . . . . . 6 ⊢ (𝑎 = 𝑁 → (𝐴 Xrm 𝑎) = (𝐴 Xrm 𝑁)) | |
6 | 5 | eqeq2d 2788 | . . . . 5 ⊢ (𝑎 = 𝑁 → ((𝐴 Xrm 𝑁) = (𝐴 Xrm 𝑎) ↔ (𝐴 Xrm 𝑁) = (𝐴 Xrm 𝑁))) |
7 | oveq2 6932 | . . . . . 6 ⊢ (𝑎 = 𝑁 → (𝐴 Yrm 𝑎) = (𝐴 Yrm 𝑁)) | |
8 | 7 | eqeq2d 2788 | . . . . 5 ⊢ (𝑎 = 𝑁 → ((𝐴 Yrm 𝑁) = (𝐴 Yrm 𝑎) ↔ (𝐴 Yrm 𝑁) = (𝐴 Yrm 𝑁))) |
9 | 6, 8 | anbi12d 624 | . . . 4 ⊢ (𝑎 = 𝑁 → (((𝐴 Xrm 𝑁) = (𝐴 Xrm 𝑎) ∧ (𝐴 Yrm 𝑁) = (𝐴 Yrm 𝑎)) ↔ ((𝐴 Xrm 𝑁) = (𝐴 Xrm 𝑁) ∧ (𝐴 Yrm 𝑁) = (𝐴 Yrm 𝑁)))) |
10 | 9 | rspcev 3511 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ ((𝐴 Xrm 𝑁) = (𝐴 Xrm 𝑁) ∧ (𝐴 Yrm 𝑁) = (𝐴 Yrm 𝑁))) → ∃𝑎 ∈ ℤ ((𝐴 Xrm 𝑁) = (𝐴 Xrm 𝑎) ∧ (𝐴 Yrm 𝑁) = (𝐴 Yrm 𝑎))) |
11 | 1, 4, 10 | syl2anc 579 | . 2 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ∃𝑎 ∈ ℤ ((𝐴 Xrm 𝑁) = (𝐴 Xrm 𝑎) ∧ (𝐴 Yrm 𝑁) = (𝐴 Yrm 𝑎))) |
12 | simpl 476 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → 𝐴 ∈ (ℤ≥‘2)) | |
13 | frmx 38451 | . . . 4 ⊢ Xrm :((ℤ≥‘2) × ℤ)⟶ℕ0 | |
14 | 13 | fovcl 7044 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) ∈ ℕ0) |
15 | frmy 38452 | . . . 4 ⊢ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ | |
16 | 15 | fovcl 7044 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑁) ∈ ℤ) |
17 | rmxycomplete 38455 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝐴 Xrm 𝑁) ∈ ℕ0 ∧ (𝐴 Yrm 𝑁) ∈ ℤ) → ((((𝐴 Xrm 𝑁)↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = 1 ↔ ∃𝑎 ∈ ℤ ((𝐴 Xrm 𝑁) = (𝐴 Xrm 𝑎) ∧ (𝐴 Yrm 𝑁) = (𝐴 Yrm 𝑎)))) | |
18 | 12, 14, 16, 17 | syl3anc 1439 | . 2 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((((𝐴 Xrm 𝑁)↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = 1 ↔ ∃𝑎 ∈ ℤ ((𝐴 Xrm 𝑁) = (𝐴 Xrm 𝑎) ∧ (𝐴 Yrm 𝑁) = (𝐴 Yrm 𝑎)))) |
19 | 11, 18 | mpbird 249 | 1 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (((𝐴 Xrm 𝑁)↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 ‘cfv 6137 (class class class)co 6924 1c1 10275 · cmul 10279 − cmin 10608 2c2 11435 ℕ0cn0 11647 ℤcz 11733 ℤ≥cuz 11997 ↑cexp 13183 Xrm crmx 38438 Yrm crmy 38439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-omul 7850 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-acn 9103 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-xnn0 11720 df-z 11734 df-dec 11851 df-uz 11998 df-q 12101 df-rp 12143 df-xneg 12262 df-xadd 12263 df-xmul 12264 df-ioo 12496 df-ioc 12497 df-ico 12498 df-icc 12499 df-fz 12649 df-fzo 12790 df-fl 12917 df-mod 12993 df-seq 13125 df-exp 13184 df-fac 13385 df-bc 13414 df-hash 13442 df-shft 14220 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-limsup 14619 df-clim 14636 df-rlim 14637 df-sum 14834 df-ef 15209 df-sin 15211 df-cos 15212 df-pi 15214 df-dvds 15397 df-gcd 15633 df-numer 15858 df-denom 15859 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-starv 16364 df-sca 16365 df-vsca 16366 df-ip 16367 df-tset 16368 df-ple 16369 df-ds 16371 df-unif 16372 df-hom 16373 df-cco 16374 df-rest 16480 df-topn 16481 df-0g 16499 df-gsum 16500 df-topgen 16501 df-pt 16502 df-prds 16505 df-xrs 16559 df-qtop 16564 df-imas 16565 df-xps 16567 df-mre 16643 df-mrc 16644 df-acs 16646 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-submnd 17733 df-mulg 17939 df-cntz 18144 df-cmn 18592 df-psmet 20145 df-xmet 20146 df-met 20147 df-bl 20148 df-mopn 20149 df-fbas 20150 df-fg 20151 df-cnfld 20154 df-top 21117 df-topon 21134 df-topsp 21156 df-bases 21169 df-cld 21242 df-ntr 21243 df-cls 21244 df-nei 21321 df-lp 21359 df-perf 21360 df-cn 21450 df-cnp 21451 df-haus 21538 df-tx 21785 df-hmeo 21978 df-fil 22069 df-fm 22161 df-flim 22162 df-flf 22163 df-xms 22544 df-ms 22545 df-tms 22546 df-cncf 23100 df-limc 24078 df-dv 24079 df-log 24751 df-squarenn 38379 df-pell1qr 38380 df-pell14qr 38381 df-pell1234qr 38382 df-pellfund 38383 df-rmx 38440 df-rmy 38441 |
This theorem is referenced by: rmxyneg 38458 rmxdbl 38477 jm2.19lem1 38529 jm2.27c 38547 rmxdiophlem 38555 |
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