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Mirrors > Home > MPE Home > Th. List > Mathboxes > rmxluc | Structured version Visualization version GIF version |
Description: The X sequence is a Lucas (second-order integer recurrence) sequence. Part 3 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 14-Oct-2014.) |
Ref | Expression |
---|---|
rmxluc | ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (𝑁 + 1)) = (((2 · 𝐴) · (𝐴 Xrm 𝑁)) − (𝐴 Xrm (𝑁 − 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2zm 12019 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
2 | frmx 39503 | . . . . . . . 8 ⊢ Xrm :((ℤ≥‘2) × ℤ)⟶ℕ0 | |
3 | 2 | fovcl 7273 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑁 − 1) ∈ ℤ) → (𝐴 Xrm (𝑁 − 1)) ∈ ℕ0) |
4 | 3 | nn0cnd 11951 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑁 − 1) ∈ ℤ) → (𝐴 Xrm (𝑁 − 1)) ∈ ℂ) |
5 | 1, 4 | sylan2 594 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (𝑁 − 1)) ∈ ℂ) |
6 | peano2z 12017 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
7 | 2 | fovcl 7273 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑁 + 1) ∈ ℤ) → (𝐴 Xrm (𝑁 + 1)) ∈ ℕ0) |
8 | 7 | nn0cnd 11951 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑁 + 1) ∈ ℤ) → (𝐴 Xrm (𝑁 + 1)) ∈ ℂ) |
9 | 6, 8 | sylan2 594 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (𝑁 + 1)) ∈ ℂ) |
10 | 5, 9 | addcomd 10836 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm (𝑁 − 1)) + (𝐴 Xrm (𝑁 + 1))) = ((𝐴 Xrm (𝑁 + 1)) + (𝐴 Xrm (𝑁 − 1)))) |
11 | rmxp1 39522 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (𝑁 + 1)) = (((𝐴 Xrm 𝑁) · 𝐴) + (((𝐴↑2) − 1) · (𝐴 Yrm 𝑁)))) | |
12 | rmxm1 39524 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (𝑁 − 1)) = ((𝐴 · (𝐴 Xrm 𝑁)) − (((𝐴↑2) − 1) · (𝐴 Yrm 𝑁)))) | |
13 | 11, 12 | oveq12d 7168 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm (𝑁 + 1)) + (𝐴 Xrm (𝑁 − 1))) = ((((𝐴 Xrm 𝑁) · 𝐴) + (((𝐴↑2) − 1) · (𝐴 Yrm 𝑁))) + ((𝐴 · (𝐴 Xrm 𝑁)) − (((𝐴↑2) − 1) · (𝐴 Yrm 𝑁))))) |
14 | 2 | fovcl 7273 | . . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) ∈ ℕ0) |
15 | 14 | nn0cnd 11951 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) ∈ ℂ) |
16 | eluzelcn 12249 | . . . . . . . 8 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℂ) | |
17 | 16 | adantr 483 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → 𝐴 ∈ ℂ) |
18 | 15, 17 | mulcld 10655 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm 𝑁) · 𝐴) ∈ ℂ) |
19 | rmspecnonsq 39497 | . . . . . . . . . 10 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ (ℕ ∖ ◻NN)) | |
20 | 19 | eldifad 3948 | . . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℕ) |
21 | 20 | nncnd 11648 | . . . . . . . 8 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℂ) |
22 | 21 | adantr 483 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴↑2) − 1) ∈ ℂ) |
23 | frmy 39504 | . . . . . . . . 9 ⊢ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ | |
24 | 23 | fovcl 7273 | . . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑁) ∈ ℤ) |
25 | 24 | zcnd 12082 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑁) ∈ ℂ) |
26 | 22, 25 | mulcld 10655 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (((𝐴↑2) − 1) · (𝐴 Yrm 𝑁)) ∈ ℂ) |
27 | 17, 15 | mulcld 10655 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 · (𝐴 Xrm 𝑁)) ∈ ℂ) |
28 | 18, 26, 27 | ppncand 11031 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((((𝐴 Xrm 𝑁) · 𝐴) + (((𝐴↑2) − 1) · (𝐴 Yrm 𝑁))) + ((𝐴 · (𝐴 Xrm 𝑁)) − (((𝐴↑2) − 1) · (𝐴 Yrm 𝑁)))) = (((𝐴 Xrm 𝑁) · 𝐴) + (𝐴 · (𝐴 Xrm 𝑁)))) |
29 | 15, 17 | mulcomd 10656 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm 𝑁) · 𝐴) = (𝐴 · (𝐴 Xrm 𝑁))) |
30 | 29 | oveq1d 7165 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (((𝐴 Xrm 𝑁) · 𝐴) + (𝐴 · (𝐴 Xrm 𝑁))) = ((𝐴 · (𝐴 Xrm 𝑁)) + (𝐴 · (𝐴 Xrm 𝑁)))) |
31 | 2cnd 11709 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → 2 ∈ ℂ) | |
32 | 31, 17, 15 | mulassd 10658 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((2 · 𝐴) · (𝐴 Xrm 𝑁)) = (2 · (𝐴 · (𝐴 Xrm 𝑁)))) |
33 | 27 | 2timesd 11874 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (2 · (𝐴 · (𝐴 Xrm 𝑁))) = ((𝐴 · (𝐴 Xrm 𝑁)) + (𝐴 · (𝐴 Xrm 𝑁)))) |
34 | 32, 33 | eqtr2d 2857 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 · (𝐴 Xrm 𝑁)) + (𝐴 · (𝐴 Xrm 𝑁))) = ((2 · 𝐴) · (𝐴 Xrm 𝑁))) |
35 | 28, 30, 34 | 3eqtrd 2860 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((((𝐴 Xrm 𝑁) · 𝐴) + (((𝐴↑2) − 1) · (𝐴 Yrm 𝑁))) + ((𝐴 · (𝐴 Xrm 𝑁)) − (((𝐴↑2) − 1) · (𝐴 Yrm 𝑁)))) = ((2 · 𝐴) · (𝐴 Xrm 𝑁))) |
36 | 10, 13, 35 | 3eqtrd 2860 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm (𝑁 − 1)) + (𝐴 Xrm (𝑁 + 1))) = ((2 · 𝐴) · (𝐴 Xrm 𝑁))) |
37 | 2cn 11706 | . . . . . 6 ⊢ 2 ∈ ℂ | |
38 | mulcl 10615 | . . . . . 6 ⊢ ((2 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (2 · 𝐴) ∈ ℂ) | |
39 | 37, 17, 38 | sylancr 589 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (2 · 𝐴) ∈ ℂ) |
40 | 39, 15 | mulcld 10655 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((2 · 𝐴) · (𝐴 Xrm 𝑁)) ∈ ℂ) |
41 | 40, 5, 9 | subaddd 11009 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((((2 · 𝐴) · (𝐴 Xrm 𝑁)) − (𝐴 Xrm (𝑁 − 1))) = (𝐴 Xrm (𝑁 + 1)) ↔ ((𝐴 Xrm (𝑁 − 1)) + (𝐴 Xrm (𝑁 + 1))) = ((2 · 𝐴) · (𝐴 Xrm 𝑁)))) |
42 | 36, 41 | mpbird 259 | . 2 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (((2 · 𝐴) · (𝐴 Xrm 𝑁)) − (𝐴 Xrm (𝑁 − 1))) = (𝐴 Xrm (𝑁 + 1))) |
43 | 42 | eqcomd 2827 | 1 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (𝑁 + 1)) = (((2 · 𝐴) · (𝐴 Xrm 𝑁)) − (𝐴 Xrm (𝑁 − 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 1c1 10532 + caddc 10534 · cmul 10536 − cmin 10864 ℕcn 11632 2c2 11686 ℕ0cn0 11891 ℤcz 11975 ℤ≥cuz 12237 ↑cexp 13423 ◻NNcsquarenn 39426 Xrm crmx 39490 Yrm crmy 39491 |
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-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 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 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 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-int 4870 df-iun 4914 df-iin 4915 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-se 5510 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-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-acn 9365 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-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-pi 15420 df-dvds 15602 df-gcd 15838 df-numer 16069 df-denom 16070 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459 df-log 25134 df-squarenn 39431 df-pell1qr 39432 df-pell14qr 39433 df-pell1234qr 39434 df-pellfund 39435 df-rmx 39492 df-rmy 39493 |
This theorem is referenced by: jm2.18 39578 |
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