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Mirrors > Home > MPE Home > Th. List > Mathboxes > rmyluc2 | Structured version Visualization version GIF version |
Description: Lucas sequence property of Y with better output ordering. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
Ref | Expression |
---|---|
rmyluc2 | ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (𝑁 + 1)) = (((2 · 𝐴) · (𝐴 Yrm 𝑁)) − (𝐴 Yrm (𝑁 − 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmyluc 37973 | . 2 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (𝑁 + 1)) = ((2 · ((𝐴 Yrm 𝑁) · 𝐴)) − (𝐴 Yrm (𝑁 − 1)))) | |
2 | frmy 37950 | . . . . . . . 8 ⊢ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ | |
3 | 2 | fovcl 6918 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑁) ∈ ℤ) |
4 | 3 | zcnd 11646 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑁) ∈ ℂ) |
5 | eluzelcn 11862 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℂ) | |
6 | 5 | adantr 472 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → 𝐴 ∈ ℂ) |
7 | 4, 6 | mulcomd 10224 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm 𝑁) · 𝐴) = (𝐴 · (𝐴 Yrm 𝑁))) |
8 | 7 | oveq2d 6817 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (2 · ((𝐴 Yrm 𝑁) · 𝐴)) = (2 · (𝐴 · (𝐴 Yrm 𝑁)))) |
9 | 2cnd 11256 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → 2 ∈ ℂ) | |
10 | 9, 6, 4 | mulassd 10226 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((2 · 𝐴) · (𝐴 Yrm 𝑁)) = (2 · (𝐴 · (𝐴 Yrm 𝑁)))) |
11 | 8, 10 | eqtr4d 2785 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (2 · ((𝐴 Yrm 𝑁) · 𝐴)) = ((2 · 𝐴) · (𝐴 Yrm 𝑁))) |
12 | 11 | oveq1d 6816 | . 2 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((2 · ((𝐴 Yrm 𝑁) · 𝐴)) − (𝐴 Yrm (𝑁 − 1))) = (((2 · 𝐴) · (𝐴 Yrm 𝑁)) − (𝐴 Yrm (𝑁 − 1)))) |
13 | 1, 12 | eqtrd 2782 | 1 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (𝑁 + 1)) = (((2 · 𝐴) · (𝐴 Yrm 𝑁)) − (𝐴 Yrm (𝑁 − 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1620 ∈ wcel 2127 ‘cfv 6037 (class class class)co 6801 ℂcc 10097 1c1 10100 + caddc 10102 · cmul 10104 − cmin 10429 2c2 11233 ℤcz 11540 ℤ≥cuz 11850 Yrm crmy 37936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-rep 4911 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-inf2 8699 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 ax-pre-sup 10177 ax-addf 10178 ax-mulf 10179 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-fal 1626 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rmo 3046 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-int 4616 df-iun 4662 df-iin 4663 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-se 5214 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-isom 6046 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-of 7050 df-om 7219 df-1st 7321 df-2nd 7322 df-supp 7452 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-1o 7717 df-2o 7718 df-oadd 7721 df-omul 7722 df-er 7899 df-map 8013 df-pm 8014 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8429 df-fi 8470 df-sup 8501 df-inf 8502 df-oi 8568 df-card 8926 df-acn 8929 df-cda 9153 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-div 10848 df-nn 11184 df-2 11242 df-3 11243 df-4 11244 df-5 11245 df-6 11246 df-7 11247 df-8 11248 df-9 11249 df-n0 11456 df-xnn0 11527 df-z 11541 df-dec 11657 df-uz 11851 df-q 11953 df-rp 11997 df-xneg 12110 df-xadd 12111 df-xmul 12112 df-ioo 12343 df-ioc 12344 df-ico 12345 df-icc 12346 df-fz 12491 df-fzo 12631 df-fl 12758 df-mod 12834 df-seq 12967 df-exp 13026 df-fac 13226 df-bc 13255 df-hash 13283 df-shft 13977 df-cj 14009 df-re 14010 df-im 14011 df-sqrt 14145 df-abs 14146 df-limsup 14372 df-clim 14389 df-rlim 14390 df-sum 14587 df-ef 14968 df-sin 14970 df-cos 14971 df-pi 14973 df-dvds 15154 df-gcd 15390 df-numer 15616 df-denom 15617 df-struct 16032 df-ndx 16033 df-slot 16034 df-base 16036 df-sets 16037 df-ress 16038 df-plusg 16127 df-mulr 16128 df-starv 16129 df-sca 16130 df-vsca 16131 df-ip 16132 df-tset 16133 df-ple 16134 df-ds 16137 df-unif 16138 df-hom 16139 df-cco 16140 df-rest 16256 df-topn 16257 df-0g 16275 df-gsum 16276 df-topgen 16277 df-pt 16278 df-prds 16281 df-xrs 16335 df-qtop 16340 df-imas 16341 df-xps 16343 df-mre 16419 df-mrc 16420 df-acs 16422 df-mgm 17414 df-sgrp 17456 df-mnd 17467 df-submnd 17508 df-mulg 17713 df-cntz 17921 df-cmn 18366 df-psmet 19911 df-xmet 19912 df-met 19913 df-bl 19914 df-mopn 19915 df-fbas 19916 df-fg 19917 df-cnfld 19920 df-top 20872 df-topon 20889 df-topsp 20910 df-bases 20923 df-cld 20996 df-ntr 20997 df-cls 20998 df-nei 21075 df-lp 21113 df-perf 21114 df-cn 21204 df-cnp 21205 df-haus 21292 df-tx 21538 df-hmeo 21731 df-fil 21822 df-fm 21914 df-flim 21915 df-flf 21916 df-xms 22297 df-ms 22298 df-tms 22299 df-cncf 22853 df-limc 23800 df-dv 23801 df-log 24473 df-squarenn 37876 df-pell1qr 37877 df-pell14qr 37878 df-pell1234qr 37879 df-pellfund 37880 df-rmx 37937 df-rmy 37938 |
This theorem is referenced by: jm2.17a 37998 jm2.17b 37999 jm2.17c 38000 |
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