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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > frmy | Structured version Visualization version GIF version |
Description: The Y sequence is an integer. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
Ref | Expression |
---|---|
frmy | ⊢ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmxyelxp 42900 | . . . 4 ⊢ ((𝑎 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℤ) → (◡(𝑐 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑐) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑐))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑏)) ∈ (ℕ0 × ℤ)) | |
2 | xp2nd 8045 | . . . 4 ⊢ ((◡(𝑐 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑐) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑐))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑏)) ∈ (ℕ0 × ℤ) → (2nd ‘(◡(𝑐 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑐) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑐))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑏))) ∈ ℤ) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ ((𝑎 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℤ) → (2nd ‘(◡(𝑐 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑐) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑐))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑏))) ∈ ℤ) |
4 | 3 | rgen2 3196 | . 2 ⊢ ∀𝑎 ∈ (ℤ≥‘2)∀𝑏 ∈ ℤ (2nd ‘(◡(𝑐 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑐) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑐))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑏))) ∈ ℤ |
5 | df-rmy 42890 | . . 3 ⊢ Yrm = (𝑎 ∈ (ℤ≥‘2), 𝑏 ∈ ℤ ↦ (2nd ‘(◡(𝑐 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑐) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑐))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑏)))) | |
6 | 5 | fmpo 8091 | . 2 ⊢ (∀𝑎 ∈ (ℤ≥‘2)∀𝑏 ∈ ℤ (2nd ‘(◡(𝑐 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑐) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑐))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑏))) ∈ ℤ ↔ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ) |
7 | 4, 6 | mpbi 230 | 1 ⊢ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∈ wcel 2105 ∀wral 3058 ↦ cmpt 5230 × cxp 5686 ◡ccnv 5687 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 1st c1st 8010 2nd c2nd 8011 1c1 11153 + caddc 11155 · cmul 11157 − cmin 11489 2c2 12318 ℕ0cn0 12523 ℤcz 12610 ℤ≥cuz 12875 ↑cexp 14098 √csqrt 15268 Yrm crmy 42888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-omul 8509 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-acn 9979 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-dvds 16287 df-gcd 16528 df-numer 16768 df-denom 16769 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-limc 25915 df-dv 25916 df-log 26612 df-squarenn 42828 df-pell1qr 42829 df-pell14qr 42830 df-pell1234qr 42831 df-pellfund 42832 df-rmy 42890 |
This theorem is referenced by: rmxycomplete 42905 rmxynorm 42906 rmxyneg 42908 rmxyadd 42909 rmxy1 42910 rmxy0 42911 rmxp1 42920 rmxm1 42922 rmym1 42923 rmxluc 42924 rmyluc 42925 rmyluc2 42926 rmxdbl 42927 rmydbl 42928 rmxypos 42935 ltrmynn0 42936 ltrmxnn0 42937 ltrmy 42940 rmyeq0 42941 rmyeq 42942 lermy 42943 rmynn 42944 rmynn0 42945 rmyabs 42946 jm2.24nn 42947 jm2.17a 42948 jm2.17b 42949 jm2.17c 42950 jm2.24 42951 rmygeid 42952 jm2.18 42976 jm2.19lem1 42977 jm2.19lem2 42978 jm2.19 42981 jm2.22 42983 jm2.23 42984 jm2.20nn 42985 jm2.25 42987 jm2.26a 42988 jm2.26lem3 42989 jm2.26 42990 jm2.15nn0 42991 jm2.16nn0 42992 jm2.27a 42993 jm2.27c 42995 rmydioph 43002 jm3.1lem1 43005 jm3.1 43008 expdiophlem1 43009 |
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