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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 40347 | . . . 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 7760 | . . . 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 3116 | . 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 40338 | . . 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 7804 | . 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 233 | 1 ⊢ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∈ wcel 2114 ∀wral 3054 ↦ cmpt 5120 × cxp 5533 ◡ccnv 5534 ⟶wf 6346 ‘cfv 6350 (class class class)co 7183 1st c1st 7725 2nd c2nd 7726 1c1 10629 + caddc 10631 · cmul 10633 − cmin 10961 2c2 11784 ℕ0cn0 11989 ℤcz 12075 ℤ≥cuz 12337 ↑cexp 13534 √csqrt 14695 Yrm crmy 40336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-inf2 9190 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 ax-pre-sup 10706 ax-addf 10707 ax-mulf 10708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 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 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-of 7438 df-om 7613 df-1st 7727 df-2nd 7728 df-supp 7870 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-2o 8145 df-oadd 8148 df-omul 8149 df-er 8333 df-map 8452 df-pm 8453 df-ixp 8521 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-fsupp 8920 df-fi 8961 df-sup 8992 df-inf 8993 df-oi 9060 df-card 9454 df-acn 9457 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-div 11389 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-5 11795 df-6 11796 df-7 11797 df-8 11798 df-9 11799 df-n0 11990 df-xnn0 12062 df-z 12076 df-dec 12193 df-uz 12338 df-q 12444 df-rp 12486 df-xneg 12603 df-xadd 12604 df-xmul 12605 df-ioo 12838 df-ioc 12839 df-ico 12840 df-icc 12841 df-fz 12995 df-fzo 13138 df-fl 13266 df-mod 13342 df-seq 13474 df-exp 13535 df-fac 13739 df-bc 13768 df-hash 13796 df-shft 14529 df-cj 14561 df-re 14562 df-im 14563 df-sqrt 14697 df-abs 14698 df-limsup 14931 df-clim 14948 df-rlim 14949 df-sum 15149 df-ef 15526 df-sin 15528 df-cos 15529 df-pi 15531 df-dvds 15713 df-gcd 15951 df-numer 16188 df-denom 16189 df-struct 16601 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-mulr 16695 df-starv 16696 df-sca 16697 df-vsca 16698 df-ip 16699 df-tset 16700 df-ple 16701 df-ds 16703 df-unif 16704 df-hom 16705 df-cco 16706 df-rest 16812 df-topn 16813 df-0g 16831 df-gsum 16832 df-topgen 16833 df-pt 16834 df-prds 16837 df-xrs 16891 df-qtop 16896 df-imas 16897 df-xps 16899 df-mre 16973 df-mrc 16974 df-acs 16976 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-submnd 18086 df-mulg 18356 df-cntz 18578 df-cmn 19039 df-psmet 20222 df-xmet 20223 df-met 20224 df-bl 20225 df-mopn 20226 df-fbas 20227 df-fg 20228 df-cnfld 20231 df-top 21658 df-topon 21675 df-topsp 21697 df-bases 21710 df-cld 21783 df-ntr 21784 df-cls 21785 df-nei 21862 df-lp 21900 df-perf 21901 df-cn 21991 df-cnp 21992 df-haus 22079 df-tx 22326 df-hmeo 22519 df-fil 22610 df-fm 22702 df-flim 22703 df-flf 22704 df-xms 23086 df-ms 23087 df-tms 23088 df-cncf 23643 df-limc 24631 df-dv 24632 df-log 25313 df-squarenn 40276 df-pell1qr 40277 df-pell14qr 40278 df-pell1234qr 40279 df-pellfund 40280 df-rmy 40338 |
This theorem is referenced by: rmxycomplete 40352 rmxynorm 40353 rmxyneg 40355 rmxyadd 40356 rmxy1 40357 rmxy0 40358 rmxp1 40367 rmxm1 40369 rmym1 40370 rmxluc 40371 rmyluc 40372 rmyluc2 40373 rmxdbl 40374 rmydbl 40375 rmxypos 40382 ltrmynn0 40383 ltrmxnn0 40384 ltrmy 40387 rmyeq0 40388 rmyeq 40389 lermy 40390 rmynn 40391 rmynn0 40392 rmyabs 40393 jm2.24nn 40394 jm2.17a 40395 jm2.17b 40396 jm2.17c 40397 jm2.24 40398 rmygeid 40399 jm2.18 40423 jm2.19lem1 40424 jm2.19lem2 40425 jm2.19 40428 jm2.22 40430 jm2.23 40431 jm2.20nn 40432 jm2.25 40434 jm2.26a 40435 jm2.26lem3 40436 jm2.26 40437 jm2.15nn0 40438 jm2.16nn0 40439 jm2.27a 40440 jm2.27c 40442 rmydioph 40449 jm3.1lem1 40452 jm3.1 40455 expdiophlem1 40456 |
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