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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > frmx | Structured version Visualization version GIF version |
Description: The X sequence is a nonnegative integer. See rmxnn 41322 for a strengthening. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
Ref | Expression |
---|---|
frmx | ⊢ Xrm :((ℤ≥‘2) × ℤ)⟶ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmxyelxp 41283 | . . . 4 ⊢ ((𝑎 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℤ) → (◡(𝑐 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑐) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑐))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑏)) ∈ (ℕ0 × ℤ)) | |
2 | xp1st 7957 | . . . 4 ⊢ ((◡(𝑐 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑐) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑐))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑏)) ∈ (ℕ0 × ℤ) → (1st ‘(◡(𝑐 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑐) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑐))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑏))) ∈ ℕ0) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ ((𝑎 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℤ) → (1st ‘(◡(𝑐 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑐) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑐))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑏))) ∈ ℕ0) |
4 | 3 | rgen2 3191 | . 2 ⊢ ∀𝑎 ∈ (ℤ≥‘2)∀𝑏 ∈ ℤ (1st ‘(◡(𝑐 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑐) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑐))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑏))) ∈ ℕ0 |
5 | df-rmx 41272 | . . 3 ⊢ Xrm = (𝑎 ∈ (ℤ≥‘2), 𝑏 ∈ ℤ ↦ (1st ‘(◡(𝑐 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑐) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑐))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑏)))) | |
6 | 5 | fmpo 8004 | . 2 ⊢ (∀𝑎 ∈ (ℤ≥‘2)∀𝑏 ∈ ℤ (1st ‘(◡(𝑐 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑐) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑐))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑏))) ∈ ℕ0 ↔ Xrm :((ℤ≥‘2) × ℤ)⟶ℕ0) |
7 | 4, 6 | mpbi 229 | 1 ⊢ Xrm :((ℤ≥‘2) × ℤ)⟶ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∈ wcel 2107 ∀wral 3061 ↦ cmpt 5192 × cxp 5635 ◡ccnv 5636 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 1st c1st 7923 2nd c2nd 7924 1c1 11060 + caddc 11062 · cmul 11064 − cmin 11393 2c2 12216 ℕ0cn0 12421 ℤcz 12507 ℤ≥cuz 12771 ↑cexp 13976 √csqrt 15127 Xrm crmx 41270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-oadd 8420 df-omul 8421 df-er 8654 df-map 8773 df-pm 8774 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-fi 9355 df-sup 9386 df-inf 9387 df-oi 9454 df-card 9883 df-acn 9886 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-xnn0 12494 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13434 df-fzo 13577 df-fl 13706 df-mod 13784 df-seq 13916 df-exp 13977 df-fac 14183 df-bc 14212 df-hash 14240 df-shft 14961 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-limsup 15362 df-clim 15379 df-rlim 15380 df-sum 15580 df-ef 15958 df-sin 15960 df-cos 15961 df-pi 15963 df-dvds 16145 df-gcd 16383 df-numer 16618 df-denom 16619 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-hom 17165 df-cco 17166 df-rest 17312 df-topn 17313 df-0g 17331 df-gsum 17332 df-topgen 17333 df-pt 17334 df-prds 17337 df-xrs 17392 df-qtop 17397 df-imas 17398 df-xps 17400 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-mulg 18881 df-cntz 19105 df-cmn 19572 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-fbas 20816 df-fg 20817 df-cnfld 20820 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-cld 22393 df-ntr 22394 df-cls 22395 df-nei 22472 df-lp 22510 df-perf 22511 df-cn 22601 df-cnp 22602 df-haus 22689 df-tx 22936 df-hmeo 23129 df-fil 23220 df-fm 23312 df-flim 23313 df-flf 23314 df-xms 23696 df-ms 23697 df-tms 23698 df-cncf 24264 df-limc 25253 df-dv 25254 df-log 25935 df-squarenn 41211 df-pell1qr 41212 df-pell14qr 41213 df-pell1234qr 41214 df-pellfund 41215 df-rmx 41272 |
This theorem is referenced by: rmxycomplete 41288 rmxynorm 41289 rmxyneg 41291 rmxyadd 41292 rmxy1 41293 rmxy0 41294 rmyp1 41304 rmxm1 41305 rmym1 41306 rmxluc 41307 rmyluc 41308 rmxdbl 41310 rmydbl 41311 rmxypos 41318 ltrmynn0 41319 ltrmxnn0 41320 lermxnn0 41321 rmxnn 41322 jm2.24nn 41330 jm2.24 41334 jm2.18 41359 jm2.19lem1 41360 jm2.19lem2 41361 jm2.22 41366 jm2.23 41367 jm2.20nn 41368 jm2.25 41370 jm2.26a 41371 jm2.26lem3 41372 jm2.26 41373 jm2.27c 41378 rmxdiophlem 41386 jm3.1 41391 expdiophlem1 41392 |
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