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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 42179 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 42140 | . . . 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 8000 | . . . 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 3189 | . 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 42129 | . . 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 8047 | . 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 395 ∈ wcel 2098 ∀wral 3053 ↦ cmpt 5221 × cxp 5664 ◡ccnv 5665 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 1st c1st 7966 2nd c2nd 7967 1c1 11107 + caddc 11109 · cmul 11111 − cmin 11441 2c2 12264 ℕ0cn0 12469 ℤcz 12555 ℤ≥cuz 12819 ↑cexp 14024 √csqrt 15177 Xrm crmx 42127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-omul 8466 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-acn 9933 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-xnn0 12542 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-sin 16010 df-cos 16011 df-pi 16013 df-dvds 16195 df-gcd 16433 df-numer 16670 df-denom 16671 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-mulg 18986 df-cntz 19223 df-cmn 19692 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-fbas 21225 df-fg 21226 df-cnfld 21229 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-cld 22845 df-ntr 22846 df-cls 22847 df-nei 22924 df-lp 22962 df-perf 22963 df-cn 23053 df-cnp 23054 df-haus 23141 df-tx 23388 df-hmeo 23581 df-fil 23672 df-fm 23764 df-flim 23765 df-flf 23766 df-xms 24148 df-ms 24149 df-tms 24150 df-cncf 24720 df-limc 25717 df-dv 25718 df-log 26407 df-squarenn 42068 df-pell1qr 42069 df-pell14qr 42070 df-pell1234qr 42071 df-pellfund 42072 df-rmx 42129 |
This theorem is referenced by: rmxycomplete 42145 rmxynorm 42146 rmxyneg 42148 rmxyadd 42149 rmxy1 42150 rmxy0 42151 rmyp1 42161 rmxm1 42162 rmym1 42163 rmxluc 42164 rmyluc 42165 rmxdbl 42167 rmydbl 42168 rmxypos 42175 ltrmynn0 42176 ltrmxnn0 42177 lermxnn0 42178 rmxnn 42179 jm2.24nn 42187 jm2.24 42191 jm2.18 42216 jm2.19lem1 42217 jm2.19lem2 42218 jm2.22 42223 jm2.23 42224 jm2.20nn 42225 jm2.25 42227 jm2.26a 42228 jm2.26lem3 42229 jm2.26 42230 jm2.27c 42235 rmxdiophlem 42243 jm3.1 42248 expdiophlem1 42249 |
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