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| Mirrors > Home > MPE Home > Th. List > decnncl | Structured version Visualization version GIF version | ||
| Description: Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
| Ref | Expression |
|---|---|
| decnncl.1 | ⊢ 𝐴 ∈ ℕ0 |
| decnncl.2 | ⊢ 𝐵 ∈ ℕ |
| Ref | Expression |
|---|---|
| decnncl | ⊢ ;𝐴𝐵 ∈ ℕ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfdec10 12736 | . 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
| 2 | 10nn0 12751 | . . 3 ⊢ ;10 ∈ ℕ0 | |
| 3 | decnncl.1 | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
| 4 | decnncl.2 | . . 3 ⊢ 𝐵 ∈ ℕ | |
| 5 | 2, 3, 4 | numnncl 12743 | . 2 ⊢ ((;10 · 𝐴) + 𝐵) ∈ ℕ |
| 6 | 1, 5 | eqeltri 2837 | 1 ⊢ ;𝐴𝐵 ∈ ℕ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ℕcn 12266 ℕ0cn0 12526 ;cdc 12733 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-dec 12734 |
| This theorem is referenced by: 11prm 17152 13prm 17153 17prm 17154 19prm 17155 23prm 17156 37prm 17158 43prm 17159 83prm 17160 139prm 17161 163prm 17162 317prm 17163 631prm 17164 1259lem1 17168 1259lem2 17169 1259lem3 17170 1259lem4 17171 1259lem5 17172 1259prm 17173 2503lem1 17174 2503lem2 17175 2503lem3 17176 2503prm 17177 4001lem1 17178 4001lem2 17179 4001lem3 17180 4001lem4 17181 4001prm 17182 ocndx 17425 ocid 17426 dsndx 17429 dsid 17430 dsndxnn 17431 unifndx 17439 unifid 17440 unifndxnn 17441 slotsdifunifndx 17445 odrngstr 17447 homndx 17455 homid 17456 ccondx 17457 ccoid 17458 slotsdifocndx 17462 imasvalstr 17496 prdsvalstr 17497 catstr 18005 ipostr 18574 sradsOLD 21192 cnfldstr 21366 cnfldstrOLD 21381 tuslemOLD 24276 tmslemOLD 24495 mcubic 26890 cubic2 26891 cubic 26892 quart1cl 26897 quart1lem 26898 quart1 26899 quartlem1 26900 quartlem2 26901 log2ub 26992 log2le1 26993 birthday 26997 bposlem8 27335 bposlem9 27336 pntlemd 27638 pntlema 27640 pntlemb 27641 pntlemf 27649 pntlemo 27651 itvndx 28445 lngndx 28446 itvid 28447 lngid 28448 slotsinbpsd 28449 slotslnbpsd 28450 lngndxnitvndx 28451 trkgstr 28452 ttgvalOLD 28884 ttglemOLD 28886 ttgdsOLD 28895 eengstr 28995 edgfid 29005 edgfndx 29006 edgfndxnn 29007 edgfndxidOLD 29009 baseltedgfOLD 29011 eufndx 33293 eufid 33294 12gcd5e1 42004 60gcd7e1 42006 420gcd8e4 42007 12lcm5e60 42009 60lcm7e420 42011 420lcm8e840 42012 lcmineqlem 42053 3lexlogpow5ineq1 42055 3lexlogpow5ineq2 42056 3lexlogpow5ineq4 42057 3lexlogpow2ineq1 42059 3lexlogpow2ineq2 42060 3lexlogpow5ineq5 42061 aks4d1p1p5 42076 aks4d1p1 42077 257prm 47548 fmtno4prmfac 47559 fmtno4prmfac193 47560 fmtno4nprmfac193 47561 fmtno5nprm 47570 139prmALT 47583 127prm 47586 3exp4mod41 47603 41prothprmlem2 47605 2exp340mod341 47720 341fppr2 47721 bgoldbtbndlem1 47792 tgblthelfgott 47802 tgoldbachlt 47803 tgoldbach 47804 |
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