decnncl - Metamath Proof Explorer

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Theorem decnncl 11518

Description: Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)

**Hypotheses**
Ref	Expression
decnncl.1	⊢ 𝐴 ∈ ℕ₀
decnncl.2	⊢ 𝐵 ∈ ℕ

**Assertion**
Ref	Expression
decnncl	⊢ ;𝐴𝐵 ∈ ℕ

**Proof of Theorem decnncl**
Step	Hyp	Ref	Expression
1		dfdec10 11497	. 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
2		10nn0 11516	. . 3 ⊢ ;10 ∈ ℕ₀
3		decnncl.1	. . 3 ⊢ 𝐴 ∈ ℕ₀
4		decnncl.2	. . 3 ⊢ 𝐵 ∈ ℕ
5	2, 3, 4	numnncl 11507	. 2 ⊢ ((;10 · 𝐴) + 𝐵) ∈ ℕ
6	1, 5	eqeltri 2697	1 ⊢ ;𝐴𝐵 ∈ ℕ

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 1990 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ℕcn 11020 ℕ₀cn0 11292 cdc 11493

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-dec 11494

This theorem is referenced by: 11prm 15822 13prm 15823 17prm 15824 19prm 15825 23prm 15826 37prm 15828 43prm 15829 83prm 15830 139prm 15831 163prm 15832 317prm 15833 631prm 15834 1259lem1 15838 1259lem2 15839 1259lem3 15840 1259lem4 15841 1259lem5 15842 1259prm 15843 2503lem1 15844 2503lem2 15845 2503lem3 15846 2503prm 15847 4001lem1 15848 4001lem2 15849 4001lem3 15850 4001lem4 15851 4001prm 15852 ocndx 16060 ocid 16061 dsndx 16062 dsid 16063 unifndx 16064 unifid 16065 odrngstr 16066 ressds 16073 homndx 16074 homid 16075 ccondx 16076 ccoid 16077 resshom 16078 ressco 16079 imasvalstr 16112 prdsvalstr 16113 oppchomfval 16374 oppcbas 16378 rescco 16492 catstr 16617 ipostr 17153 mgpds 18499 srads 19186 cnfldstr 19748 ressunif 22066 tuslem 22071 tmslem 22287 mcubic 24574 cubic2 24575 cubic 24576 quart1cl 24581 quart1lem 24582 quart1 24583 quartlem1 24584 quartlem2 24585 log2ub 24676 log2le1 24677 birthday 24681 bposlem8 25016 bposlem9 25017 pntlemd 25283 pntlema 25285 pntlemb 25286 pntlemf 25294 pntlemo 25296 itvndx 25339 lngndx 25340 itvid 25341 lngid 25342 trkgstr 25343 ttgval 25755 ttglem 25756 ttgds 25761 eengstr 25860 edgfid 25869 edgfndxnn 25870 edgfndxid 25871 baseltedgf 25872 257prm 41473 fmtno4prmfac 41484 fmtno4prmfac193 41485 fmtno4nprmfac193 41486 fmtno5nprm 41495 139prmALT 41511 127prm 41515 3exp4mod41 41533 41prothprmlem2 41535 bgoldbtbndlem1 41693 tgblthelfgott 41703 tgoldbachlt 41704 tgoldbach 41705 bgoldbachltOLD 41707 tgblthelfgottOLD 41709 tgoldbachltOLD 41710 tgoldbachOLD 41712

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