decnncl - Metamath Proof Explorer

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Theorem decnncl 11394

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 11373	. 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
2		10nn0 11392	. . 3 ⊢ ;10 ∈ ℕ₀
3		decnncl.1	. . 3 ⊢ 𝐴 ∈ ℕ₀
4		decnncl.2	. . 3 ⊢ 𝐵 ∈ ℕ
5	2, 3, 4	numnncl 11383	. 2 ⊢ ((;10 · 𝐴) + 𝐵) ∈ ℕ
6	1, 5	eqeltri 2684	1 ⊢ ;𝐴𝐵 ∈ ℕ

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 1977 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ℕcn 10897 ℕ₀cn0 11169 cdc 11369

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-dec 11370

This theorem is referenced by: 11prm 15660 13prm 15661 17prm 15662 19prm 15663 23prm 15664 37prm 15666 43prm 15667 83prm 15668 139prm 15669 163prm 15670 317prm 15671 631prm 15672 1259lem1 15676 1259lem2 15677 1259lem3 15678 1259lem4 15679 1259lem5 15680 1259prm 15681 2503lem1 15682 2503lem2 15683 2503lem3 15684 2503prm 15685 4001lem1 15686 4001lem2 15687 4001lem3 15688 4001lem4 15689 4001prm 15690 ocndx 15883 ocid 15884 dsndx 15885 dsid 15886 unifndx 15887 unifid 15888 odrngstr 15889 ressds 15896 homndx 15897 homid 15898 ccondx 15899 ccoid 15900 resshom 15901 ressco 15902 imasvalstr 15935 prdsvalstr 15936 oppchomfval 16197 oppcbas 16201 rescco 16315 catstr 16440 ipostr 16976 mgpds 18322 srads 19007 cnfldstr 19569 ressunif 21876 tuslem 21881 tmslem 22097 mcubic 24374 cubic2 24375 cubic 24376 quart1cl 24381 quart1lem 24382 quart1 24383 quartlem1 24384 quartlem2 24385 log2ub 24476 log2le1 24477 birthday 24481 bposlem8 24816 bposlem9 24817 pntlemd 25083 pntlema 25085 pntlemb 25086 pntlemf 25094 pntlemo 25096 itvndx 25139 lngndx 25140 itvid 25141 lngid 25142 trkgstr 25143 ttgval 25555 ttglem 25556 ttgds 25561 eengstr 25660 edgfndxnn 25669 edgfndxid 25670 baseltedgf 25671 struct2griedg 25705 uhgrstrrepe 25745 257prm 40011 fmtno4prmfac 40022 fmtno4prmfac193 40023 fmtno4nprmfac193 40024 fmtno5nprm 40033 139prmALT 40049 127prm 40053 3exp4mod41 40071 41prothprmlem2 40073 bgoldbtbndlem1 40221 tgblthelfgott 40229 tgoldbachlt 40230 tgoldbach 40232 bgoldbachltOLD 40234 tgblthelfgottOLD 40236 tgoldbachltOLD 40237 tgoldbachOLD 40239

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