decnncl - Metamath Proof Explorer

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Theorem decnncl 12110

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 12093	. 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
2		10nn0 12108	. . 3 ⊢ ;10 ∈ ℕ₀
3		decnncl.1	. . 3 ⊢ 𝐴 ∈ ℕ₀
4		decnncl.2	. . 3 ⊢ 𝐵 ∈ ℕ
5	2, 3, 4	numnncl 12100	. 2 ⊢ ((;10 · 𝐴) + 𝐵) ∈ ℕ
6	1, 5	eqeltri 2907	1 ⊢ ;𝐴𝐵 ∈ ℕ

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 2108 (class class class)co 7148 0cc0 10529 1c1 10530 + caddc 10532 · cmul 10534 ℕcn 11630 ℕ₀cn0 11889 cdc 12090

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-om 7573 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-ltxr 10672 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-dec 12091

This theorem is referenced by: 11prm 16440 13prm 16441 17prm 16442 19prm 16443 23prm 16444 37prm 16446 43prm 16447 83prm 16448 139prm 16449 163prm 16450 317prm 16451 631prm 16452 1259lem1 16456 1259lem2 16457 1259lem3 16458 1259lem4 16459 1259lem5 16460 1259prm 16461 2503lem1 16462 2503lem2 16463 2503lem3 16464 2503prm 16465 4001lem1 16466 4001lem2 16467 4001lem3 16468 4001lem4 16469 4001prm 16470 ocndx 16665 ocid 16666 dsndx 16667 dsid 16668 unifndx 16669 unifid 16670 odrngstr 16671 ressds 16678 homndx 16679 homid 16680 ccondx 16681 ccoid 16682 resshom 16683 ressco 16684 imasvalstr 16717 prdsvalstr 16718 oppchomfval 16976 oppcbas 16980 rescco 17094 catstr 17219 ipostr 17755 mgpds 19241 srads 19950 cnfldstr 20539 ressunif 22863 tuslem 22868 tmslem 23084 mcubic 25417 cubic2 25418 cubic 25419 quart1cl 25424 quart1lem 25425 quart1 25426 quartlem1 25427 quartlem2 25428 log2ub 25519 log2le1 25520 birthday 25524 bposlem8 25859 bposlem9 25860 pntlemd 26162 pntlema 26164 pntlemb 26165 pntlemf 26173 pntlemo 26175 itvndx 26218 lngndx 26219 itvid 26220 lngid 26221 trkgstr 26222 ttgval 26653 ttglem 26654 ttgds 26659 eengstr 26758 edgfid 26768 edgfndxnn 26769 edgfndxid 26770 baseltedgf 26771 257prm 43714 fmtno4prmfac 43725 fmtno4prmfac193 43726 fmtno4nprmfac193 43727 fmtno5nprm 43736 139prmALT 43750 127prm 43754 3exp4mod41 43772 41prothprmlem2 43774 2exp340mod341 43889 341fppr2 43890 bgoldbtbndlem1 43961 tgblthelfgott 43971 tgoldbachlt 43972 tgoldbach 43973