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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 12090 | . 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
2 | 10nn0 12105 | . . 3 ⊢ ;10 ∈ ℕ0 | |
3 | decnncl.1 | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
4 | decnncl.2 | . . 3 ⊢ 𝐵 ∈ ℕ | |
5 | 2, 3, 4 | numnncl 12097 | . 2 ⊢ ((;10 · 𝐴) + 𝐵) ∈ ℕ |
6 | 1, 5 | eqeltri 2909 | 1 ⊢ ;𝐴𝐵 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 (class class class)co 7145 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ℕcn 11627 ℕ0cn0 11886 ;cdc 12087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-dec 12088 |
This theorem is referenced by: 11prm 16438 13prm 16439 17prm 16440 19prm 16441 23prm 16442 37prm 16444 43prm 16445 83prm 16446 139prm 16447 163prm 16448 317prm 16449 631prm 16450 1259lem1 16454 1259lem2 16455 1259lem3 16456 1259lem4 16457 1259lem5 16458 1259prm 16459 2503lem1 16460 2503lem2 16461 2503lem3 16462 2503prm 16463 4001lem1 16464 4001lem2 16465 4001lem3 16466 4001lem4 16467 4001prm 16468 ocndx 16663 ocid 16664 dsndx 16665 dsid 16666 unifndx 16667 unifid 16668 odrngstr 16669 ressds 16676 homndx 16677 homid 16678 ccondx 16679 ccoid 16680 resshom 16681 ressco 16682 imasvalstr 16715 prdsvalstr 16716 oppchomfval 16974 oppcbas 16978 rescco 17092 catstr 17217 ipostr 17753 mgpds 19180 srads 19889 cnfldstr 20477 ressunif 22800 tuslem 22805 tmslem 23021 mcubic 25352 cubic2 25353 cubic 25354 quart1cl 25359 quart1lem 25360 quart1 25361 quartlem1 25362 quartlem2 25363 log2ub 25455 log2le1 25456 birthday 25460 bposlem8 25795 bposlem9 25796 pntlemd 26098 pntlema 26100 pntlemb 26101 pntlemf 26109 pntlemo 26111 itvndx 26154 lngndx 26155 itvid 26156 lngid 26157 trkgstr 26158 ttgval 26589 ttglem 26590 ttgds 26595 eengstr 26694 edgfid 26704 edgfndxnn 26705 edgfndxid 26706 baseltedgf 26707 257prm 43570 fmtno4prmfac 43581 fmtno4prmfac193 43582 fmtno4nprmfac193 43583 fmtno5nprm 43592 139prmALT 43606 127prm 43610 3exp4mod41 43628 41prothprmlem2 43630 2exp340mod341 43745 341fppr2 43746 bgoldbtbndlem1 43817 tgblthelfgott 43827 tgoldbachlt 43828 tgoldbach 43829 |
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