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Mirrors > Home > MPE Home > Th. List > nnmulcld | Structured version Visualization version GIF version |
Description: Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnge1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
nnmulcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℕ) |
Ref | Expression |
---|---|
nnmulcld | ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | nnmulcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
3 | nnmulcl 11649 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 (class class class)co 7145 · cmul 10530 ℕcn 11626 |
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 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-addass 10590 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rrecex 10597 ax-cnre 10598 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 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 7148 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-nn 11627 |
This theorem is referenced by: bcm1k 13663 bcp1n 13664 permnn 13674 trireciplem 15205 efaddlem 15434 eftlub 15450 eirrlem 15545 modmulconst 15629 isprm5 16039 crth 16103 phimullem 16104 pcqmul 16178 pcaddlem 16212 pcbc 16224 oddprmdvds 16227 pockthlem 16229 pockthg 16230 vdwlem3 16307 vdwlem6 16310 vdwlem9 16313 torsubg 18903 ablfacrp 19117 dgrcolem1 24790 aalioulem5 24852 aaliou3lem2 24859 log2cnv 25449 log2tlbnd 25450 log2ublem2 25452 log2ub 25454 lgamgulmlem4 25536 wilthlem2 25573 ftalem7 25583 basellem5 25589 mumul 25685 fsumfldivdiaglem 25693 dvdsmulf1o 25698 sgmmul 25704 chtublem 25714 bcmono 25780 bposlem3 25789 bposlem5 25791 gausslemma2dlem1a 25868 lgsquadlem2 25884 lgsquadlem3 25885 lgsquad2lem2 25888 2sqlem6 25926 2sqmod 25939 rplogsumlem1 25987 rplogsumlem2 25988 dchrisum0fmul 26009 vmalogdivsum2 26041 pntrsumbnd2 26070 pntpbnd1 26089 pntpbnd2 26090 ostth2lem2 26137 oddpwdc 31511 eulerpartlemgh 31535 subfaclim 32332 bcprod 32867 faclim2 32877 nnadddir 39041 jm2.27c 39482 relexpmulnn 39932 mccllem 41754 limsup10exlem 41929 wallispilem5 42231 wallispi2lem1 42233 wallispi2 42235 stirlinglem3 42238 stirlinglem8 42243 stirlinglem15 42250 dirkertrigeqlem3 42262 hoicvrrex 42715 deccarry 43388 fmtnoprmfac2 43606 sfprmdvdsmersenne 43645 lighneallem3 43649 proththdlem 43655 fppr2odd 43773 blennnt2 44577 |
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