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Mirrors > Home > MPE Home > Th. List > nnmulcl | Structured version Visualization version Unicode version |
Description: Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) |
Ref | Expression |
---|---|
nnmulcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6658 | . . . . 5 | |
2 | 1 | eleq1d 2686 | . . . 4 |
3 | 2 | imbi2d 330 | . . 3 |
4 | oveq2 6658 | . . . . 5 | |
5 | 4 | eleq1d 2686 | . . . 4 |
6 | 5 | imbi2d 330 | . . 3 |
7 | oveq2 6658 | . . . . 5 | |
8 | 7 | eleq1d 2686 | . . . 4 |
9 | 8 | imbi2d 330 | . . 3 |
10 | oveq2 6658 | . . . . 5 | |
11 | 10 | eleq1d 2686 | . . . 4 |
12 | 11 | imbi2d 330 | . . 3 |
13 | nncn 11028 | . . . 4 | |
14 | mulid1 10037 | . . . . . 6 | |
15 | 14 | eleq1d 2686 | . . . . 5 |
16 | 15 | biimprd 238 | . . . 4 |
17 | 13, 16 | mpcom 38 | . . 3 |
18 | nnaddcl 11042 | . . . . . . . 8 | |
19 | 18 | ancoms 469 | . . . . . . 7 |
20 | nncn 11028 | . . . . . . . . 9 | |
21 | ax-1cn 9994 | . . . . . . . . . . 11 | |
22 | adddi 10025 | . . . . . . . . . . 11 | |
23 | 21, 22 | mp3an3 1413 | . . . . . . . . . 10 |
24 | 14 | oveq2d 6666 | . . . . . . . . . . 11 |
25 | 24 | adantr 481 | . . . . . . . . . 10 |
26 | 23, 25 | eqtrd 2656 | . . . . . . . . 9 |
27 | 13, 20, 26 | syl2an 494 | . . . . . . . 8 |
28 | 27 | eleq1d 2686 | . . . . . . 7 |
29 | 19, 28 | syl5ibr 236 | . . . . . 6 |
30 | 29 | exp4b 632 | . . . . 5 |
31 | 30 | pm2.43b 55 | . . . 4 |
32 | 31 | a2d 29 | . . 3 |
33 | 3, 6, 9, 12, 17, 32 | nnind 11038 | . 2 |
34 | 33 | impcom 446 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 (class class class)co 6650 cc 9934 c1 9937 caddc 9939 cmul 9941 cn 11020 |
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-rrecex 10008 ax-cnre 10009 |
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-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-nn 11021 |
This theorem is referenced by: nnmulcli 11044 nndivtr 11062 nnmulcld 11068 nn0mulcl 11329 qaddcl 11804 qmulcl 11806 modmulnn 12688 nnexpcl 12873 nnsqcl 12933 expmulnbnd 12996 faccl 13070 facdiv 13074 faclbnd3 13079 faclbnd4lem3 13082 faclbnd5 13085 bcrpcl 13095 trirecip 14595 fprodnncl 14685 nnrisefaccl 14750 lcmgcdlem 15319 lcmgcdnn 15324 pcmptcl 15595 prmreclem1 15620 prmreclem6 15625 4sqlem12 15660 vdwlem3 15687 vdwlem9 15693 vdwlem10 15694 mulgnnass 17576 mulgnnassOLD 17577 ovolunlem1a 23264 ovolunlem1 23265 mbfi1fseqlem3 23484 mbfi1fseqlem4 23485 elqaalem2 24075 elqaalem3 24076 log2cnv 24671 log2tlbnd 24672 log2ublem2 24674 log2ub 24676 basellem1 24807 basellem2 24808 basellem3 24809 basellem4 24810 basellem5 24811 basellem6 24812 basellem7 24813 basellem8 24814 basellem9 24815 efnnfsumcl 24829 efchtdvds 24885 mumullem1 24905 mumullem2 24906 fsumdvdscom 24911 dvdsflf1o 24913 chtublem 24936 pcbcctr 25001 bclbnd 25005 bposlem1 25009 bposlem2 25010 bposlem3 25011 bposlem4 25012 bposlem5 25013 bposlem6 25014 lgseisenlem1 25100 lgseisenlem2 25101 lgseisenlem3 25102 lgseisenlem4 25103 lgsquadlem1 25105 lgsquadlem2 25106 chebbnd1lem1 25158 chebbnd1lem3 25160 dchrisumlem1 25178 mulogsum 25221 pntrsumo1 25254 pntrsumbnd 25255 ostth2lem1 25307 subfaclim 31170 jm2.17a 37527 jm2.17b 37528 jm2.17c 37529 acongrep 37547 acongeq 37550 jm2.27a 37572 jm2.27c 37574 |
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