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Mirrors > Home > ILE Home > Th. List > mulclpi | Unicode version |
Description: Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.) |
Ref | Expression |
---|---|
mulclpi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulpiord 7093 | . 2 | |
2 | pinn 7085 | . . . 4 | |
3 | pinn 7085 | . . . 4 | |
4 | nnmcl 6345 | . . . 4 | |
5 | 2, 3, 4 | syl2an 287 | . . 3 |
6 | elni2 7090 | . . . . . . 7 | |
7 | 6 | simprbi 273 | . . . . . 6 |
8 | 7 | adantl 275 | . . . . 5 |
9 | 3 | adantl 275 | . . . . . 6 |
10 | 2 | adantr 274 | . . . . . 6 |
11 | elni2 7090 | . . . . . . . 8 | |
12 | 11 | simprbi 273 | . . . . . . 7 |
13 | 12 | adantr 274 | . . . . . 6 |
14 | nnmordi 6380 | . . . . . 6 | |
15 | 9, 10, 13, 14 | syl21anc 1200 | . . . . 5 |
16 | 8, 15 | mpd 13 | . . . 4 |
17 | ne0i 3339 | . . . 4 | |
18 | 16, 17 | syl 14 | . . 3 |
19 | elni 7084 | . . 3 | |
20 | 5, 18, 19 | sylanbrc 413 | . 2 |
21 | 1, 20 | eqeltrd 2194 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1465 wne 2285 c0 3333 com 4474 (class class class)co 5742 comu 6279 cnpi 7048 cmi 7050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-oadd 6285 df-omul 6286 df-ni 7080 df-mi 7082 |
This theorem is referenced by: mulasspig 7108 distrpig 7109 ltmpig 7115 enqer 7134 enqdc 7137 addcmpblnq 7143 mulcmpblnq 7144 addpipqqslem 7145 mulpipq2 7147 mulpipqqs 7149 ordpipqqs 7150 addclnq 7151 mulclnq 7152 addcomnqg 7157 addassnqg 7158 mulassnqg 7160 mulcanenq 7161 distrnqg 7163 recexnq 7166 nqtri3or 7172 ltdcnq 7173 ltsonq 7174 ltanqg 7176 ltmnqg 7177 1lt2nq 7182 ltexnqq 7184 archnqq 7193 addcmpblnq0 7219 mulcmpblnq0 7220 mulcanenq0ec 7221 addclnq0 7227 mulclnq0 7228 nqpnq0nq 7229 nqnq0a 7230 nqnq0m 7231 nq0m0r 7232 distrnq0 7235 addassnq0lemcl 7237 |
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