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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 7252 | . 2 | |
2 | pinn 7244 | . . . 4 | |
3 | pinn 7244 | . . . 4 | |
4 | nnmcl 6443 | . . . 4 | |
5 | 2, 3, 4 | syl2an 287 | . . 3 |
6 | elni2 7249 | . . . . . . 7 | |
7 | 6 | simprbi 273 | . . . . . 6 |
8 | 7 | adantl 275 | . . . . 5 |
9 | 3 | adantl 275 | . . . . . 6 |
10 | 2 | adantr 274 | . . . . . 6 |
11 | elni2 7249 | . . . . . . . 8 | |
12 | 11 | simprbi 273 | . . . . . . 7 |
13 | 12 | adantr 274 | . . . . . 6 |
14 | nnmordi 6478 | . . . . . 6 | |
15 | 9, 10, 13, 14 | syl21anc 1226 | . . . . 5 |
16 | 8, 15 | mpd 13 | . . . 4 |
17 | ne0i 3413 | . . . 4 | |
18 | 16, 17 | syl 14 | . . 3 |
19 | elni 7243 | . . 3 | |
20 | 5, 18, 19 | sylanbrc 414 | . 2 |
21 | 1, 20 | eqeltrd 2241 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2135 wne 2334 c0 3407 com 4564 (class class class)co 5839 comu 6376 cnpi 7207 cmi 7209 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4094 ax-sep 4097 ax-nul 4105 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-iinf 4562 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2726 df-sbc 2950 df-csb 3044 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-iun 3865 df-br 3980 df-opab 4041 df-mpt 4042 df-tr 4078 df-id 4268 df-iord 4341 df-on 4343 df-suc 4346 df-iom 4565 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-f1 5190 df-fo 5191 df-f1o 5192 df-fv 5193 df-ov 5842 df-oprab 5843 df-mpo 5844 df-1st 6103 df-2nd 6104 df-recs 6267 df-irdg 6332 df-oadd 6382 df-omul 6383 df-ni 7239 df-mi 7241 |
This theorem is referenced by: mulasspig 7267 distrpig 7268 ltmpig 7274 enqer 7293 enqdc 7296 addcmpblnq 7302 mulcmpblnq 7303 addpipqqslem 7304 mulpipq2 7306 mulpipqqs 7308 ordpipqqs 7309 addclnq 7310 mulclnq 7311 addcomnqg 7316 addassnqg 7317 mulassnqg 7319 mulcanenq 7320 distrnqg 7322 recexnq 7325 nqtri3or 7331 ltdcnq 7332 ltsonq 7333 ltanqg 7335 ltmnqg 7336 1lt2nq 7341 ltexnqq 7343 archnqq 7352 addcmpblnq0 7378 mulcmpblnq0 7379 mulcanenq0ec 7380 addclnq0 7386 mulclnq0 7387 nqpnq0nq 7388 nqnq0a 7389 nqnq0m 7390 nq0m0r 7391 distrnq0 7394 addassnq0lemcl 7396 |
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