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Mirrors > Home > ILE Home > Th. List > nnmcl | Unicode version |
Description: Closure of multiplication of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
nnmcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5782 | . . . . 5 | |
2 | 1 | eleq1d 2208 | . . . 4 |
3 | 2 | imbi2d 229 | . . 3 |
4 | oveq2 5782 | . . . . 5 | |
5 | 4 | eleq1d 2208 | . . . 4 |
6 | oveq2 5782 | . . . . 5 | |
7 | 6 | eleq1d 2208 | . . . 4 |
8 | oveq2 5782 | . . . . 5 | |
9 | 8 | eleq1d 2208 | . . . 4 |
10 | nnm0 6371 | . . . . 5 | |
11 | peano1 4508 | . . . . 5 | |
12 | 10, 11 | eqeltrdi 2230 | . . . 4 |
13 | nnacl 6376 | . . . . . . . 8 | |
14 | 13 | expcom 115 | . . . . . . 7 |
15 | 14 | adantr 274 | . . . . . 6 |
16 | nnmsuc 6373 | . . . . . . 7 | |
17 | 16 | eleq1d 2208 | . . . . . 6 |
18 | 15, 17 | sylibrd 168 | . . . . 5 |
19 | 18 | expcom 115 | . . . 4 |
20 | 5, 7, 9, 12, 19 | finds2 4515 | . . 3 |
21 | 3, 20 | vtoclga 2752 | . 2 |
22 | 21 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 c0 3363 csuc 4287 com 4504 (class class class)co 5774 coa 6310 comu 6311 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-oadd 6317 df-omul 6318 |
This theorem is referenced by: nnmcli 6379 nndi 6382 nnmass 6383 nnmsucr 6384 nnmordi 6412 nnmord 6413 nnmword 6414 mulclpi 7136 enq0tr 7242 addcmpblnq0 7251 mulcmpblnq0 7252 mulcanenq0ec 7253 addclnq0 7259 mulclnq0 7260 nqpnq0nq 7261 distrnq0 7267 addassnq0lemcl 7269 addassnq0 7270 |
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