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Mirrors > Home > ILE Home > Th. List > nnmulcl | Unicode version |
Description: Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) |
Ref | Expression |
---|---|
nnmulcl |
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 | 5 | imbi2d 229 | . . 3 |
7 | oveq2 5782 | . . . . 5 | |
8 | 7 | eleq1d 2208 | . . . 4 |
9 | 8 | imbi2d 229 | . . 3 |
10 | oveq2 5782 | . . . . 5 | |
11 | 10 | eleq1d 2208 | . . . 4 |
12 | 11 | imbi2d 229 | . . 3 |
13 | nncn 8728 | . . . 4 | |
14 | mulid1 7763 | . . . . . 6 | |
15 | 14 | eleq1d 2208 | . . . . 5 |
16 | 15 | biimprd 157 | . . . 4 |
17 | 13, 16 | mpcom 36 | . . 3 |
18 | nnaddcl 8740 | . . . . . . . 8 | |
19 | 18 | ancoms 266 | . . . . . . 7 |
20 | nncn 8728 | . . . . . . . . 9 | |
21 | ax-1cn 7713 | . . . . . . . . . . 11 | |
22 | adddi 7752 | . . . . . . . . . . 11 | |
23 | 21, 22 | mp3an3 1304 | . . . . . . . . . 10 |
24 | 14 | oveq2d 5790 | . . . . . . . . . . 11 |
25 | 24 | adantr 274 | . . . . . . . . . 10 |
26 | 23, 25 | eqtrd 2172 | . . . . . . . . 9 |
27 | 13, 20, 26 | syl2an 287 | . . . . . . . 8 |
28 | 27 | eleq1d 2208 | . . . . . . 7 |
29 | 19, 28 | syl5ibr 155 | . . . . . 6 |
30 | 29 | exp4b 364 | . . . . 5 |
31 | 30 | pm2.43b 52 | . . . 4 |
32 | 31 | a2d 26 | . . 3 |
33 | 3, 6, 9, 12, 17, 32 | nnind 8736 | . 2 |
34 | 33 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 (class class class)co 5774 cc 7618 c1 7621 caddc 7623 cmul 7625 cn 8720 |
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-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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-1rid 7727 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 df-inn 8721 |
This theorem is referenced by: nnmulcli 8742 nndivtr 8762 nnmulcld 8769 nn0mulcl 9013 qaddcl 9427 qmulcl 9429 modqmulnn 10115 nnexpcl 10306 nnsqcl 10362 faccl 10481 facdiv 10484 faclbnd3 10489 bcrpcl 10499 trirecip 11270 lcmgcdlem 11758 lcmgcdnn 11763 |
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