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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5463 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | 1 | eleq1d 2103 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | 2 | imbi2d 219 |
. . 3
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4 | oveq2 5463 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | eleq1d 2103 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 5 | imbi2d 219 |
. . 3
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7 | oveq2 5463 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 7 | eleq1d 2103 |
. . . 4
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9 | 8 | imbi2d 219 |
. . 3
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10 | oveq2 5463 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 10 | eleq1d 2103 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 11 | imbi2d 219 |
. . 3
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13 | nncn 7703 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | mulid1 6822 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 14 | eleq1d 2103 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 15 | biimprd 147 |
. . . 4
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17 | 13, 16 | mpcom 32 |
. . 3
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18 | nnaddcl 7715 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 18 | ancoms 255 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | nncn 7703 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | ax-1cn 6776 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() | |
22 | adddi 6811 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 21, 22 | mp3an3 1220 |
. . . . . . . . . 10
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24 | 14 | oveq2d 5471 |
. . . . . . . . . . 11
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25 | 24 | adantr 261 |
. . . . . . . . . 10
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26 | 23, 25 | eqtrd 2069 |
. . . . . . . . 9
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27 | 13, 20, 26 | syl2an 273 |
. . . . . . . 8
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28 | 27 | eleq1d 2103 |
. . . . . . 7
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29 | 19, 28 | syl5ibr 145 |
. . . . . 6
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30 | 29 | exp4b 349 |
. . . . 5
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31 | 30 | pm2.43b 46 |
. . . 4
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32 | 31 | a2d 23 |
. . 3
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33 | 3, 6, 9, 12, 17, 32 | nnind 7711 |
. 2
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34 | 33 | impcom 116 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-mulcom 6784 ax-addass 6785 ax-mulass 6786 ax-distr 6787 ax-1rid 6790 ax-cnre 6794 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-br 3756 df-iota 4810 df-fv 4853 df-ov 5458 df-inn 7696 |
This theorem is referenced by: nnmulcli 7717 nndivtr 7736 nnmulcld 7742 nn0mulcl 7994 qaddcl 8346 qmulcl 8348 nnexpcl 8922 nnsqcl 8976 |
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