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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 5775 | . . . . 5 | |
2 | 1 | eleq1d 2206 | . . . 4 |
3 | 2 | imbi2d 229 | . . 3 |
4 | oveq2 5775 | . . . . 5 | |
5 | 4 | eleq1d 2206 | . . . 4 |
6 | 5 | imbi2d 229 | . . 3 |
7 | oveq2 5775 | . . . . 5 | |
8 | 7 | eleq1d 2206 | . . . 4 |
9 | 8 | imbi2d 229 | . . 3 |
10 | oveq2 5775 | . . . . 5 | |
11 | 10 | eleq1d 2206 | . . . 4 |
12 | 11 | imbi2d 229 | . . 3 |
13 | nncn 8721 | . . . 4 | |
14 | mulid1 7756 | . . . . . 6 | |
15 | 14 | eleq1d 2206 | . . . . 5 |
16 | 15 | biimprd 157 | . . . 4 |
17 | 13, 16 | mpcom 36 | . . 3 |
18 | nnaddcl 8733 | . . . . . . . 8 | |
19 | 18 | ancoms 266 | . . . . . . 7 |
20 | nncn 8721 | . . . . . . . . 9 | |
21 | ax-1cn 7706 | . . . . . . . . . . 11 | |
22 | adddi 7745 | . . . . . . . . . . 11 | |
23 | 21, 22 | mp3an3 1304 | . . . . . . . . . 10 |
24 | 14 | oveq2d 5783 | . . . . . . . . . . 11 |
25 | 24 | adantr 274 | . . . . . . . . . 10 |
26 | 23, 25 | eqtrd 2170 | . . . . . . . . 9 |
27 | 13, 20, 26 | syl2an 287 | . . . . . . . 8 |
28 | 27 | eleq1d 2206 | . . . . . . 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 8729 | . 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 5767 cc 7611 c1 7614 caddc 7616 cmul 7618 cn 8713 |
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 2119 ax-sep 4041 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-1rid 7720 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 df-inn 8714 |
This theorem is referenced by: nnmulcli 8735 nndivtr 8755 nnmulcld 8762 nn0mulcl 9006 qaddcl 9420 qmulcl 9422 modqmulnn 10108 nnexpcl 10299 nnsqcl 10355 faccl 10474 facdiv 10477 faclbnd3 10482 bcrpcl 10492 trirecip 11263 lcmgcdlem 11747 lcmgcdnn 11752 |
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