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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 5540 | . . . . 5 | |
2 | 1 | eleq1d 2147 | . . . 4 |
3 | 2 | imbi2d 228 | . . 3 |
4 | oveq2 5540 | . . . . 5 | |
5 | 4 | eleq1d 2147 | . . . 4 |
6 | 5 | imbi2d 228 | . . 3 |
7 | oveq2 5540 | . . . . 5 | |
8 | 7 | eleq1d 2147 | . . . 4 |
9 | 8 | imbi2d 228 | . . 3 |
10 | oveq2 5540 | . . . . 5 | |
11 | 10 | eleq1d 2147 | . . . 4 |
12 | 11 | imbi2d 228 | . . 3 |
13 | nncn 8047 | . . . 4 | |
14 | mulid1 7116 | . . . . . 6 | |
15 | 14 | eleq1d 2147 | . . . . 5 |
16 | 15 | biimprd 156 | . . . 4 |
17 | 13, 16 | mpcom 36 | . . 3 |
18 | nnaddcl 8059 | . . . . . . . 8 | |
19 | 18 | ancoms 264 | . . . . . . 7 |
20 | nncn 8047 | . . . . . . . . 9 | |
21 | ax-1cn 7069 | . . . . . . . . . . 11 | |
22 | adddi 7105 | . . . . . . . . . . 11 | |
23 | 21, 22 | mp3an3 1257 | . . . . . . . . . 10 |
24 | 14 | oveq2d 5548 | . . . . . . . . . . 11 |
25 | 24 | adantr 270 | . . . . . . . . . 10 |
26 | 23, 25 | eqtrd 2113 | . . . . . . . . 9 |
27 | 13, 20, 26 | syl2an 283 | . . . . . . . 8 |
28 | 27 | eleq1d 2147 | . . . . . . 7 |
29 | 19, 28 | syl5ibr 154 | . . . . . 6 |
30 | 29 | exp4b 359 | . . . . 5 |
31 | 30 | pm2.43b 51 | . . . 4 |
32 | 31 | a2d 26 | . . 3 |
33 | 3, 6, 9, 12, 17, 32 | nnind 8055 | . 2 |
34 | 33 | impcom 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 (class class class)co 5532 cc 6979 c1 6982 caddc 6984 cmul 6986 cn 8039 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-1rid 7083 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 df-inn 8040 |
This theorem is referenced by: nnmulcli 8061 nndivtr 8080 nnmulcld 8087 nn0mulcl 8324 qaddcl 8720 qmulcl 8722 modqmulnn 9344 nnexpcl 9489 nnsqcl 9545 faccl 9662 facdiv 9665 faclbnd3 9670 bcrpcl 9680 lcmgcdlem 10459 lcmgcdnn 10464 |
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