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Mirrors > Home > ILE Home > Th. List > nnmulcld | Unicode version |
Description: Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnge1d.1 |
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nnmulcld.2 |
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Ref | Expression |
---|---|
nnmulcld |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1d.1 |
. 2
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2 | nnmulcld.2 |
. 2
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3 | nnmulcl 8337 |
. 2
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4 | 1, 2, 3 | syl2anc 403 |
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 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-mulcom 7349 ax-addass 7350 ax-mulass 7351 ax-distr 7352 ax-1rid 7355 ax-cnre 7359 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-iota 4934 df-fv 4977 df-ov 5594 df-inn 8317 |
This theorem is referenced by: qbtwnre 9557 bcval 9992 bcm1k 10003 bcp1n 10004 permnn 10014 cvg1nlemcxze 10242 cvg1nlemf 10243 cvg1nlemcau 10244 cvg1nlemres 10245 modmulconst 10608 lcmval 10825 oddpwdclemxy 10927 oddpwdclemdc 10931 sqpweven 10933 2sqpwodd 10934 crth 10980 phimullem 10981 evenennn 10986 |
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