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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 8443 |
. 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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-cnex 7436 ax-resscn 7437 ax-1cn 7438 ax-1re 7439 ax-icn 7440 ax-addcl 7441 ax-addrcl 7442 ax-mulcl 7443 ax-mulcom 7446 ax-addass 7447 ax-mulass 7448 ax-distr 7449 ax-1rid 7452 ax-cnre 7456 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-br 3846 df-iota 4980 df-fv 5023 df-ov 5655 df-inn 8423 |
This theorem is referenced by: qbtwnre 9668 bcval 10157 bcm1k 10168 bcp1n 10169 permnn 10179 cvg1nlemcxze 10415 cvg1nlemf 10416 cvg1nlemcau 10417 cvg1nlemres 10418 trireciplem 10894 efaddlem 10964 eftlub 10980 eirraplem 11064 modmulconst 11106 lcmval 11323 oddpwdclemxy 11425 oddpwdclemdc 11429 sqpweven 11431 2sqpwodd 11432 crth 11478 phimullem 11479 evenennn 11484 |
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