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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 8969 |
. 2
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4 | 1, 2, 3 | syl2anc 411 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-sep 4136 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-1rid 7947 ax-cnre 7951 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-iota 5196 df-fv 5243 df-ov 5898 df-inn 8949 |
This theorem is referenced by: qbtwnre 10286 bcval 10760 bcm1k 10771 bcp1n 10772 permnn 10782 cvg1nlemcxze 11022 cvg1nlemf 11023 cvg1nlemcau 11024 cvg1nlemres 11025 trireciplem 11539 efaddlem 11713 eftlub 11729 eirraplem 11815 modmulconst 11861 lcmval 12094 oddpwdclemxy 12200 oddpwdclemdc 12204 sqpweven 12206 2sqpwodd 12207 crth 12255 phimullem 12256 modprm0 12285 pcqmul 12334 pcaddlem 12370 pcbc 12382 oddprmdvds 12385 pockthlem 12387 pockthg 12388 4sqlem13m 12434 4sqlem14 12435 4sqlem17 12438 4sqlem18 12439 evenennn 12443 lgseisenlem2 14904 2sqlem6 14920 |
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