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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 8765 |
. 2
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4 | 1, 2, 3 | syl2anc 409 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-1rid 7751 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 df-inn 8745 |
This theorem is referenced by: qbtwnre 10065 bcval 10527 bcm1k 10538 bcp1n 10539 permnn 10549 cvg1nlemcxze 10786 cvg1nlemf 10787 cvg1nlemcau 10788 cvg1nlemres 10789 trireciplem 11301 efaddlem 11417 eftlub 11433 eirraplem 11519 modmulconst 11561 lcmval 11780 oddpwdclemxy 11883 oddpwdclemdc 11887 sqpweven 11889 2sqpwodd 11890 crth 11936 phimullem 11937 evenennn 11942 |
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