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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 | |
nnmulcld.2 |
Ref | Expression |
---|---|
nnmulcld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1d.1 | . 2 | |
2 | nnmulcld.2 | . 2 | |
3 | nnmulcl 8734 | . 2 | |
4 | 1, 2, 3 | syl2anc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1480 (class class class)co 5767 cmul 7618 cn 8713 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-1rid 7720 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 df-inn 8714 |
This theorem is referenced by: qbtwnre 10027 bcval 10488 bcm1k 10499 bcp1n 10500 permnn 10510 cvg1nlemcxze 10747 cvg1nlemf 10748 cvg1nlemcau 10749 cvg1nlemres 10750 trireciplem 11262 efaddlem 11369 eftlub 11385 eirraplem 11472 modmulconst 11514 lcmval 11733 oddpwdclemxy 11836 oddpwdclemdc 11840 sqpweven 11842 2sqpwodd 11843 crth 11889 phimullem 11890 evenennn 11895 |
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