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Mirrors > Home > ILE Home > Th. List > zmulcld | Unicode version |
Description: Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
zred.1 | |
zaddcld.1 |
Ref | Expression |
---|---|
zmulcld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zred.1 | . 2 | |
2 | zaddcld.1 | . 2 | |
3 | zmulcl 9075 | . 2 | |
4 | 1, 2, 3 | syl2anc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1465 (class class class)co 5742 cmul 7593 cz 9022 |
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-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 |
This theorem is referenced by: qapne 9399 qtri3or 9988 2tnp1ge0ge0 10042 flhalf 10043 intfracq 10061 zmodcl 10085 modqmul1 10118 addmodlteq 10139 sqoddm1div8 10412 eirraplem 11410 dvdscmulr 11449 dvdsmulcr 11450 modmulconst 11452 dvds2ln 11453 dvdsmod 11487 even2n 11498 2tp1odd 11508 ltoddhalfle 11517 m1expo 11524 m1exp1 11525 divalglemqt 11543 modremain 11553 flodddiv4 11558 gcdaddm 11599 gcdmultipled 11608 bezoutlemnewy 11611 bezoutlemstep 11612 bezoutlembi 11620 mulgcd 11631 dvdsmulgcd 11640 bezoutr 11647 lcmval 11671 lcmcllem 11675 lcmgcdlem 11685 mulgcddvds 11702 rpmulgcd2 11703 divgcdcoprm0 11709 cncongr1 11711 cncongr2 11712 prmind2 11728 exprmfct 11745 2sqpwodd 11781 hashdvds 11824 phimullem 11828 oddennn 11832 |
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