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Mirrors > Home > ILE Home > Th. List > zmulcl | Unicode version |
Description: Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.) |
Ref | Expression |
---|---|
zmulcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elznn0 9037 | . 2 | |
2 | elznn0 9037 | . 2 | |
3 | nn0mulcl 8981 | . . . . . . . . 9 | |
4 | 3 | orcd 707 | . . . . . . . 8 |
5 | 4 | a1i 9 | . . . . . . 7 |
6 | remulcl 7716 | . . . . . . 7 | |
7 | 5, 6 | jctild 314 | . . . . . 6 |
8 | nn0mulcl 8981 | . . . . . . . . 9 | |
9 | recn 7721 | . . . . . . . . . . 11 | |
10 | recn 7721 | . . . . . . . . . . 11 | |
11 | mulneg1 8125 | . . . . . . . . . . 11 | |
12 | 9, 10, 11 | syl2an 287 | . . . . . . . . . 10 |
13 | 12 | eleq1d 2186 | . . . . . . . . 9 |
14 | 8, 13 | syl5ib 153 | . . . . . . . 8 |
15 | olc 685 | . . . . . . . 8 | |
16 | 14, 15 | syl6 33 | . . . . . . 7 |
17 | 16, 6 | jctild 314 | . . . . . 6 |
18 | nn0mulcl 8981 | . . . . . . . . 9 | |
19 | mulneg2 8126 | . . . . . . . . . . 11 | |
20 | 9, 10, 19 | syl2an 287 | . . . . . . . . . 10 |
21 | 20 | eleq1d 2186 | . . . . . . . . 9 |
22 | 18, 21 | syl5ib 153 | . . . . . . . 8 |
23 | 22, 15 | syl6 33 | . . . . . . 7 |
24 | 23, 6 | jctild 314 | . . . . . 6 |
25 | nn0mulcl 8981 | . . . . . . . . 9 | |
26 | mul2neg 8128 | . . . . . . . . . . 11 | |
27 | 9, 10, 26 | syl2an 287 | . . . . . . . . . 10 |
28 | 27 | eleq1d 2186 | . . . . . . . . 9 |
29 | 25, 28 | syl5ib 153 | . . . . . . . 8 |
30 | orc 686 | . . . . . . . 8 | |
31 | 29, 30 | syl6 33 | . . . . . . 7 |
32 | 31, 6 | jctild 314 | . . . . . 6 |
33 | 7, 17, 24, 32 | ccased 934 | . . . . 5 |
34 | elznn0 9037 | . . . . 5 | |
35 | 33, 34 | syl6ibr 161 | . . . 4 |
36 | 35 | imp 123 | . . 3 |
37 | 36 | an4s 562 | . 2 |
38 | 1, 2, 37 | syl2anb 289 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 682 wceq 1316 wcel 1465 (class class class)co 5742 cc 7586 cr 7587 cmul 7593 cneg 7902 cn0 8945 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: zdivmul 9109 msqznn 9119 zmulcld 9147 uz2mulcl 9370 qaddcl 9395 qmulcl 9397 qreccl 9402 fzctr 9878 flqmulnn0 10040 zexpcl 10276 iexpcyc 10365 zesq 10378 dvdsmul1 11442 dvdsmul2 11443 muldvds1 11445 muldvds2 11446 dvdscmul 11447 dvdsmulc 11448 dvds2ln 11453 dvdstr 11457 dvdsmultr1 11458 dvdsmultr2 11460 3dvdsdec 11489 3dvds2dec 11490 oexpneg 11501 mulsucdiv2z 11509 divalgb 11549 divalgmod 11551 ndvdsi 11557 absmulgcd 11632 gcdmultiple 11635 gcdmultiplez 11636 dvdsmulgcd 11640 rpmulgcd 11641 lcmcllem 11675 rpmul 11706 cncongr1 11711 cncongr2 11712 |
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