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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 8366 | . 2 | |
2 | elznn0 8366 | . 2 | |
3 | nn0mulcl 8324 | . . . . . . . . 9 | |
4 | 3 | orcd 684 | . . . . . . . 8 |
5 | 4 | a1i 9 | . . . . . . 7 |
6 | remulcl 7101 | . . . . . . 7 | |
7 | 5, 6 | jctild 309 | . . . . . 6 |
8 | nn0mulcl 8324 | . . . . . . . . 9 | |
9 | recn 7106 | . . . . . . . . . . 11 | |
10 | recn 7106 | . . . . . . . . . . 11 | |
11 | mulneg1 7499 | . . . . . . . . . . 11 | |
12 | 9, 10, 11 | syl2an 283 | . . . . . . . . . 10 |
13 | 12 | eleq1d 2147 | . . . . . . . . 9 |
14 | 8, 13 | syl5ib 152 | . . . . . . . 8 |
15 | olc 664 | . . . . . . . 8 | |
16 | 14, 15 | syl6 33 | . . . . . . 7 |
17 | 16, 6 | jctild 309 | . . . . . 6 |
18 | nn0mulcl 8324 | . . . . . . . . 9 | |
19 | mulneg2 7500 | . . . . . . . . . . 11 | |
20 | 9, 10, 19 | syl2an 283 | . . . . . . . . . 10 |
21 | 20 | eleq1d 2147 | . . . . . . . . 9 |
22 | 18, 21 | syl5ib 152 | . . . . . . . 8 |
23 | 22, 15 | syl6 33 | . . . . . . 7 |
24 | 23, 6 | jctild 309 | . . . . . 6 |
25 | nn0mulcl 8324 | . . . . . . . . 9 | |
26 | mul2neg 7502 | . . . . . . . . . . 11 | |
27 | 9, 10, 26 | syl2an 283 | . . . . . . . . . 10 |
28 | 27 | eleq1d 2147 | . . . . . . . . 9 |
29 | 25, 28 | syl5ib 152 | . . . . . . . 8 |
30 | orc 665 | . . . . . . . 8 | |
31 | 29, 30 | syl6 33 | . . . . . . 7 |
32 | 31, 6 | jctild 309 | . . . . . 6 |
33 | 7, 17, 24, 32 | ccased 906 | . . . . 5 |
34 | elznn0 8366 | . . . . 5 | |
35 | 33, 34 | syl6ibr 160 | . . . 4 |
36 | 35 | imp 122 | . . 3 |
37 | 36 | an4s 552 | . 2 |
38 | 1, 2, 37 | syl2anb 285 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wo 661 wceq 1284 wcel 1433 (class class class)co 5532 cc 6979 cr 6980 cmul 6986 cneg 7280 cn0 8288 cz 8351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 |
This theorem is referenced by: zdivmul 8437 msqznn 8447 zmulcld 8475 uz2mulcl 8695 qaddcl 8720 qmulcl 8722 qreccl 8727 fzctr 9144 flqmulnn0 9301 zexpcl 9491 iexpcyc 9579 zesq 9591 dvdsmul1 10217 dvdsmul2 10218 muldvds1 10220 muldvds2 10221 dvdscmul 10222 dvdsmulc 10223 dvds2ln 10228 dvdstr 10232 dvdsmultr1 10233 dvdsmultr2 10235 3dvdsdec 10264 3dvds2dec 10265 oexpneg 10276 mulsucdiv2z 10285 divalgb 10325 divalgmod 10327 ndvdsi 10333 absmulgcd 10406 gcdmultiple 10409 gcdmultiplez 10410 dvdsmulgcd 10414 rpmulgcd 10415 lcmcllem 10449 rpmul 10480 cncongr1 10485 cncongr2 10486 |
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