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Mirrors > Home > MPE Home > Th. List > zmulcl | Structured version Visualization version Unicode version |
Description: Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.) |
Ref | Expression |
---|---|
zmulcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elznn0 10941 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | elznn0 10941 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | nn0mulcl 10895 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 3 | orcd 398 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 4 | a1i 11 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | remulcl 9610 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 5, 6 | jctild 550 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | nn0mulcl 10895 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | recn 9615 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | recn 9615 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | mulneg1 10043 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 9, 10, 11 | syl2an 484 |
. . . . . . . . . 10
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13 | 12 | eleq1d 2513 |
. . . . . . . . 9
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14 | 8, 13 | syl5ib 227 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | olc 390 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 14, 15 | syl6 34 |
. . . . . . 7
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17 | 16, 6 | jctild 550 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | nn0mulcl 10895 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | mulneg2 10044 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 9, 10, 19 | syl2an 484 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 20 | eleq1d 2513 |
. . . . . . . . 9
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22 | 18, 21 | syl5ib 227 |
. . . . . . . 8
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23 | 22, 15 | syl6 34 |
. . . . . . 7
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24 | 23, 6 | jctild 550 |
. . . . . 6
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25 | nn0mulcl 10895 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | mul2neg 10046 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 9, 10, 26 | syl2an 484 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 27 | eleq1d 2513 |
. . . . . . . . 9
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29 | 25, 28 | syl5ib 227 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | orc 391 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 29, 30 | syl6 34 |
. . . . . . 7
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32 | 31, 6 | jctild 550 |
. . . . . 6
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33 | 7, 17, 24, 32 | ccased 959 |
. . . . 5
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34 | elznn0 10941 |
. . . . 5
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35 | 33, 34 | syl6ibr 235 |
. . . 4
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36 | 35 | imp 435 |
. . 3
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37 | 36 | an4s 839 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | 1, 2, 37 | syl2anb 486 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1672 ax-4 1685 ax-5 1761 ax-6 1808 ax-7 1854 ax-8 1892 ax-9 1899 ax-10 1918 ax-11 1923 ax-12 1936 ax-13 2091 ax-ext 2431 ax-sep 4496 ax-nul 4505 ax-pow 4553 ax-pr 4611 ax-un 6570 ax-resscn 9582 ax-1cn 9583 ax-icn 9584 ax-addcl 9585 ax-addrcl 9586 ax-mulcl 9587 ax-mulrcl 9588 ax-mulcom 9589 ax-addass 9590 ax-mulass 9591 ax-distr 9592 ax-i2m1 9593 ax-1ne0 9594 ax-1rid 9595 ax-rnegex 9596 ax-rrecex 9597 ax-cnre 9598 ax-pre-lttri 9599 ax-pre-lttrn 9600 ax-pre-ltadd 9601 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 987 df-3an 988 df-tru 1450 df-ex 1667 df-nf 1671 df-sb 1801 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2623 df-nel 2624 df-ral 2741 df-rex 2742 df-reu 2743 df-rab 2745 df-v 3014 df-sbc 3235 df-csb 3331 df-dif 3374 df-un 3376 df-in 3378 df-ss 3385 df-pss 3387 df-nul 3699 df-if 3849 df-pw 3920 df-sn 3936 df-pr 3938 df-tp 3940 df-op 3942 df-uni 4168 df-iun 4249 df-br 4374 df-opab 4433 df-mpt 4434 df-tr 4469 df-eprel 4722 df-id 4726 df-po 4732 df-so 4733 df-fr 4770 df-we 4772 df-xp 4817 df-rel 4818 df-cnv 4819 df-co 4820 df-dm 4821 df-rn 4822 df-res 4823 df-ima 4824 df-pred 5358 df-ord 5404 df-on 5405 df-lim 5406 df-suc 5407 df-iota 5524 df-fun 5562 df-fn 5563 df-f 5564 df-f1 5565 df-fo 5566 df-f1o 5567 df-fv 5568 df-riota 6237 df-ov 6278 df-oprab 6279 df-mpt2 6280 df-om 6680 df-wrecs 7014 df-recs 7076 df-rdg 7114 df-er 7349 df-en 7556 df-dom 7557 df-sdom 7558 df-pnf 9663 df-mnf 9664 df-ltxr 9666 df-sub 9848 df-neg 9849 df-nn 10598 df-n0 10859 df-z 10927 |
This theorem is referenced by: zdivmul 10997 msqznn 11006 zmulcld 11035 uz2mulcl 11225 qaddcl 11269 qmulcl 11271 qreccl 11273 fzctr 11894 flmulnn0 12053 zexpcl 12280 iexpcyc 12372 zesq 12388 cshweqrep 12918 fprodzcl 14018 zrisefaccl 14083 zfallfaccl 14084 dvdsmul1 14334 dvdsmul2 14335 muldvds1 14337 muldvds2 14338 dvdscmul 14339 dvdsmulc 14340 dvdscmulr 14341 dvdsmulcr 14342 dvds2ln 14343 dvdstr 14347 dvdsmultr1 14348 dvdsmultr2 14350 oexpneg 14378 divalglem0 14381 divalglem2 14383 divalglem2OLD 14384 divalglem4 14385 divalglem8 14390 divalgb 14395 divalgmod 14397 ndvdsi 14401 gcdaddmlem 14502 absmulgcd 14525 gcdmultiple 14528 gcdmultiplez 14529 dvdsmulgcd 14532 rpmulgcd 14533 lcmcllem 14571 coprmdvds 14669 rpmul 14685 eulerthlem2 14740 modprminv 14760 modprminveq 14761 modprm0 14766 pythagtriplem4 14779 pcpremul 14803 pcmul 14811 gzmulcl 14892 pgpfac1lem2 17718 zsubrg 19031 dvdsrzring 19062 mulgrhm 19079 domnchr 19113 znfld 19141 znunit 19144 mbfi1fseqlem5 22688 dvexp3 22941 basellem2 24019 basellem5 24022 dvdsflf1o 24127 chtub 24151 bposlem1 24223 bposlem5 24227 bposlem6 24228 lgslem3 24237 lgsval4a 24257 lgsneg 24258 lgsdir2 24267 lgsdchr 24287 lgseisenlem1 24288 lgseisenlem2 24289 lgseisenlem3 24290 lgsquadlem1 24293 lgsquad2lem2 24298 chebbnd1lem1 24318 chebbnd1lem3 24320 gxnn0mul 26016 fzmul 32070 mzpclall 35570 mzpindd 35589 acongrep 35831 acongeq 35834 jm2.18 35844 jm2.21 35850 jm2.26a 35856 jm2.26 35858 jm2.16nn0 35860 jm2.27a 35861 jm2.27c 35863 jm3.1lem3 35875 fourierswlem 38150 oexpnegALTV 38896 oexpnegnz 38897 tgblthelfgott 38998 2zrngmmgm 40270 zlmodzxzequa 40613 zlmodzxzequap 40616 |
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