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Theorem zmulcl 10908
Description: Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
Assertion
Ref Expression
zmulcl  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )

Proof of Theorem zmulcl
StepHypRef Expression
1 elznn0 10875 . 2  |-  ( M  e.  ZZ  <->  ( M  e.  RR  /\  ( M  e.  NN0  \/  -u M  e.  NN0 ) ) )
2 elznn0 10875 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
3 nn0mulcl 10828 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  x.  N
)  e.  NN0 )
43orcd 390 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( M  x.  N )  e.  NN0  \/  -u ( M  x.  N
)  e.  NN0 )
)
54a1i 11 . . . . . . 7  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( M  e. 
NN0  /\  N  e.  NN0 )  ->  ( ( M  x.  N )  e.  NN0  \/  -u ( M  x.  N )  e.  NN0 ) ) )
6 remulcl 9566 . . . . . . 7  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  x.  N
)  e.  RR )
75, 6jctild 541 . . . . . 6  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( M  e. 
NN0  /\  N  e.  NN0 )  ->  ( ( M  x.  N )  e.  RR  /\  ( ( M  x.  N )  e.  NN0  \/  -u ( M  x.  N )  e.  NN0 ) ) ) )
8 nn0mulcl 10828 . . . . . . . . 9  |-  ( (
-u M  e.  NN0  /\  N  e.  NN0 )  ->  ( -u M  x.  N )  e.  NN0 )
9 recn 9571 . . . . . . . . . . 11  |-  ( M  e.  RR  ->  M  e.  CC )
10 recn 9571 . . . . . . . . . . 11  |-  ( N  e.  RR  ->  N  e.  CC )
11 mulneg1 9989 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( -u M  x.  N )  =  -u ( M  x.  N
) )
129, 10, 11syl2an 475 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( -u M  x.  N )  =  -u ( M  x.  N
) )
1312eleq1d 2523 . . . . . . . . 9  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  x.  N )  e.  NN0  <->  -u ( M  x.  N )  e.  NN0 ) )
148, 13syl5ib 219 . . . . . . . 8  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  e.  NN0  /\  N  e. 
NN0 )  ->  -u ( M  x.  N )  e.  NN0 ) )
15 olc 382 . . . . . . . 8  |-  ( -u ( M  x.  N
)  e.  NN0  ->  ( ( M  x.  N
)  e.  NN0  \/  -u ( M  x.  N
)  e.  NN0 )
)
1614, 15syl6 33 . . . . . . 7  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  e.  NN0  /\  N  e. 
NN0 )  ->  (
( M  x.  N
)  e.  NN0  \/  -u ( M  x.  N
)  e.  NN0 )
) )
1716, 6jctild 541 . . . . . 6  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  e.  NN0  /\  N  e. 
NN0 )  ->  (
( M  x.  N
)  e.  RR  /\  ( ( M  x.  N )  e.  NN0  \/  -u ( M  x.  N
)  e.  NN0 )
) ) )
18 nn0mulcl 10828 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( M  x.  -u N
)  e.  NN0 )
19 mulneg2 9990 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  x.  -u N
)  =  -u ( M  x.  N )
)
209, 10, 19syl2an 475 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  x.  -u N
)  =  -u ( M  x.  N )
)
2120eleq1d 2523 . . . . . . . . 9  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( M  x.  -u N )  e.  NN0  <->  -u ( M  x.  N )  e.  NN0 ) )
2218, 21syl5ib 219 . . . . . . . 8  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( M  e. 
NN0  /\  -u N  e. 
NN0 )  ->  -u ( M  x.  N )  e.  NN0 ) )
2322, 15syl6 33 . . . . . . 7  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( M  e. 
NN0  /\  -u N  e. 
NN0 )  ->  (
( M  x.  N
)  e.  NN0  \/  -u ( M  x.  N
)  e.  NN0 )
) )
2423, 6jctild 541 . . . . . 6  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( M  e. 
NN0  /\  -u N  e. 
NN0 )  ->  (
( M  x.  N
)  e.  RR  /\  ( ( M  x.  N )  e.  NN0  \/  -u ( M  x.  N
)  e.  NN0 )
) ) )
25 nn0mulcl 10828 . . . . . . . . 9  |-  ( (
-u M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( -u M  x.  -u N )  e. 
NN0 )
26 mul2neg 9992 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( -u M  x.  -u N )  =  ( M  x.  N ) )
279, 10, 26syl2an 475 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( -u M  x.  -u N )  =  ( M  x.  N ) )
2827eleq1d 2523 . . . . . . . . 9  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  x.  -u N )  e. 
NN0 
<->  ( M  x.  N
)  e.  NN0 )
)
2925, 28syl5ib 219 . . . . . . . 8  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( M  x.  N )  e.  NN0 ) )
30 orc 383 . . . . . . . 8  |-  ( ( M  x.  N )  e.  NN0  ->  ( ( M  x.  N )  e.  NN0  \/  -u ( M  x.  N )  e.  NN0 ) )
3129, 30syl6 33 . . . . . . 7  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  e.  NN0  /\  -u N  e.  NN0 )  ->  (
( M  x.  N
)  e.  NN0  \/  -u ( M  x.  N
)  e.  NN0 )
) )
3231, 6jctild 541 . . . . . 6  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  e.  NN0  /\  -u N  e.  NN0 )  ->  (
( M  x.  N
)  e.  RR  /\  ( ( M  x.  N )  e.  NN0  \/  -u ( M  x.  N
)  e.  NN0 )
) ) )
337, 17, 24, 32ccased 945 . . . . 5  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( ( M  e.  NN0  \/  -u M  e.  NN0 )  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) )  -> 
( ( M  x.  N )  e.  RR  /\  ( ( M  x.  N )  e.  NN0  \/  -u ( M  x.  N
)  e.  NN0 )
) ) )
34 elznn0 10875 . . . . 5  |-  ( ( M  x.  N )  e.  ZZ  <->  ( ( M  x.  N )  e.  RR  /\  ( ( M  x.  N )  e.  NN0  \/  -u ( M  x.  N )  e.  NN0 ) ) )
3533, 34syl6ibr 227 . . . 4  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( ( M  e.  NN0  \/  -u M  e.  NN0 )  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) )  -> 
( M  x.  N
)  e.  ZZ ) )
3635imp 427 . . 3  |-  ( ( ( M  e.  RR  /\  N  e.  RR )  /\  ( ( M  e.  NN0  \/  -u M  e.  NN0 )  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )  ->  ( M  x.  N )  e.  ZZ )
3736an4s 824 . 2  |-  ( ( ( M  e.  RR  /\  ( M  e.  NN0  \/  -u M  e.  NN0 ) )  /\  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )  ->  ( M  x.  N )  e.  ZZ )
381, 2, 37syl2anb 477 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823  (class class class)co 6270   CCcc 9479   RRcr 9480    x. cmul 9486   -ucneg 9797   NN0cn0 10791   ZZcz 10860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861
This theorem is referenced by:  zdivmul  10931  msqznn  10940  zmulcld  10971  uz2mulcl  11160  qaddcl  11199  qmulcl  11201  qreccl  11203  fzctr  11791  flmulnn0  11942  zexpcl  12163  iexpcyc  12254  zesq  12271  cshweqrep  12780  fprodzcl  13843  dvdsmul1  14089  dvdsmul2  14090  muldvds1  14092  muldvds2  14093  dvdscmul  14094  dvdsmulc  14095  dvdscmulr  14096  dvdsmulcr  14097  dvds2ln  14098  dvdstr  14102  dvdsmultr1  14103  dvdsmultr2  14105  oexpneg  14133  divalglem0  14135  divalglem2  14137  divalglem4  14138  divalglem8  14142  divalgb  14146  divalgmod  14148  ndvdsi  14152  gcdaddmlem  14250  absmulgcd  14269  gcdmultiple  14272  gcdmultiplez  14273  dvdsmulgcd  14276  rpmulgcd  14277  coprmdvds  14327  rpmul  14348  eulerthlem2  14396  modprminv  14410  modprminveq  14411  modprm0  14414  pythagtriplem4  14427  pcpremul  14451  pcmul  14459  gzmulcl  14540  pgpfac1lem2  17321  zsubrg  18666  dvdsrzring  18696  mulgrhm  18710  domnchr  18744  znfld  18772  znunit  18775  mbfi1fseqlem5  22292  dvexp3  22545  basellem2  23553  basellem5  23556  dvdsflf1o  23661  chtub  23685  bposlem1  23757  bposlem5  23761  bposlem6  23762  lgslem3  23771  lgsval4a  23791  lgsneg  23792  lgsdir2  23801  lgsdchr  23821  lgseisenlem1  23822  lgseisenlem2  23823  lgseisenlem3  23824  lgsquadlem1  23827  lgsquad2lem2  23832  chebbnd1lem1  23852  chebbnd1lem3  23854  gxnn0mul  25477  zrisefaccl  29383  zfallfaccl  29384  fzmul  30473  mzpclall  30899  mzpindd  30918  acongrep  31157  acongeq  31160  jm2.18  31169  jm2.21  31175  jm2.26a  31181  jm2.26  31183  jm2.16nn0  31185  jm2.27a  31186  jm2.27c  31188  jm3.1lem3  31200  lcmcllem  31443  fourierswlem  32252  oexpnegALTV  32583  oexpnegnz  32584  2zrngmmgm  33006  zlmodzxzequa  33351  zlmodzxzequap  33354
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