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Theorem zmulcl 10288
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 10260 . 2  |-  ( M  e.  ZZ  <->  ( M  e.  RR  /\  ( M  e.  NN0  \/  -u M  e.  NN0 ) ) )
2 elznn0 10260 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
3 nn0mulcl 10220 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  x.  N
)  e.  NN0 )
43orcd 382 . . . . . . . 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 9039 . . . . . . 7  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  x.  N
)  e.  RR )
75, 6jctild 528 . . . . . 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 10220 . . . . . . . . 9  |-  ( (
-u M  e.  NN0  /\  N  e.  NN0 )  ->  ( -u M  x.  N )  e.  NN0 )
9 recn 9044 . . . . . . . . . . 11  |-  ( M  e.  RR  ->  M  e.  CC )
10 recn 9044 . . . . . . . . . . 11  |-  ( N  e.  RR  ->  N  e.  CC )
11 mulneg1 9434 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( -u M  x.  N )  =  -u ( M  x.  N
) )
129, 10, 11syl2an 464 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( -u M  x.  N )  =  -u ( M  x.  N
) )
1312eleq1d 2478 . . . . . . . . 9  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  x.  N )  e.  NN0  <->  -u ( M  x.  N )  e.  NN0 ) )
148, 13syl5ib 211 . . . . . . . 8  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  e.  NN0  /\  N  e. 
NN0 )  ->  -u ( M  x.  N )  e.  NN0 ) )
15 olc 374 . . . . . . . 8  |-  ( -u ( M  x.  N
)  e.  NN0  ->  ( ( M  x.  N
)  e.  NN0  \/  -u ( M  x.  N
)  e.  NN0 )
)
1614, 15syl6 31 . . . . . . 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 528 . . . . . 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 10220 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( M  x.  -u N
)  e.  NN0 )
19 mulneg2 9435 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  x.  -u N
)  =  -u ( M  x.  N )
)
209, 10, 19syl2an 464 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  x.  -u N
)  =  -u ( M  x.  N )
)
2120eleq1d 2478 . . . . . . . . 9  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( M  x.  -u N )  e.  NN0  <->  -u ( M  x.  N )  e.  NN0 ) )
2218, 21syl5ib 211 . . . . . . . 8  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( M  e. 
NN0  /\  -u N  e. 
NN0 )  ->  -u ( M  x.  N )  e.  NN0 ) )
2322, 15syl6 31 . . . . . . 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 528 . . . . . 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 10220 . . . . . . . . 9  |-  ( (
-u M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( -u M  x.  -u N )  e. 
NN0 )
26 mul2neg 9437 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( -u M  x.  -u N )  =  ( M  x.  N ) )
279, 10, 26syl2an 464 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( -u M  x.  -u N )  =  ( M  x.  N ) )
2827eleq1d 2478 . . . . . . . . 9  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  x.  -u N )  e. 
NN0 
<->  ( M  x.  N
)  e.  NN0 )
)
2925, 28syl5ib 211 . . . . . . . 8  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( M  x.  N )  e.  NN0 ) )
30 orc 375 . . . . . . . 8  |-  ( ( M  x.  N )  e.  NN0  ->  ( ( M  x.  N )  e.  NN0  \/  -u ( M  x.  N )  e.  NN0 ) )
3129, 30syl6 31 . . . . . . 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 528 . . . . . 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 914 . . . . 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 10260 . . . . 5  |-  ( ( M  x.  N )  e.  ZZ  <->  ( ( M  x.  N )  e.  RR  /\  ( ( M  x.  N )  e.  NN0  \/  -u ( M  x.  N )  e.  NN0 ) ) )
3533, 34syl6ibr 219 . . . 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 419 . . 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 800 . 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 466 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721  (class class class)co 6048   CCcc 8952   RRcr 8953    x. cmul 8959   -ucneg 9256   NN0cn0 10185   ZZcz 10246
This theorem is referenced by:  zdivmul  10306  msqznn  10315  zmulcld  10345  uz2mulcl  10517  qaddcl  10554  qmulcl  10556  qreccl  10558  fzctr  11080  flmulnn0  11192  zexpcl  11359  iexpcyc  11448  zesq  11465  dvdsmul1  12834  dvdsmul2  12835  muldvds1  12837  muldvds2  12838  dvdscmul  12839  dvdsmulc  12840  dvdscmulr  12841  dvdsmulcr  12842  dvds2ln  12843  dvdstr  12846  dvdsmultr1  12847  dvdsmultr2  12848  oexpneg  12874  divalglem0  12876  divalglem2  12878  divalglem4  12879  divalglem8  12883  divalgb  12887  divalgmod  12889  ndvdsi  12893  gcdaddmlem  12991  absmulgcd  13010  gcdmultiple  13013  gcdmultiplez  13014  dvdsmulgcd  13017  rpmulgcd  13018  coprmdvds  13065  rpmul  13086  eulerthlem2  13134  pythagtriplem4  13156  pcpremul  13180  pcmul  13188  gzmulcl  13269  pgpfac1lem2  15596  zsubrg  16715  dvdsrz  16730  mulgrhm  16750  domnchr  16776  znfld  16804  znunit  16807  mbfi1fseqlem5  19572  dvexp3  19823  basellem2  20825  basellem5  20828  dvdsflf1o  20933  chtub  20957  bposlem1  21029  bposlem5  21033  bposlem6  21034  lgslem3  21043  lgsval4a  21063  lgsneg  21064  lgsdir2  21073  lgsdchr  21093  lgseisenlem1  21094  lgseisenlem2  21095  lgseisenlem3  21096  lgsquadlem1  21099  lgsquad2lem2  21104  chebbnd1lem1  21124  chebbnd1lem3  21126  gxnn0mul  21826  fprodzcl  25241  zrisefaccl  25296  zfallfaccl  25297  fzmul  26342  mzpclall  26682  mzpindd  26701  acongrep  26943  acongeq  26946  jm2.18  26957  jm2.21  26963  jm2.26a  26969  jm2.26  26971  jm2.16nn0  26973  jm2.27a  26974  jm2.27c  26976  jm3.1lem3  26988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-ltxr 9089  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247
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