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Theorem zmulcl 10974
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 10941 . 2  |-  ( M  e.  ZZ  <->  ( M  e.  RR  /\  ( M  e.  NN0  \/  -u M  e.  NN0 ) ) )
2 elznn0 10941 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
3 nn0mulcl 10895 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  x.  N
)  e.  NN0 )
43orcd 398 . . . . . . . 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 9610 . . . . . . 7  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  x.  N
)  e.  RR )
75, 6jctild 550 . . . . . 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 10895 . . . . . . . . 9  |-  ( (
-u M  e.  NN0  /\  N  e.  NN0 )  ->  ( -u M  x.  N )  e.  NN0 )
9 recn 9615 . . . . . . . . . . 11  |-  ( M  e.  RR  ->  M  e.  CC )
10 recn 9615 . . . . . . . . . . 11  |-  ( N  e.  RR  ->  N  e.  CC )
11 mulneg1 10043 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( -u M  x.  N )  =  -u ( M  x.  N
) )
129, 10, 11syl2an 484 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( -u M  x.  N )  =  -u ( M  x.  N
) )
1312eleq1d 2513 . . . . . . . . 9  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  x.  N )  e.  NN0  <->  -u ( M  x.  N )  e.  NN0 ) )
148, 13syl5ib 227 . . . . . . . 8  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  e.  NN0  /\  N  e. 
NN0 )  ->  -u ( M  x.  N )  e.  NN0 ) )
15 olc 390 . . . . . . . 8  |-  ( -u ( M  x.  N
)  e.  NN0  ->  ( ( M  x.  N
)  e.  NN0  \/  -u ( M  x.  N
)  e.  NN0 )
)
1614, 15syl6 34 . . . . . . 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 550 . . . . . 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 10895 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( M  x.  -u N
)  e.  NN0 )
19 mulneg2 10044 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  x.  -u N
)  =  -u ( M  x.  N )
)
209, 10, 19syl2an 484 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  x.  -u N
)  =  -u ( M  x.  N )
)
2120eleq1d 2513 . . . . . . . . 9  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( M  x.  -u N )  e.  NN0  <->  -u ( M  x.  N )  e.  NN0 ) )
2218, 21syl5ib 227 . . . . . . . 8  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( M  e. 
NN0  /\  -u N  e. 
NN0 )  ->  -u ( M  x.  N )  e.  NN0 ) )
2322, 15syl6 34 . . . . . . 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 550 . . . . . 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 10895 . . . . . . . . 9  |-  ( (
-u M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( -u M  x.  -u N )  e. 
NN0 )
26 mul2neg 10046 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( -u M  x.  -u N )  =  ( M  x.  N ) )
279, 10, 26syl2an 484 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( -u M  x.  -u N )  =  ( M  x.  N ) )
2827eleq1d 2513 . . . . . . . . 9  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  x.  -u N )  e. 
NN0 
<->  ( M  x.  N
)  e.  NN0 )
)
2925, 28syl5ib 227 . . . . . . . 8  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( -u M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( M  x.  N )  e.  NN0 ) )
30 orc 391 . . . . . . . 8  |-  ( ( M  x.  N )  e.  NN0  ->  ( ( M  x.  N )  e.  NN0  \/  -u ( M  x.  N )  e.  NN0 ) )
3129, 30syl6 34 . . . . . . 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 550 . . . . . 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 959 . . . . 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 10941 . . . . 5  |-  ( ( M  x.  N )  e.  ZZ  <->  ( ( M  x.  N )  e.  RR  /\  ( ( M  x.  N )  e.  NN0  \/  -u ( M  x.  N )  e.  NN0 ) ) )
3533, 34syl6ibr 235 . . . 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 435 . . 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 839 . 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 486 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 374    /\ wa 375    = wceq 1447    e. wcel 1890  (class class class)co 6275   CCcc 9523   RRcr 9524    x. cmul 9530   -ucneg 9847   NN0cn0 10858   ZZcz 10926
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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