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Theorem zmulcl 9300
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 9262 . 2 |- ( M e. ZZ <-> ( M e. RR /\ ( M e. NN0 \/ -u M e. NN0 ) ) )
2 elznn0 9262 . 2 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )
3 nn0mulcl 9206 . . . . . . . . 9 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M x. N ) e. NN0 )
43orcd 733 . . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( M x. N ) e. NN0 \/ -u ( M x. N ) e. NN0 ) )
54a1i 9 . . . . . . 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 7934 . . . . . . 7 |- ( ( M e. RR /\ N e. RR ) -> ( M x. N ) e. RR )
75, 6jctild 316 . . . . . 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 9206 . . . . . . . . 9 |- ( ( -u M e. NN0 /\ N e. NN0 ) -> ( -u M x. N ) e. NN0 )
9 recn 7939 . . . . . . . . . . 11 |- ( M e. RR -> M e. CC )
10 recn 7939 . . . . . . . . . . 11 |- ( N e. RR -> N e. CC )
11 mulneg1 8346 . . . . . . . . . . 11 |- ( ( M e. CC /\ N e. CC ) -> ( -u M x. N ) = -u ( M x. N ) )
129, 10, 11syl2an 289 . . . . . . . . . 10 |- ( ( M e. RR /\ N e. RR ) -> ( -u M x. N ) = -u ( M x. N ) )
1312eleq1d 2246 . . . . . . . . 9 |- ( ( M e. RR /\ N e. RR ) -> ( ( -u M x. N ) e. NN0 <-> -u ( M x. N ) e. NN0 ) )
148, 13imbitrid 154 . . . . . . . 8 |- ( ( M e. RR /\ N e. RR ) -> ( ( -u M e. NN0 /\ N e. NN0 ) -> -u ( M x. N ) e. NN0 ) )
15 olc 711 . . . . . . . 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 316 . . . . . 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 9206 . . . . . . . . 9 |- ( ( M e. NN0 /\ -u N e. NN0 ) -> ( M x. -u N ) e. NN0 )
19 mulneg2 8347 . . . . . . . . . . 11 |- ( ( M e. CC /\ N e. CC ) -> ( M x. -u N ) = -u ( M x. N ) )
209, 10, 19syl2an 289 . . . . . . . . . 10 |- ( ( M e. RR /\ N e. RR ) -> ( M x. -u N ) = -u ( M x. N ) )
2120eleq1d 2246 . . . . . . . . 9 |- ( ( M e. RR /\ N e. RR ) -> ( ( M x. -u N ) e. NN0 <-> -u ( M x. N ) e. NN0 ) )
2218, 21imbitrid 154 . . . . . . . 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 316 . . . . . 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 9206 . . . . . . . . 9 |- ( ( -u M e. NN0 /\ -u N e. NN0 ) -> ( -u M x. -u N ) e. NN0 )
26 mul2neg 8349 . . . . . . . . . . 11 |- ( ( M e. CC /\ N e. CC ) -> ( -u M x. -u N ) = ( M x. N ) )
279, 10, 26syl2an 289 . . . . . . . . . 10 |- ( ( M e. RR /\ N e. RR ) -> ( -u M x. -u N ) = ( M x. N ) )
2827eleq1d 2246 . . . . . . . . 9 |- ( ( M e. RR /\ N e. RR ) -> ( ( -u M x. -u N ) e. NN0 <-> ( M x. N ) e. NN0 ) )
2925, 28imbitrid 154 . . . . . . . 8 |- ( ( M e. RR /\ N e. RR ) -> ( ( -u M e. NN0 /\ -u N e. NN0 ) -> ( M x. N ) e. NN0 ) )
30 orc 712 . . . . . . . 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 316 . . . . . 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 965 . . . . 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 9262 . . . . 5 |- ( ( M x. N ) e. ZZ <-> ( ( M x. N ) e. RR /\ ( ( M x. N ) e. NN0 \/ -u ( M x. N ) e. NN0 ) ) )
3533, 34syl6ibr 162 . . . 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 124 . . 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 588 . 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 291 1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   \/ wo 708   = wceq 1353   e. wcel 2148  (class class class)co 5870   CCcc 7804   RRcr 7805   x. cmul 7811   -ucneg 8123   NN0cn0 9170   ZZcz 9247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-pow 4172  ax-pr 4207  ax-setind 4534  ax-cnex 7897  ax-resscn 7898  ax-1cn 7899  ax-1re 7900  ax-icn 7901  ax-addcl 7902  ax-addrcl 7903  ax-mulcl 7904  ax-mulrcl 7905  ax-addcom 7906  ax-mulcom 7907  ax-addass 7908  ax-mulass 7909  ax-distr 7910  ax-i2m1 7911  ax-1rid 7913  ax-0id 7914  ax-rnegex 7915  ax-cnre 7917
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-int 3844  df-br 4002  df-opab 4063  df-id 4291  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-iota 5175  df-fun 5215  df-fv 5221  df-riota 5826  df-ov 5873  df-oprab 5874  df-mpo 5875  df-sub 8124  df-neg 8125  df-inn 8914  df-n0 9171  df-z 9248
This theorem is referenced by:  zdivmul  9337  msqznn  9347  zmulcld  9375  uz2mulcl  9602  qaddcl  9629  qmulcl  9631  qreccl  9636  fzctr  10126  flqmulnn0  10292  zexpcl  10528  iexpcyc  10617  zesq  10631  fprodzcl  11608  dvdsmul1  11811  dvdsmul2  11812  muldvds1  11814  muldvds2  11815  dvdscmul  11816  dvdsmulc  11817  dvds2ln  11822  dvdstr  11826  dvdsmultr1  11829  dvdsmultr2  11831  3dvdsdec  11860  3dvds2dec  11861  oexpneg  11872  mulsucdiv2z  11880  divalgb  11920  divalgmod  11922  ndvdsi  11928  absmulgcd  12008  gcdmultiple  12011  gcdmultiplez  12012  dvdsmulgcd  12016  rpmulgcd  12017  lcmcllem  12057  rpmul  12088  cncongr1  12093  cncongr2  12094  modprminv  12239  modprminveq  12240  modprm0  12244  pythagtriplem4  12258  pcpremul  12283  pcmul  12291  gzmulcl  12366  lgslem3  14185  lgsval  14187  lgsval2lem  14193  lgsval4a  14205  lgsneg  14207  lgsdir2  14216  lgsdir  14218  lgsdilem2  14219  lgsdi  14220  lgsne0  14221
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