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Theorem zmulcl 9107
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 9069 . 2 |- ( M e. ZZ <-> ( M e. RR /\ ( M e. NN0 \/ -u M e. NN0 ) ) )
2 elznn0 9069 . 2 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )
3 nn0mulcl 9013 . . . . . . . . 9 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M x. N ) e. NN0 )
43orcd 722 . . . . . . . 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 7748 . . . . . . 7 |- ( ( M e. RR /\ N e. RR ) -> ( M x. N ) e. RR )
75, 6jctild 314 . . . . . 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 9013 . . . . . . . . 9 |- ( ( -u M e. NN0 /\ N e. NN0 ) -> ( -u M x. N ) e. NN0 )
9 recn 7753 . . . . . . . . . . 11 |- ( M e. RR -> M e. CC )
10 recn 7753 . . . . . . . . . . 11 |- ( N e. RR -> N e. CC )
11 mulneg1 8157 . . . . . . . . . . 11 |- ( ( M e. CC /\ N e. CC ) -> ( -u M x. N ) = -u ( M x. N ) )
129, 10, 11syl2an 287 . . . . . . . . . 10 |- ( ( M e. RR /\ N e. RR ) -> ( -u M x. N ) = -u ( M x. N ) )
1312eleq1d 2208 . . . . . . . . 9 |- ( ( M e. RR /\ N e. RR ) -> ( ( -u M x. N ) e. NN0 <-> -u ( M x. N ) e. NN0 ) )
148, 13syl5ib 153 . . . . . . . 8 |- ( ( M e. RR /\ N e. RR ) -> ( ( -u M e. NN0 /\ N e. NN0 ) -> -u ( M x. N ) e. NN0 ) )
15 olc 700 . . . . . . . 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 314 . . . . . 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 9013 . . . . . . . . 9 |- ( ( M e. NN0 /\ -u N e. NN0 ) -> ( M x. -u N ) e. NN0 )
19 mulneg2 8158 . . . . . . . . . . 11 |- ( ( M e. CC /\ N e. CC ) -> ( M x. -u N ) = -u ( M x. N ) )
209, 10, 19syl2an 287 . . . . . . . . . 10 |- ( ( M e. RR /\ N e. RR ) -> ( M x. -u N ) = -u ( M x. N ) )
2120eleq1d 2208 . . . . . . . . 9 |- ( ( M e. RR /\ N e. RR ) -> ( ( M x. -u N ) e. NN0 <-> -u ( M x. N ) e. NN0 ) )
2218, 21syl5ib 153 . . . . . . . 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 314 . . . . . 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 9013 . . . . . . . . 9 |- ( ( -u M e. NN0 /\ -u N e. NN0 ) -> ( -u M x. -u N ) e. NN0 )
26 mul2neg 8160 . . . . . . . . . . 11 |- ( ( M e. CC /\ N e. CC ) -> ( -u M x. -u N ) = ( M x. N ) )
279, 10, 26syl2an 287 . . . . . . . . . 10 |- ( ( M e. RR /\ N e. RR ) -> ( -u M x. -u N ) = ( M x. N ) )
2827eleq1d 2208 . . . . . . . . 9 |- ( ( M e. RR /\ N e. RR ) -> ( ( -u M x. -u N ) e. NN0 <-> ( M x. N ) e. NN0 ) )
2925, 28syl5ib 153 . . . . . . . 8 |- ( ( M e. RR /\ N e. RR ) -> ( ( -u M e. NN0 /\ -u N e. NN0 ) -> ( M x. N ) e. NN0 ) )
30 orc 701 . . . . . . . 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 314 . . . . . 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 949 . . . . 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 9069 . . . . 5 |- ( ( M x. N ) e. ZZ <-> ( ( M x. N ) e. RR /\ ( ( M x. N ) e. NN0 \/ -u ( M x. N ) e. NN0 ) ) )
3533, 34syl6ibr 161 . . . 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 123 . . 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 577 . 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 289 1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   \/ wo 697   = wceq 1331   e. wcel 1480  (class class class)co 5774   CCcc 7618   RRcr 7619   x. cmul 7625   -ucneg 7934   NN0cn0 8977   ZZcz 9054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055
This theorem is referenced by:  zdivmul  9141  msqznn  9151  zmulcld  9179  uz2mulcl  9402  qaddcl  9427  qmulcl  9429  qreccl  9434  fzctr  9910  flqmulnn0  10072  zexpcl  10308  iexpcyc  10397  zesq  10410  dvdsmul1  11515  dvdsmul2  11516  muldvds1  11518  muldvds2  11519  dvdscmul  11520  dvdsmulc  11521  dvds2ln  11526  dvdstr  11530  dvdsmultr1  11531  dvdsmultr2  11533  3dvdsdec  11562  3dvds2dec  11563  oexpneg  11574  mulsucdiv2z  11582  divalgb  11622  divalgmod  11624  ndvdsi  11630  absmulgcd  11705  gcdmultiple  11708  gcdmultiplez  11709  dvdsmulgcd  11713  rpmulgcd  11714  lcmcllem  11748  rpmul  11779  cncongr1  11784  cncongr2  11785
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