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Theorem zmulcl 9265
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 9227 . 2 |- ( M e. ZZ <-> ( M e. RR /\ ( M e. NN0 \/ -u M e. NN0 ) ) )
2 elznn0 9227 . 2 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )
3 nn0mulcl 9171 . . . . . . . . 9 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M x. N ) e. NN0 )
43orcd 728 . . . . . . . 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 7902 . . . . . . 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 9171 . . . . . . . . 9 |- ( ( -u M e. NN0 /\ N e. NN0 ) -> ( -u M x. N ) e. NN0 )
9 recn 7907 . . . . . . . . . . 11 |- ( M e. RR -> M e. CC )
10 recn 7907 . . . . . . . . . . 11 |- ( N e. RR -> N e. CC )
11 mulneg1 8314 . . . . . . . . . . 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 2239 . . . . . . . . 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 706 . . . . . . . 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 9171 . . . . . . . . 9 |- ( ( M e. NN0 /\ -u N e. NN0 ) -> ( M x. -u N ) e. NN0 )
19 mulneg2 8315 . . . . . . . . . . 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 2239 . . . . . . . . 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 9171 . . . . . . . . 9 |- ( ( -u M e. NN0 /\ -u N e. NN0 ) -> ( -u M x. -u N ) e. NN0 )
26 mul2neg 8317 . . . . . . . . . . 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 2239 . . . . . . . . 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 707 . . . . . . . 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 960 . . . . 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 9227 . . . . 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 583 . 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 703   = wceq 1348   e. wcel 2141  (class class class)co 5853   CCcc 7772   RRcr 7773   x. cmul 7779   -ucneg 8091   NN0cn0 9135   ZZcz 9212
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213
This theorem is referenced by:  zdivmul  9302  msqznn  9312  zmulcld  9340  uz2mulcl  9567  qaddcl  9594  qmulcl  9596  qreccl  9601  fzctr  10089  flqmulnn0  10255  zexpcl  10491  iexpcyc  10580  zesq  10594  fprodzcl  11572  dvdsmul1  11775  dvdsmul2  11776  muldvds1  11778  muldvds2  11779  dvdscmul  11780  dvdsmulc  11781  dvds2ln  11786  dvdstr  11790  dvdsmultr1  11793  dvdsmultr2  11795  3dvdsdec  11824  3dvds2dec  11825  oexpneg  11836  mulsucdiv2z  11844  divalgb  11884  divalgmod  11886  ndvdsi  11892  absmulgcd  11972  gcdmultiple  11975  gcdmultiplez  11976  dvdsmulgcd  11980  rpmulgcd  11981  lcmcllem  12021  rpmul  12052  cncongr1  12057  cncongr2  12058  modprminv  12203  modprminveq  12204  modprm0  12208  pythagtriplem4  12222  pcpremul  12247  pcmul  12255  gzmulcl  12330  lgslem3  13697  lgsval  13699  lgsval2lem  13705  lgsval4a  13717  lgsneg  13719  lgsdir2  13728  lgsdir  13730  lgsdilem2  13731  lgsdi  13732  lgsne0  13733
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