Theorem zmulcl 9306

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

Step	Hyp	Ref	Expression
1		elznn0 9268	. 2 |- ( M e. ZZ <-> ( M e. RR /\ ( M e. NN0 \/ -u M e. NN0 ) ) )
2		elznn0 9268	. 2 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )
3		nn0mulcl 9212	. . . . . . . . 9 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M x. N ) e. NN0 )
4	3	orcd 733	. . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( M x. N ) e. NN0 \/ -u ( M x. N ) e. NN0 ) )
5	4	a1i 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 7939	. . . . . . 7 |- ( ( M e. RR /\ N e. RR ) -> ( M x. N ) e. RR )
7	5, 6	jctild 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 9212	. . . . . . . . 9 |- ( ( -u M e. NN0 /\ N e. NN0 ) -> ( -u M x. N ) e. NN0 )
9		recn 7944	. . . . . . . . . . 11 |- ( M e. RR -> M e. CC )
10		recn 7944	. . . . . . . . . . 11 |- ( N e. RR -> N e. CC )
11		mulneg1 8352	. . . . . . . . . . 11 |- ( ( M e. CC /\ N e. CC ) -> ( -u M x. N ) = -u ( M x. N ) )
12	9, 10, 11	syl2an 289	. . . . . . . . . 10 |- ( ( M e. RR /\ N e. RR ) -> ( -u M x. N ) = -u ( M x. N ) )
13	12	eleq1d 2246	. . . . . . . . 9 |- ( ( M e. RR /\ N e. RR ) -> ( ( -u M x. N ) e. NN0 <-> -u ( M x. N ) e. NN0 ) )
14	8, 13	imbitrid 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 ) )
16	14, 15	syl6 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 ) ) )
17	16, 6	jctild 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 9212	. . . . . . . . 9 |- ( ( M e. NN0 /\ -u N e. NN0 ) -> ( M x. -u N ) e. NN0 )
19		mulneg2 8353	. . . . . . . . . . 11 |- ( ( M e. CC /\ N e. CC ) -> ( M x. -u N ) = -u ( M x. N ) )
20	9, 10, 19	syl2an 289	. . . . . . . . . 10 |- ( ( M e. RR /\ N e. RR ) -> ( M x. -u N ) = -u ( M x. N ) )
21	20	eleq1d 2246	. . . . . . . . 9 |- ( ( M e. RR /\ N e. RR ) -> ( ( M x. -u N ) e. NN0 <-> -u ( M x. N ) e. NN0 ) )
22	18, 21	imbitrid 154	. . . . . . . 8 |- ( ( M e. RR /\ N e. RR ) -> ( ( M e. NN0 /\ -u N e. NN0 ) -> -u ( M x. N ) e. NN0 ) )
23	22, 15	syl6 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 ) ) )
24	23, 6	jctild 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 9212	. . . . . . . . 9 |- ( ( -u M e. NN0 /\ -u N e. NN0 ) -> ( -u M x. -u N ) e. NN0 )
26		mul2neg 8355	. . . . . . . . . . 11 |- ( ( M e. CC /\ N e. CC ) -> ( -u M x. -u N ) = ( M x. N ) )
27	9, 10, 26	syl2an 289	. . . . . . . . . 10 |- ( ( M e. RR /\ N e. RR ) -> ( -u M x. -u N ) = ( M x. N ) )
28	27	eleq1d 2246	. . . . . . . . 9 |- ( ( M e. RR /\ N e. RR ) -> ( ( -u M x. -u N ) e. NN0 <-> ( M x. N ) e. NN0 ) )
29	25, 28	imbitrid 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 ) )
31	29, 30	syl6 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 ) ) )
32	31, 6	jctild 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 ) ) ) )
33	7, 17, 24, 32	ccased 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 9268	. . . . 5 |- ( ( M x. N ) e. ZZ <-> ( ( M x. N ) e. RR /\ ( ( M x. N ) e. NN0 \/ -u ( M x. N ) e. NN0 ) ) )
35	33, 34	imbitrrdi 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 ) )
36	35	imp 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 )
37	36	an4s 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 )
38	1, 2, 37	syl2anb 291	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 708   = wceq 1353   e. wcel 2148  (class class class)co 5875   CCcc 7809   RRcr 7810   x. cmul 7816   -ucneg 8129   NN0cn0 9176   ZZcz 9253

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 4122  ax-pow 4175  ax-pr 4210  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922

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 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-sub 8130  df-neg 8131  df-inn 8920  df-n0 9177  df-z 9254

This theorem is referenced by:  zdivmul  9343  msqznn  9353  zmulcld  9381  uz2mulcl  9608  qaddcl  9635  qmulcl  9637  qreccl  9642  fzctr  10133  flqmulnn0  10299  zexpcl  10535  iexpcyc  10625  zesq  10639  fprodzcl  11617  dvdsmul1  11820  dvdsmul2  11821  muldvds1  11823  muldvds2  11824  dvdscmul  11825  dvdsmulc  11826  dvds2ln  11831  dvdstr  11835  dvdsmultr1  11838  dvdsmultr2  11840  3dvdsdec  11870  3dvds2dec  11871  oexpneg  11882  mulsucdiv2z  11890  divalgb  11930  divalgmod  11932  ndvdsi  11938  absmulgcd  12018  gcdmultiple  12021  gcdmultiplez  12022  dvdsmulgcd  12026  rpmulgcd  12027  lcmcllem  12067  rpmul  12098  cncongr1  12103  cncongr2  12104  modprminv  12249  modprminveq  12250  modprm0  12254  pythagtriplem4  12268  pcpremul  12293  pcmul  12301  gzmulcl  12376  zsubrg  13478  dvdsrzring  13496  lgslem3  14406  lgsval  14408  lgsval2lem  14414  lgsval4a  14426  lgsneg  14428  lgsdir2  14437  lgsdir  14439  lgsdilem2  14440  lgsdi  14441  lgsne0  14442  lgseisenlem1  14453  lgseisenlem2  14454  2lgsoddprmlem2  14457