Theorem zmulcl 9309

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 9271	. 2 |- ( M e. ZZ <-> ( M e. RR /\ ( M e. NN0 \/ -u M e. NN0 ) ) )
2		elznn0 9271	. 2 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )
3		nn0mulcl 9215	. . . . . . . . 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 7942	. . . . . . 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 9215	. . . . . . . . 9 |- ( ( -u M e. NN0 /\ N e. NN0 ) -> ( -u M x. N ) e. NN0 )
9		recn 7947	. . . . . . . . . . 11 |- ( M e. RR -> M e. CC )
10		recn 7947	. . . . . . . . . . 11 |- ( N e. RR -> N e. CC )
11		mulneg1 8355	. . . . . . . . . . 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 9215	. . . . . . . . 9 |- ( ( M e. NN0 /\ -u N e. NN0 ) -> ( M x. -u N ) e. NN0 )
19		mulneg2 8356	. . . . . . . . . . 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 9215	. . . . . . . . 9 |- ( ( -u M e. NN0 /\ -u N e. NN0 ) -> ( -u M x. -u N ) e. NN0 )
26		mul2neg 8358	. . . . . . . . . . 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 9271	. . . . 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 5878   CCcc 7812   RRcr 7813   x. cmul 7819   -ucneg 8132   NN0cn0 9179   ZZcz 9256

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 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-cnre 7925

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 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-sub 8133  df-neg 8134  df-inn 8923  df-n0 9180  df-z 9257

This theorem is referenced by:  zdivmul  9346  msqznn  9356  zmulcld  9384  uz2mulcl  9611  qaddcl  9638  qmulcl  9640  qreccl  9645  fzctr  10136  flqmulnn0  10302  zexpcl  10538  iexpcyc  10628  zesq  10642  fprodzcl  11620  dvdsmul1  11823  dvdsmul2  11824  muldvds1  11826  muldvds2  11827  dvdscmul  11828  dvdsmulc  11829  dvds2ln  11834  dvdstr  11838  dvdsmultr1  11841  dvdsmultr2  11843  3dvdsdec  11873  3dvds2dec  11874  oexpneg  11885  mulsucdiv2z  11893  divalgb  11933  divalgmod  11935  ndvdsi  11941  absmulgcd  12021  gcdmultiple  12024  gcdmultiplez  12025  dvdsmulgcd  12029  rpmulgcd  12030  lcmcllem  12070  rpmul  12101  cncongr1  12106  cncongr2  12107  modprminv  12252  modprminveq  12253  modprm0  12257  pythagtriplem4  12271  pcpremul  12296  pcmul  12304  gzmulcl  12379  zsubrg  13622  dvdsrzring  13640  lgslem3  14564  lgsval  14566  lgsval2lem  14572  lgsval4a  14584  lgsneg  14586  lgsdir2  14595  lgsdir  14597  lgsdilem2  14598  lgsdi  14599  lgsne0  14600  lgseisenlem1  14611  lgseisenlem2  14612  2lgsoddprmlem2  14615