Theorem dvdscmulr 15010

Description: Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
dvdscmulr	|- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( K x. M ) || ( K x. N ) <-> M || N ) )

Proof of Theorem dvdscmulr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zmulcl 11426	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K x. M ) e. ZZ )
2	1	3adant3 1081	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. ZZ )
3		zmulcl 11426	. . . . . . 7 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
4	3	3adant2 1080	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
5	2, 4	jca 554	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) e. ZZ /\ ( K x. N ) e. ZZ ) )
6	5	3coml 1272	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( ( K x. M ) e. ZZ /\ ( K x. N ) e. ZZ ) )
7	6	3adant3r 1323	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( K x. M ) e. ZZ /\ ( K x. N ) e. ZZ ) )
8		3simpa 1058	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( M e. ZZ /\ N e. ZZ ) )
9		simpr 477	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> x e. ZZ )
10		zcn 11382	. . . . . . . . . . . 12 |- ( x e. ZZ -> x e. CC )
11		zcn 11382	. . . . . . . . . . . 12 |- ( M e. ZZ -> M e. CC )
12	10, 11	anim12i 590	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ M e. ZZ ) -> ( x e. CC /\ M e. CC ) )
13		zcn 11382	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
14		zcn 11382	. . . . . . . . . . . 12 |- ( K e. ZZ -> K e. CC )
15	14	anim1i 592	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ K =/= 0 ) -> ( K e. CC /\ K =/= 0 ) )
16		mul12 10202	. . . . . . . . . . . . . . . . 17 |- ( ( K e. CC /\ x e. CC /\ M e. CC ) -> ( K x. ( x x. M ) ) = ( x x. ( K x. M ) ) )
17	16	3adant1r 1319	. . . . . . . . . . . . . . . 16 |- ( ( ( K e. CC /\ K =/= 0 ) /\ x e. CC /\ M e. CC ) -> ( K x. ( x x. M ) ) = ( x x. ( K x. M ) ) )
18	17	3expb 1266	. . . . . . . . . . . . . . 15 |- ( ( ( K e. CC /\ K =/= 0 ) /\ ( x e. CC /\ M e. CC ) ) -> ( K x. ( x x. M ) ) = ( x x. ( K x. M ) ) )
19	18	ancoms 469	. . . . . . . . . . . . . 14 |- ( ( ( x e. CC /\ M e. CC ) /\ ( K e. CC /\ K =/= 0 ) ) -> ( K x. ( x x. M ) ) = ( x x. ( K x. M ) ) )
20	19	3adant2 1080	. . . . . . . . . . . . 13 |- ( ( ( x e. CC /\ M e. CC ) /\ N e. CC /\ ( K e. CC /\ K =/= 0 ) ) -> ( K x. ( x x. M ) ) = ( x x. ( K x. M ) ) )
21	20	eqeq1d 2624	. . . . . . . . . . . 12 |- ( ( ( x e. CC /\ M e. CC ) /\ N e. CC /\ ( K e. CC /\ K =/= 0 ) ) -> ( ( K x. ( x x. M ) ) = ( K x. N ) <-> ( x x. ( K x. M ) ) = ( K x. N ) ) )
22		mulcl 10020	. . . . . . . . . . . . 13 |- ( ( x e. CC /\ M e. CC ) -> ( x x. M ) e. CC )
23		mulcan 10664	. . . . . . . . . . . . 13 |- ( ( ( x x. M ) e. CC /\ N e. CC /\ ( K e. CC /\ K =/= 0 ) ) -> ( ( K x. ( x x. M ) ) = ( K x. N ) <-> ( x x. M ) = N ) )
24	22, 23	syl3an1 1359	. . . . . . . . . . . 12 |- ( ( ( x e. CC /\ M e. CC ) /\ N e. CC /\ ( K e. CC /\ K =/= 0 ) ) -> ( ( K x. ( x x. M ) ) = ( K x. N ) <-> ( x x. M ) = N ) )
25	21, 24	bitr3d 270	. . . . . . . . . . 11 |- ( ( ( x e. CC /\ M e. CC ) /\ N e. CC /\ ( K e. CC /\ K =/= 0 ) ) -> ( ( x x. ( K x. M ) ) = ( K x. N ) <-> ( x x. M ) = N ) )
26	12, 13, 15, 25	syl3an 1368	. . . . . . . . . 10 |- ( ( ( x e. ZZ /\ M e. ZZ ) /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( x x. ( K x. M ) ) = ( K x. N ) <-> ( x x. M ) = N ) )
27	26	3expb 1266	. . . . . . . . 9 |- ( ( ( x e. ZZ /\ M e. ZZ ) /\ ( N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) ) -> ( ( x x. ( K x. M ) ) = ( K x. N ) <-> ( x x. M ) = N ) )
28	27	3impa 1259	. . . . . . . 8 |- ( ( x e. ZZ /\ M e. ZZ /\ ( N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) ) -> ( ( x x. ( K x. M ) ) = ( K x. N ) <-> ( x x. M ) = N ) )
29	28	3coml 1272	. . . . . . 7 |- ( ( M e. ZZ /\ ( N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( ( x x. ( K x. M ) ) = ( K x. N ) <-> ( x x. M ) = N ) )
30	29	3expia 1267	. . . . . 6 |- ( ( M e. ZZ /\ ( N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) ) -> ( x e. ZZ -> ( ( x x. ( K x. M ) ) = ( K x. N ) <-> ( x x. M ) = N ) ) )
31	30	3impb 1260	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( x e. ZZ -> ( ( x x. ( K x. M ) ) = ( K x. N ) <-> ( x x. M ) = N ) ) )
32	31	imp 445	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( ( x x. ( K x. M ) ) = ( K x. N ) <-> ( x x. M ) = N ) )
33	32	biimpd 219	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( ( x x. ( K x. M ) ) = ( K x. N ) -> ( x x. M ) = N ) )
34	7, 8, 9, 33	dvds1lem 14993	. 2 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( K x. M ) || ( K x. N ) -> M || N ) )
35		dvdscmul 15008	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( K x. M ) || ( K x. N ) ) )
36	35	3adant3r 1323	. 2 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( M || N -> ( K x. M ) || ( K x. N ) ) )
37	34, 36	impbid 202	1 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( K x. M ) || ( K x. N ) <-> M || N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941   ZZcz 11377   || cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-dvds 14984

This theorem is referenced by:  modmulconst  15013  bitsmod  15158  mulgcd  15265  pcpremul  15548  4sqlem17  15665  odmulg  17973  ablfacrp2  18466  ablfac1b  18469  pgpfac1lem3a  18475  znrrg  19914  fsumdvdsdiaglem  24909  oddpwdc  30416  jm2.20nn  37564