Theorem dvdsmulcr 11371

Description: Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

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

Proof of Theorem dvdsmulcr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 964	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> M e. ZZ )
2		simp3l 992	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> K e. ZZ )
3	1, 2	zmulcld 9083	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( M x. K ) e. ZZ )
4		simp2 965	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> N e. ZZ )
5	4, 2	zmulcld 9083	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( N x. K ) e. ZZ )
6	3, 5	jca 302	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( M x. K ) e. ZZ /\ ( N x. K ) e. ZZ ) )
7		3simpa 961	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( M e. ZZ /\ N e. ZZ ) )
8		simpr 109	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> x e. ZZ )
9	8	zcnd 9078	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> x e. CC )
10	1	zcnd 9078	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> M e. CC )
11	10	adantr 272	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> M e. CC )
12	2	adantr 272	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K e. ZZ )
13	12	zcnd 9078	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K e. CC )
14	9, 11, 13	mulassd 7713	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
15	14	eqeq1d 2123	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( ( ( x x. M ) x. K ) = ( N x. K ) <-> ( x x. ( M x. K ) ) = ( N x. K ) ) )
16	9, 11	mulcld 7710	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( x x. M ) e. CC )
17	4	adantr 272	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> N e. ZZ )
18	17	zcnd 9078	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> N e. CC )
19		simpl3r 1020	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K =/= 0 )
20		0z 8969	. . . . . . . . . . 11 |- 0 e. ZZ
21		zapne 9029	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ 0 e. ZZ ) -> ( K # 0 <-> K =/= 0 ) )
22	20, 21	mpan2 419	. . . . . . . . . 10 |- ( K e. ZZ -> ( K # 0 <-> K =/= 0 ) )
23	22	adantr 272	. . . . . . . . 9 |- ( ( K e. ZZ /\ K =/= 0 ) -> ( K # 0 <-> K =/= 0 ) )
24	23	3ad2ant3 987	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( K # 0 <-> K =/= 0 ) )
25	24	adantr 272	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( K # 0 <-> K =/= 0 ) )
26	19, 25	mpbird 166	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K # 0 )
27	16, 18, 13, 26	mulcanap2d 8336	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( ( ( x x. M ) x. K ) = ( N x. K ) <-> ( x x. M ) = N ) )
28	15, 27	bitr3d 189	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( ( x x. ( M x. K ) ) = ( N x. K ) <-> ( x x. M ) = N ) )
29	28	biimpd 143	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( ( x x. ( M x. K ) ) = ( N x. K ) -> ( x x. M ) = N ) )
30	6, 7, 8, 29	dvds1lem 11352	. 2 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( M x. K ) || ( N x. K ) -> M || N ) )
31		dvdsmulc 11369	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
32	31	3adant3r 1196	. 2 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
33	30, 32	impbid 128	1 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( M x. K ) || ( N x. K ) <-> M || N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 945   = wceq 1314   e. wcel 1463   =/= wne 2282   class class class wbr 3895  (class class class)co 5728   CCcc 7545   0cc0 7547   x. cmul 7552   # cap 8261   ZZcz 8958   || cdvds 11341

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663

This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-br 3896  df-opab 3950  df-id 4175  df-po 4178  df-iso 4179  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-inn 8631  df-n0 8882  df-z 8959  df-dvds 11342

This theorem is referenced by:  mulgcddvds  11621