Theorem dvdscmulr 11525

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		simp3l 1009	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> K e. ZZ )
2		simp1 981	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> M e. ZZ )
3	1, 2	zmulcld 9182	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( K x. M ) e. ZZ )
4		simp2 982	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> N e. ZZ )
5	1, 4	zmulcld 9182	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( K x. N ) e. ZZ )
6	3, 5	jca 304	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( K x. M ) e. ZZ /\ ( K x. N ) e. ZZ ) )
7	2, 4	jca 304	. . 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	1	adantr 274	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K e. ZZ )
10	9	zcnd 9177	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K e. CC )
11	8	zcnd 9177	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> x e. CC )
12	2	adantr 274	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> M e. ZZ )
13	12	zcnd 9177	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> M e. CC )
14	10, 11, 13	mul12d 7917	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( K x. ( x x. M ) ) = ( x x. ( K x. M ) ) )
15	14	eqeq1d 2148	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( ( K x. ( x x. M ) ) = ( K x. N ) <-> ( x x. ( K x. M ) ) = ( K x. N ) ) )
16	11, 13	mulcld 7789	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( x x. M ) e. CC )
17	4	adantr 274	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> N e. ZZ )
18	17	zcnd 9177	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> N e. CC )
19		simpl3r 1037	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K =/= 0 )
20		0z 9068	. . . . . . . 8 |- 0 e. ZZ
21		zapne 9128	. . . . . . . 8 |- ( ( K e. ZZ /\ 0 e. ZZ ) -> ( K # 0 <-> K =/= 0 ) )
22	9, 20, 21	sylancl 409	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( K # 0 <-> K =/= 0 ) )
23	19, 22	mpbird 166	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K # 0 )
24	16, 18, 10, 23	mulcanapd 8425	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( ( K x. ( x x. M ) ) = ( K x. N ) <-> ( x x. M ) = N ) )
25	15, 24	bitr3d 189	. . . 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 ) )
26	25	biimpd 143	. . 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 ) )
27	6, 7, 8, 26	dvds1lem 11507	. 2 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( K x. M ) || ( K x. N ) -> M || N ) )
28		dvdscmul 11523	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( K x. M ) || ( K x. N ) ) )
29	28	3adant3r 1213	. 2 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( M || N -> ( K x. M ) || ( K x. N ) ) )
30	27, 29	impbid 128	1 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( K x. M ) || ( K x. N ) <-> M || N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   =/= wne 2308   class class class wbr 3929  (class class class)co 5774   0cc0 7623   x. cmul 7628   # cap 8346   ZZcz 9057   || cdvds 11496

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7714  ax-resscn 7715  ax-1cn 7716  ax-1re 7717  ax-icn 7718  ax-addcl 7719  ax-addrcl 7720  ax-mulcl 7721  ax-mulrcl 7722  ax-addcom 7723  ax-mulcom 7724  ax-addass 7725  ax-mulass 7726  ax-distr 7727  ax-i2m1 7728  ax-0lt1 7729  ax-1rid 7730  ax-0id 7731  ax-rnegex 7732  ax-precex 7733  ax-cnre 7734  ax-pre-ltirr 7735  ax-pre-ltwlin 7736  ax-pre-lttrn 7737  ax-pre-apti 7738  ax-pre-ltadd 7739  ax-pre-mulgt0 7740  ax-pre-mulext 7741

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7805  df-mnf 7806  df-xr 7807  df-ltxr 7808  df-le 7809  df-sub 7938  df-neg 7939  df-reap 8340  df-ap 8347  df-inn 8724  df-n0 8981  df-z 9058  df-dvds 11497

This theorem is referenced by:  modmulconst  11528  mulgcd  11707  oddpwdclemxy  11850  oddpwdclemodd  11853