Theorem dvdsmulcr 11986

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 999	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> M e. ZZ )
2		simp3l 1027	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> K e. ZZ )
3	1, 2	zmulcld 9454	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( M x. K ) e. ZZ )
4		simp2 1000	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> N e. ZZ )
5	4, 2	zmulcld 9454	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( N x. K ) e. ZZ )
6	3, 5	jca 306	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( M x. K ) e. ZZ /\ ( N x. K ) e. ZZ ) )
7		3simpa 996	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( M e. ZZ /\ N e. ZZ ) )
8		simpr 110	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> x e. ZZ )
9	8	zcnd 9449	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> x e. CC )
10	1	zcnd 9449	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> M e. CC )
11	10	adantr 276	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> M e. CC )
12	2	adantr 276	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K e. ZZ )
13	12	zcnd 9449	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K e. CC )
14	9, 11, 13	mulassd 8050	. . . . . 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 2205	. . . . 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 8047	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( x x. M ) e. CC )
17	4	adantr 276	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> N e. ZZ )
18	17	zcnd 9449	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> N e. CC )
19		simpl3r 1055	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K =/= 0 )
20		0z 9337	. . . . . . . . . . 11 |- 0 e. ZZ
21		zapne 9400	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ 0 e. ZZ ) -> ( K # 0 <-> K =/= 0 ) )
22	20, 21	mpan2 425	. . . . . . . . . 10 |- ( K e. ZZ -> ( K # 0 <-> K =/= 0 ) )
23	22	adantr 276	. . . . . . . . 9 |- ( ( K e. ZZ /\ K =/= 0 ) -> ( K # 0 <-> K =/= 0 ) )
24	23	3ad2ant3 1022	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( K # 0 <-> K =/= 0 ) )
25	24	adantr 276	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> ( K # 0 <-> K =/= 0 ) )
26	19, 25	mpbird 167	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K # 0 )
27	16, 18, 13, 26	mulcanap2d 8689	. . . . 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 190	. . . 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 144	. . 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 11967	. 2 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( M x. K ) || ( N x. K ) -> M || N ) )
31		dvdsmulc 11984	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
32	31	3adant3r 1237	. 2 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
33	30, 32	impbid 129	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 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   =/= wne 2367   class class class wbr 4033  (class class class)co 5922   CCcc 7877   0cc0 7879   x. cmul 7884   # cap 8608   ZZcz 9326   || cdvds 11952

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-inn 8991  df-n0 9250  df-z 9327  df-dvds 11953

This theorem is referenced by:  mulgcddvds  12262  prmpwdvds  12524  4sqlem10  12556