Theorem dvdsmulcr 11812

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 997	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> M e. ZZ )
2		simp3l 1025	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> K e. ZZ )
3	1, 2	zmulcld 9370	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( M x. K ) e. ZZ )
4		simp2 998	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> N e. ZZ )
5	4, 2	zmulcld 9370	. . . 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 994	. . 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 9365	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> x e. CC )
10	1	zcnd 9365	. . . . . . . 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 9365	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K e. CC )
14	9, 11, 13	mulassd 7971	. . . . . 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 2186	. . . . 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 7968	. . . . . 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 9365	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> N e. CC )
19		simpl3r 1053	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K =/= 0 )
20		0z 9253	. . . . . . . . . . 11 |- 0 e. ZZ
21		zapne 9316	. . . . . . . . . . 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 1020	. . . . . . . 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 8608	. . . . 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 11793	. 2 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( M x. K ) || ( N x. K ) -> M || N ) )
31		dvdsmulc 11810	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
32	31	3adant3r 1235	. 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 978   = wceq 1353   e. wcel 2148   =/= wne 2347   class class class wbr 4000  (class class class)co 5869   CCcc 7800   0cc0 7802   x. cmul 7807   # cap 8528   ZZcz 9242   || cdvds 11778

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-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920

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-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-inn 8909  df-n0 9166  df-z 9243  df-dvds 11779

This theorem is referenced by:  mulgcddvds  12077  prmpwdvds  12336  4sqlem10  12368