Theorem dvdsmulcr 12165

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 1000	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> M e. ZZ )
2		simp3l 1028	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> K e. ZZ )
3	1, 2	zmulcld 9503	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( M x. K ) e. ZZ )
4		simp2 1001	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> N e. ZZ )
5	4, 2	zmulcld 9503	. . . 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 997	. . 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 9498	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> x e. CC )
10	1	zcnd 9498	. . . . . . . 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 9498	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K e. CC )
14	9, 11, 13	mulassd 8098	. . . . . 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 2214	. . . . 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 8095	. . . . . 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 9498	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> N e. CC )
19		simpl3r 1056	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) /\ x e. ZZ ) -> K =/= 0 )
20		0z 9385	. . . . . . . . . . 11 |- 0 e. ZZ
21		zapne 9449	. . . . . . . . . . 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 1023	. . . . . . . 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 8737	. . . . 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 12146	. 2 |- ( ( M e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( M x. K ) || ( N x. K ) -> M || N ) )
31		dvdsmulc 12163	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
32	31	3adant3r 1238	. 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 981   = wceq 1373   e. wcel 2176   =/= wne 2376   class class class wbr 4045  (class class class)co 5946   CCcc 7925   0cc0 7927   x. cmul 7932   # cap 8656   ZZcz 9374   || cdvds 12131

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-id 4341  df-po 4344  df-iso 4345  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-inn 9039  df-n0 9298  df-z 9375  df-dvds 12132

This theorem is referenced by:  mulgcddvds  12449  prmpwdvds  12711  4sqlem10  12743