Theorem dvdstr 12595

Description: The divides relation is transitive. Theorem 1.1(b) in [ApostolNT] p. 14 (transitive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
dvdstr	|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ M || N ) -> K || N ) )

Proof of Theorem dvdstr

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpa 1025	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ M e. ZZ ) )
2		3simpc 1027	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
3		3simpb 1026	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ N e. ZZ ) )
4		zmulcl 9698	. . 3 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( x x. y ) e. ZZ )
5	4	adantl 277	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x x. y ) e. ZZ )
6		oveq2 6093	. . . . 5 |- ( ( x x. K ) = M -> ( y x. ( x x. K ) ) = ( y x. M ) )
7	6	adantr 276	. . . 4 |- ( ( ( x x. K ) = M /\ ( y x. M ) = N ) -> ( y x. ( x x. K ) ) = ( y x. M ) )
8		eqeq2 2248	. . . . 5 |- ( ( y x. M ) = N -> ( ( y x. ( x x. K ) ) = ( y x. M ) <-> ( y x. ( x x. K ) ) = N ) )
9	8	adantl 277	. . . 4 |- ( ( ( x x. K ) = M /\ ( y x. M ) = N ) -> ( ( y x. ( x x. K ) ) = ( y x. M ) <-> ( y x. ( x x. K ) ) = N ) )
10	7, 9	mpbid 147	. . 3 |- ( ( ( x x. K ) = M /\ ( y x. M ) = N ) -> ( y x. ( x x. K ) ) = N )
11		zcn 9649	. . . . . . . 8 |- ( x e. ZZ -> x e. CC )
12		zcn 9649	. . . . . . . 8 |- ( y e. ZZ -> y e. CC )
13		zcn 9649	. . . . . . . 8 |- ( K e. ZZ -> K e. CC )
14		mulass 8310	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC /\ K e. CC ) -> ( ( x x. y ) x. K ) = ( x x. ( y x. K ) ) )
15		mul12 8455	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC /\ K e. CC ) -> ( x x. ( y x. K ) ) = ( y x. ( x x. K ) ) )
16	14, 15	eqtrd 2271	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC /\ K e. CC ) -> ( ( x x. y ) x. K ) = ( y x. ( x x. K ) ) )
17	11, 12, 13, 16	syl3an 1320	. . . . . . 7 |- ( ( x e. ZZ /\ y e. ZZ /\ K e. ZZ ) -> ( ( x x. y ) x. K ) = ( y x. ( x x. K ) ) )
18	17	3comr 1242	. . . . . 6 |- ( ( K e. ZZ /\ x e. ZZ /\ y e. ZZ ) -> ( ( x x. y ) x. K ) = ( y x. ( x x. K ) ) )
19	18	3expb 1235	. . . . 5 |- ( ( K e. ZZ /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x x. y ) x. K ) = ( y x. ( x x. K ) ) )
20	19	3ad2antl1 1190	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x x. y ) x. K ) = ( y x. ( x x. K ) ) )
21	20	eqeq1d 2247	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ( x x. y ) x. K ) = N <-> ( y x. ( x x. K ) ) = N ) )
22	10, 21	imbitrrid 156	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ( x x. K ) = M /\ ( y x. M ) = N ) -> ( ( x x. y ) x. K ) = N ) )
23	1, 2, 3, 5, 22	dvds2lem 12570	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ M || N ) -> K || N ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   CCcc 8177   x. cmul 8184   ZZcz 9644   || cdvds 12554

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-dvds 12555

This theorem is used by:  dvdstrd  12597  dvdsmultr1  12598  dvdsmultr2  12600  4dvdseven  12684  dvdsgcdb  12790  dvdsmulgcd  12802  gcddvdslcm  12851  lcmgcdeq  12861  lcmdvdsb  12862  mulgcddvds  12872  rpmulgcd2  12873  rpdvds  12877  exprmfct  12916  rpexp  12931  phimullem  13003  pcpremul  13072  pcdvdsb  13099  pcprmpw2  13112  mpodvdsmulf1o  16104  lgsquad2lem1  16200