Theorem dvdstr 11530

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 978	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ M e. ZZ ) )
2		3simpc 980	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
3		3simpb 979	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ N e. ZZ ) )
4		zmulcl 9107	. . 3 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( x x. y ) e. ZZ )
5	4	adantl 275	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x x. y ) e. ZZ )
6		oveq2 5782	. . . . 5 |- ( ( x x. K ) = M -> ( y x. ( x x. K ) ) = ( y x. M ) )
7	6	adantr 274	. . . 4 |- ( ( ( x x. K ) = M /\ ( y x. M ) = N ) -> ( y x. ( x x. K ) ) = ( y x. M ) )
8		eqeq2 2149	. . . . 5 |- ( ( y x. M ) = N -> ( ( y x. ( x x. K ) ) = ( y x. M ) <-> ( y x. ( x x. K ) ) = N ) )
9	8	adantl 275	. . . 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 146	. . 3 |- ( ( ( x x. K ) = M /\ ( y x. M ) = N ) -> ( y x. ( x x. K ) ) = N )
11		zcn 9059	. . . . . . . 8 |- ( x e. ZZ -> x e. CC )
12		zcn 9059	. . . . . . . 8 |- ( y e. ZZ -> y e. CC )
13		zcn 9059	. . . . . . . 8 |- ( K e. ZZ -> K e. CC )
14		mulass 7751	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC /\ K e. CC ) -> ( ( x x. y ) x. K ) = ( x x. ( y x. K ) ) )
15		mul12 7891	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC /\ K e. CC ) -> ( x x. ( y x. K ) ) = ( y x. ( x x. K ) ) )
16	14, 15	eqtrd 2172	. . . . . . . 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 1258	. . . . . . 7 |- ( ( x e. ZZ /\ y e. ZZ /\ K e. ZZ ) -> ( ( x x. y ) x. K ) = ( y x. ( x x. K ) ) )
18	17	3comr 1189	. . . . . 6 |- ( ( K e. ZZ /\ x e. ZZ /\ y e. ZZ ) -> ( ( x x. y ) x. K ) = ( y x. ( x x. K ) ) )
19	18	3expb 1182	. . . . 5 |- ( ( K e. ZZ /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x x. y ) x. K ) = ( y x. ( x x. K ) ) )
20	19	3ad2antl1 1143	. . . 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 2148	. . 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	syl5ibr 155	. 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 11505	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ M || N ) -> K || N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7618   x. cmul 7625   ZZcz 9054   || cdvds 11493

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-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-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731

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-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-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-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-dvds 11494

This theorem is referenced by:  dvdsmultr1  11531  dvdsmultr2  11533  4dvdseven  11614  dvdsgcdb  11701  dvdsmulgcd  11713  gcddvdslcm  11754  lcmgcdeq  11764  lcmdvdsb  11765  mulgcddvds  11775  rpmulgcd2  11776  rpdvds  11780  exprmfct  11818  rpexp  11831  phimullem  11901