Theorem dvdstr 10377

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 936	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ M e. ZZ ) )
2		3simpc 938	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
3		3simpb 937	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ N e. ZZ ) )
4		zmulcl 8485	. . 3 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( x x. y ) e. ZZ )
5	4	adantl 271	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x x. y ) e. ZZ )
6		oveq2 5551	. . . . 5 |- ( ( x x. K ) = M -> ( y x. ( x x. K ) ) = ( y x. M ) )
7	6	adantr 270	. . . 4 |- ( ( ( x x. K ) = M /\ ( y x. M ) = N ) -> ( y x. ( x x. K ) ) = ( y x. M ) )
8		eqeq2 2091	. . . . 5 |- ( ( y x. M ) = N -> ( ( y x. ( x x. K ) ) = ( y x. M ) <-> ( y x. ( x x. K ) ) = N ) )
9	8	adantl 271	. . . 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 145	. . 3 |- ( ( ( x x. K ) = M /\ ( y x. M ) = N ) -> ( y x. ( x x. K ) ) = N )
11		zcn 8437	. . . . . . . 8 |- ( x e. ZZ -> x e. CC )
12		zcn 8437	. . . . . . . 8 |- ( y e. ZZ -> y e. CC )
13		zcn 8437	. . . . . . . 8 |- ( K e. ZZ -> K e. CC )
14		mulass 7166	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC /\ K e. CC ) -> ( ( x x. y ) x. K ) = ( x x. ( y x. K ) ) )
15		mul12 7304	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC /\ K e. CC ) -> ( x x. ( y x. K ) ) = ( y x. ( x x. K ) ) )
16	14, 15	eqtrd 2114	. . . . . . . 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 1212	. . . . . . 7 |- ( ( x e. ZZ /\ y e. ZZ /\ K e. ZZ ) -> ( ( x x. y ) x. K ) = ( y x. ( x x. K ) ) )
18	17	3comr 1147	. . . . . 6 |- ( ( K e. ZZ /\ x e. ZZ /\ y e. ZZ ) -> ( ( x x. y ) x. K ) = ( y x. ( x x. K ) ) )
19	18	3expb 1140	. . . . 5 |- ( ( K e. ZZ /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x x. y ) x. K ) = ( y x. ( x x. K ) ) )
20	19	3ad2antl1 1101	. . . 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 2090	. . 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 154	. 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 10352	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ M || N ) -> K || N ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 920   = wceq 1285   e. wcel 1434   class class class wbr 3793  (class class class)co 5543   CCcc 7041   x. cmul 7048   ZZcz 8432   || cdvds 10340

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149

This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-dvds 10341

This theorem is referenced by:  dvdsmultr1  10378  dvdsmultr2  10380  4dvdseven  10461  dvdsgcdb  10546  dvdsmulgcd  10558  gcddvdslcm  10599  lcmgcdeq  10609  lcmdvdsb  10610  mulgcddvds  10620  rpmulgcd2  10621  rpdvds  10625  exprmfct  10663  rpexp  10676