Theorem dvdstr 12620

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 9709	. . 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 6096	. . . . 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 9660	. . . . . . . 8 |- ( x e. ZZ -> x e. CC )
12		zcn 9660	. . . . . . . 8 |- ( y e. ZZ -> y e. CC )
13		zcn 9660	. . . . . . . 8 |- ( K e. ZZ -> K e. CC )
14		mulass 8316	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC /\ K e. CC ) -> ( ( x x. y ) x. K ) = ( x x. ( y x. K ) ) )
15		mul12 8462	. . . . . . . . 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 12595	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 4131  (class class class)co 6088   CCcc 8183   x. cmul 8190   ZZcz 9655   || cdvds 12579

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 4251  ax-pow 4313  ax-pr 4348  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296

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 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-iota 5340  df-fun 5382  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-dvds 12580

This theorem is used by:  dvdstrd  12622  dvdsmultr1  12623  dvdsmultr2  12625  4dvdseven  12709  dvdsgcdb  12815  dvdsmulgcd  12827  gcddvdslcm  12876  lcmgcdeq  12886  lcmdvdsb  12887  mulgcddvds  12897  rpmulgcd2  12898  rpdvds  12902  exprmfct  12942  rpexp  12957  phimullem  13032  pcpremul  13101  pcdvdsb  13128  pcprmpw2  13141  mpodvdsmulf1o  16284  bposlem3  16313  lgsquad2lem1  16405