Theorem dvdstr 12139

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 997	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ M e. ZZ ) )
2		3simpc 999	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
3		3simpb 998	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ZZ /\ N e. ZZ ) )
4		zmulcl 9426	. . 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 5952	. . . . 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 2215	. . . . 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 9377	. . . . . . . 8 |- ( x e. ZZ -> x e. CC )
12		zcn 9377	. . . . . . . 8 |- ( y e. ZZ -> y e. CC )
13		zcn 9377	. . . . . . . 8 |- ( K e. ZZ -> K e. CC )
14		mulass 8056	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC /\ K e. CC ) -> ( ( x x. y ) x. K ) = ( x x. ( y x. K ) ) )
15		mul12 8201	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC /\ K e. CC ) -> ( x x. ( y x. K ) ) = ( y x. ( x x. K ) ) )
16	14, 15	eqtrd 2238	. . . . . . . 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 1292	. . . . . . 7 |- ( ( x e. ZZ /\ y e. ZZ /\ K e. ZZ ) -> ( ( x x. y ) x. K ) = ( y x. ( x x. K ) ) )
18	17	3comr 1214	. . . . . 6 |- ( ( K e. ZZ /\ x e. ZZ /\ y e. ZZ ) -> ( ( x x. y ) x. K ) = ( y x. ( x x. K ) ) )
19	18	3expb 1207	. . . . 5 |- ( ( K e. ZZ /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x x. y ) x. K ) = ( y x. ( x x. K ) ) )
20	19	3ad2antl1 1162	. . . 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 2214	. . 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 12114	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ M || N ) -> K || N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   class class class wbr 4044  (class class class)co 5944   CCcc 7923   x. cmul 7930   ZZcz 9372   || cdvds 12098

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-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036

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-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 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-sub 8245  df-neg 8246  df-inn 9037  df-n0 9296  df-z 9373  df-dvds 12099

This theorem is referenced by:  dvdstrd  12141  dvdsmultr1  12142  dvdsmultr2  12144  4dvdseven  12228  dvdsgcdb  12334  dvdsmulgcd  12346  gcddvdslcm  12395  lcmgcdeq  12405  lcmdvdsb  12406  mulgcddvds  12416  rpmulgcd2  12417  rpdvds  12421  exprmfct  12460  rpexp  12475  phimullem  12547  pcpremul  12616  pcdvdsb  12643  pcprmpw2  12656  mpodvdsmulf1o  15462  lgsquad2lem1  15558