Theorem dvdsgcd 12519

Description: An integer which divides each of two others also divides their gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 30-May-2014.)

Assertion

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

Proof of Theorem dvdsgcd

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

Step	Hyp	Ref	Expression
1		bezout 12518	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> E. x e. ZZ E. y e. ZZ ( M gcd N ) = ( ( M x. x ) + ( N x. y ) ) )
2	1	3adant1 1039	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> E. x e. ZZ E. y e. ZZ ( M gcd N ) = ( ( M x. x ) + ( N x. y ) ) )
3		dvds2ln 12321	. . . . . . . . 9 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) ) -> ( ( K || M /\ K || N ) -> K || ( ( x x. M ) + ( y x. N ) ) ) )
4	3	3impia 1224	. . . . . . . 8 |- ( ( ( x e. ZZ /\ y e. ZZ ) /\ ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || M /\ K || N ) ) -> K || ( ( x x. M ) + ( y x. N ) ) )
5	4	3coml 1234	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || M /\ K || N ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> K || ( ( x x. M ) + ( y x. N ) ) )
6		simp3l 1049	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || M /\ K || N ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> x e. ZZ )
7		simp12 1052	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || M /\ K || N ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> M e. ZZ )
8		zcn 9439	. . . . . . . . . 10 |- ( x e. ZZ -> x e. CC )
9		zcn 9439	. . . . . . . . . 10 |- ( M e. ZZ -> M e. CC )
10		mulcom 8116	. . . . . . . . . 10 |- ( ( x e. CC /\ M e. CC ) -> ( x x. M ) = ( M x. x ) )
11	8, 9, 10	syl2an 289	. . . . . . . . 9 |- ( ( x e. ZZ /\ M e. ZZ ) -> ( x x. M ) = ( M x. x ) )
12	6, 7, 11	syl2anc 411	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || M /\ K || N ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x x. M ) = ( M x. x ) )
13		simp3r 1050	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || M /\ K || N ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> y e. ZZ )
14		simp13 1053	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || M /\ K || N ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> N e. ZZ )
15		zcn 9439	. . . . . . . . . 10 |- ( y e. ZZ -> y e. CC )
16		zcn 9439	. . . . . . . . . 10 |- ( N e. ZZ -> N e. CC )
17		mulcom 8116	. . . . . . . . . 10 |- ( ( y e. CC /\ N e. CC ) -> ( y x. N ) = ( N x. y ) )
18	15, 16, 17	syl2an 289	. . . . . . . . 9 |- ( ( y e. ZZ /\ N e. ZZ ) -> ( y x. N ) = ( N x. y ) )
19	13, 14, 18	syl2anc 411	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || M /\ K || N ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( y x. N ) = ( N x. y ) )
20	12, 19	oveq12d 6012	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || M /\ K || N ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x x. M ) + ( y x. N ) ) = ( ( M x. x ) + ( N x. y ) ) )
21	5, 20	breqtrd 4108	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || M /\ K || N ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> K || ( ( M x. x ) + ( N x. y ) ) )
22		breq2 4086	. . . . . 6 |- ( ( M gcd N ) = ( ( M x. x ) + ( N x. y ) ) -> ( K || ( M gcd N ) <-> K || ( ( M x. x ) + ( N x. y ) ) ) )
23	21, 22	syl5ibrcom 157	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || M /\ K || N ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( M gcd N ) = ( ( M x. x ) + ( N x. y ) ) -> K || ( M gcd N ) ) )
24	23	3expia 1229	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || M /\ K || N ) ) -> ( ( x e. ZZ /\ y e. ZZ ) -> ( ( M gcd N ) = ( ( M x. x ) + ( N x. y ) ) -> K || ( M gcd N ) ) ) )
25	24	rexlimdvv 2655	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K || M /\ K || N ) ) -> ( E. x e. ZZ E. y e. ZZ ( M gcd N ) = ( ( M x. x ) + ( N x. y ) ) -> K || ( M gcd N ) ) )
26	25	ex 115	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ K || N ) -> ( E. x e. ZZ E. y e. ZZ ( M gcd N ) = ( ( M x. x ) + ( N x. y ) ) -> K || ( M gcd N ) ) ) )
27	2, 26	mpid 42	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ K || N ) -> K || ( M gcd N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4082  (class class class)co 5994   CCcc 7985   + caddc 7990   x. cmul 7992   ZZcz 9434   || cdvds 12284   gcd cgcd 12460

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106  ax-caucvg 8107

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-sup 7139  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-n0 9358  df-z 9435  df-uz 9711  df-q 9803  df-rp 9838  df-fz 10193  df-fzo 10327  df-fl 10477  df-mod 10532  df-seqfrec 10657  df-exp 10748  df-cj 11339  df-re 11340  df-im 11341  df-rsqrt 11495  df-abs 11496  df-dvds 12285  df-gcd 12461

This theorem is referenced by:  dvdsgcdb  12520  dfgcd2  12521  mulgcd  12523  ncoprmgcdne1b  12597  mulgcddvds  12602  rpmulgcd2  12603  rpexp  12661  pythagtriplem4  12777  pcgcd1  12837  pockthlem  12865  lgsne0  15702  lgsquad2lem2  15746