Theorem dvdsgcd 12704

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 12703	. . 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 1042	. 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 12506	. . . . . . . . 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 1227	. . . . . . . 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 1237	. . . . . . 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 1052	. . . . . . . . 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 1055	. . . . . . . . 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 9581	. . . . . . . . . 10 |- ( x e. ZZ -> x e. CC )
9		zcn 9581	. . . . . . . . . 10 |- ( M e. ZZ -> M e. CC )
10		mulcom 8255	. . . . . . . . . 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 1053	. . . . . . . . 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 1056	. . . . . . . . 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 9581	. . . . . . . . . 10 |- ( y e. ZZ -> y e. CC )
16		zcn 9581	. . . . . . . . . 10 |- ( N e. ZZ -> N e. CC )
17		mulcom 8255	. . . . . . . . . 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 6067	. . . . . . 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 4134	. . . . . 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 4112	. . . . . 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 1232	. . . 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 2667	. . 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 1005   = wceq 1398   e. wcel 2203   E.wrex 2521   class class class wbr 4108  (class class class)co 6049   CCcc 8124   + caddc 8129   x. cmul 8131   ZZcz 9576   || cdvds 12469   gcd cgcd 12645

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-sup 7274  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-fz 10342  df-fzo 10476  df-fl 10629  df-mod 10684  df-seqfrec 10809  df-exp 10900  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-dvds 12470  df-gcd 12646

This theorem is referenced by:  dvdsgcdb  12705  dfgcd2  12706  mulgcd  12708  ncoprmgcdne1b  12782  mulgcddvds  12787  rpmulgcd2  12788  rpexp  12846  pythagtriplem4  12962  pcgcd1  13022  pockthlem  13050  lgsne0  15903  lgsquad2lem2  15947