Theorem dvdsgcd 12733

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 12732	. . 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 12535	. . . . . . . . 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 9599	. . . . . . . . . 10 |- ( x e. ZZ -> x e. CC )
9		zcn 9599	. . . . . . . . . 10 |- ( M e. ZZ -> M e. CC )
10		mulcom 8272	. . . . . . . . . 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 9599	. . . . . . . . . 10 |- ( y e. ZZ -> y e. CC )
16		zcn 9599	. . . . . . . . . 10 |- ( N e. ZZ -> N e. CC )
17		mulcom 8272	. . . . . . . . . 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 6076	. . . . . . 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 4140	. . . . . 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 4118	. . . . . 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 2669	. . 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 2205   E.wrex 2523   class class class wbr 4114  (class class class)co 6058   CCcc 8141   + caddc 8146   x. cmul 8148   ZZcz 9594   || cdvds 12498   gcd cgcd 12674

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-gcd 12675

This theorem is referenced by:  dvdsgcdb  12734  dfgcd2  12735  mulgcd  12737  ncoprmgcdne1b  12811  mulgcddvds  12816  rpmulgcd2  12817  rpexp  12875  pythagtriplem4  12991  pcgcd1  13051  pockthlem  13079  lgsne0  16023  lgsquad2lem2  16067