Theorem dvdsgcd 12027

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 12026	. . 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 1016	. 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 11845	. . . . . . . . 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 1201	. . . . . . . 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 1211	. . . . . . 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 1026	. . . . . . . . 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 1029	. . . . . . . . 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 9272	. . . . . . . . . 10 |- ( x e. ZZ -> x e. CC )
9		zcn 9272	. . . . . . . . . 10 |- ( M e. ZZ -> M e. CC )
10		mulcom 7954	. . . . . . . . . 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 1027	. . . . . . . . 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 1030	. . . . . . . . 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 9272	. . . . . . . . . 10 |- ( y e. ZZ -> y e. CC )
16		zcn 9272	. . . . . . . . . 10 |- ( N e. ZZ -> N e. CC )
17		mulcom 7954	. . . . . . . . . 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 5906	. . . . . . 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 4041	. . . . . 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 4019	. . . . . 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 1206	. . . 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 2611	. . 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 979   = wceq 1363   e. wcel 2158   E.wrex 2466   class class class wbr 4015  (class class class)co 5888   CCcc 7823   + caddc 7828   x. cmul 7830   ZZcz 9267   || cdvds 11808   gcd cgcd 11957

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-frec 6406  df-sup 6997  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-fz 10023  df-fzo 10157  df-fl 10284  df-mod 10337  df-seqfrec 10460  df-exp 10534  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-dvds 11809  df-gcd 11958

This theorem is referenced by:  dvdsgcdb  12028  dfgcd2  12029  mulgcd  12031  ncoprmgcdne1b  12103  mulgcddvds  12108  rpmulgcd2  12109  rpexp  12167  pythagtriplem4  12282  pcgcd1  12341  pockthlem  12368  lgsne0  14735