Theorem dvdsgcd 12015

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 12014	. . 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 1015	. 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 11833	. . . . . . . . 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 1200	. . . . . . . 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 1210	. . . . . . 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 1025	. . . . . . . . 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 1028	. . . . . . . . 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 9260	. . . . . . . . . 10 |- ( x e. ZZ -> x e. CC )
9		zcn 9260	. . . . . . . . . 10 |- ( M e. ZZ -> M e. CC )
10		mulcom 7942	. . . . . . . . . 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 1026	. . . . . . . . 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 1029	. . . . . . . . 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 9260	. . . . . . . . . 10 |- ( y e. ZZ -> y e. CC )
16		zcn 9260	. . . . . . . . . 10 |- ( N e. ZZ -> N e. CC )
17		mulcom 7942	. . . . . . . . . 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 5895	. . . . . . 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 4031	. . . . . 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 4009	. . . . . 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 1205	. . . 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 2601	. . 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 978   = wceq 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4005  (class class class)co 5877   CCcc 7811   + caddc 7816   x. cmul 7818   ZZcz 9255   || cdvds 11796   gcd cgcd 11945

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-sup 6985  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797  df-gcd 11946

This theorem is referenced by:  dvdsgcdb  12016  dfgcd2  12017  mulgcd  12019  ncoprmgcdne1b  12091  mulgcddvds  12096  rpmulgcd2  12097  rpexp  12155  pythagtriplem4  12270  pcgcd1  12329  pockthlem  12356  lgsne0  14524