Theorem gcddiv 12535

Description: Division law for GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
gcddiv	|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ ( C || A /\ C || B ) ) -> ( ( A gcd B ) / C ) = ( ( A / C ) gcd ( B / C ) ) )

Proof of Theorem gcddiv

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnz 9461	. . . . . . 7 |- ( C e. NN -> C e. ZZ )
2	1	3ad2ant3 1044	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> C e. ZZ )
3		simp1 1021	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> A e. ZZ )
4		divides 12295	. . . . . 6 |- ( ( C e. ZZ /\ A e. ZZ ) -> ( C || A <-> E. a e. ZZ ( a x. C ) = A ) )
5	2, 3, 4	syl2anc 411	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( C || A <-> E. a e. ZZ ( a x. C ) = A ) )
6		simp2 1022	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> B e. ZZ )
7		divides 12295	. . . . . 6 |- ( ( C e. ZZ /\ B e. ZZ ) -> ( C || B <-> E. b e. ZZ ( b x. C ) = B ) )
8	2, 6, 7	syl2anc 411	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( C || B <-> E. b e. ZZ ( b x. C ) = B ) )
9	5, 8	anbi12d 473	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( ( C || A /\ C || B ) <-> ( E. a e. ZZ ( a x. C ) = A /\ E. b e. ZZ ( b x. C ) = B ) ) )
10		reeanv 2701	. . . 4 |- ( E. a e. ZZ E. b e. ZZ ( ( a x. C ) = A /\ ( b x. C ) = B ) <-> ( E. a e. ZZ ( a x. C ) = A /\ E. b e. ZZ ( b x. C ) = B ) )
11	9, 10	bitr4di 198	. . 3 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( ( C || A /\ C || B ) <-> E. a e. ZZ E. b e. ZZ ( ( a x. C ) = A /\ ( b x. C ) = B ) ) )
12		gcdcl 12482	. . . . . . . . . . . 12 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a gcd b ) e. NN0 )
13	12	nn0cnd 9420	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a gcd b ) e. CC )
14	13	3adant3 1041	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( a gcd b ) e. CC )
15		nncn 9114	. . . . . . . . . . 11 |- ( C e. NN -> C e. CC )
16	15	3ad2ant3 1044	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> C e. CC )
17		simp3 1023	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> C e. NN )
18	17	nnap0d 9152	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> C # 0 )
19	14, 16, 18	divcanap4d 8939	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( ( a gcd b ) x. C ) / C ) = ( a gcd b ) )
20		nnnn0 9372	. . . . . . . . . . 11 |- ( C e. NN -> C e. NN0 )
21		mulgcdr 12534	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN0 ) -> ( ( a x. C ) gcd ( b x. C ) ) = ( ( a gcd b ) x. C ) )
22	20, 21	syl3an3 1306	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( a x. C ) gcd ( b x. C ) ) = ( ( a gcd b ) x. C ) )
23	22	oveq1d 6015	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( ( a x. C ) gcd ( b x. C ) ) / C ) = ( ( ( a gcd b ) x. C ) / C ) )
24		zcn 9447	. . . . . . . . . . . 12 |- ( a e. ZZ -> a e. CC )
25	24	3ad2ant1 1042	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> a e. CC )
26	25, 16, 18	divcanap4d 8939	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( a x. C ) / C ) = a )
27		zcn 9447	. . . . . . . . . . . 12 |- ( b e. ZZ -> b e. CC )
28	27	3ad2ant2 1043	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> b e. CC )
29	28, 16, 18	divcanap4d 8939	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( b x. C ) / C ) = b )
30	26, 29	oveq12d 6018	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( ( a x. C ) / C ) gcd ( ( b x. C ) / C ) ) = ( a gcd b ) )
31	19, 23, 30	3eqtr4d 2272	. . . . . . . 8 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( ( a x. C ) gcd ( b x. C ) ) / C ) = ( ( ( a x. C ) / C ) gcd ( ( b x. C ) / C ) ) )
32		oveq12 6009	. . . . . . . . . 10 |- ( ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( a x. C ) gcd ( b x. C ) ) = ( A gcd B ) )
33	32	oveq1d 6015	. . . . . . . . 9 |- ( ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( ( a x. C ) gcd ( b x. C ) ) / C ) = ( ( A gcd B ) / C ) )
34		oveq1 6007	. . . . . . . . . 10 |- ( ( a x. C ) = A -> ( ( a x. C ) / C ) = ( A / C ) )
35		oveq1 6007	. . . . . . . . . 10 |- ( ( b x. C ) = B -> ( ( b x. C ) / C ) = ( B / C ) )
36	34, 35	oveqan12d 6019	. . . . . . . . 9 |- ( ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( ( a x. C ) / C ) gcd ( ( b x. C ) / C ) ) = ( ( A / C ) gcd ( B / C ) ) )
37	33, 36	eqeq12d 2244	. . . . . . . 8 |- ( ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( ( ( a x. C ) gcd ( b x. C ) ) / C ) = ( ( ( a x. C ) / C ) gcd ( ( b x. C ) / C ) ) <-> ( ( A gcd B ) / C ) = ( ( A / C ) gcd ( B / C ) ) ) )
38	31, 37	syl5ibcom 155	. . . . . . 7 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( A gcd B ) / C ) = ( ( A / C ) gcd ( B / C ) ) ) )
39	38	3expa 1227	. . . . . 6 |- ( ( ( a e. ZZ /\ b e. ZZ ) /\ C e. NN ) -> ( ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( A gcd B ) / C ) = ( ( A / C ) gcd ( B / C ) ) ) )
40	39	expcom 116	. . . . 5 |- ( C e. NN -> ( ( a e. ZZ /\ b e. ZZ ) -> ( ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( A gcd B ) / C ) = ( ( A / C ) gcd ( B / C ) ) ) ) )
41	40	rexlimdvv 2655	. . . 4 |- ( C e. NN -> ( E. a e. ZZ E. b e. ZZ ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( A gcd B ) / C ) = ( ( A / C ) gcd ( B / C ) ) ) )
42	41	3ad2ant3 1044	. . 3 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( E. a e. ZZ E. b e. ZZ ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( A gcd B ) / C ) = ( ( A / C ) gcd ( B / C ) ) ) )
43	11, 42	sylbid 150	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( ( C || A /\ C || B ) -> ( ( A gcd B ) / C ) = ( ( A / C ) gcd ( B / C ) ) ) )
44	43	imp 124	1 |- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ ( C || A /\ C || B ) ) -> ( ( A gcd B ) / C ) = ( ( A / C ) gcd ( B / C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4082  (class class class)co 6000   CCcc 7993   x. cmul 8000   / cdiv 8815   NNcn 9106   NN0cn0 9365   ZZcz 9442   || cdvds 12293   gcd cgcd 12469

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 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

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 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-sup 7147  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-fzo 10335  df-fl 10485  df-mod 10540  df-seqfrec 10665  df-exp 10756  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-dvds 12294  df-gcd 12470

This theorem is referenced by:  sqgcd  12545  divgcdodd  12660  divnumden  12713  hashgcdlem  12755  pythagtriplem19  12800