Theorem gcddiv 11974

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 9231	. . . . . . 7 |- ( C e. NN -> C e. ZZ )
2	1	3ad2ant3 1015	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> C e. ZZ )
3		simp1 992	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> A e. ZZ )
4		divides 11751	. . . . . 6 |- ( ( C e. ZZ /\ A e. ZZ ) -> ( C || A <-> E. a e. ZZ ( a x. C ) = A ) )
5	2, 3, 4	syl2anc 409	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( C || A <-> E. a e. ZZ ( a x. C ) = A ) )
6		simp2 993	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> B e. ZZ )
7		divides 11751	. . . . . 6 |- ( ( C e. ZZ /\ B e. ZZ ) -> ( C || B <-> E. b e. ZZ ( b x. C ) = B ) )
8	2, 6, 7	syl2anc 409	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> ( C || B <-> E. b e. ZZ ( b x. C ) = B ) )
9	5, 8	anbi12d 470	. . . 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 2639	. . . 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 197	. . 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 11921	. . . . . . . . . . . 12 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a gcd b ) e. NN0 )
13	12	nn0cnd 9190	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a gcd b ) e. CC )
14	13	3adant3 1012	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( a gcd b ) e. CC )
15		nncn 8886	. . . . . . . . . . 11 |- ( C e. NN -> C e. CC )
16	15	3ad2ant3 1015	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> C e. CC )
17		simp3 994	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> C e. NN )
18	17	nnap0d 8924	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> C # 0 )
19	14, 16, 18	divcanap4d 8713	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( ( a gcd b ) x. C ) / C ) = ( a gcd b ) )
20		nnnn0 9142	. . . . . . . . . . 11 |- ( C e. NN -> C e. NN0 )
21		mulgcdr 11973	. . . . . . . . . . 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 1268	. . . . . . . . . 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 5868	. . . . . . . . 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 9217	. . . . . . . . . . . 12 |- ( a e. ZZ -> a e. CC )
25	24	3ad2ant1 1013	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> a e. CC )
26	25, 16, 18	divcanap4d 8713	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( a x. C ) / C ) = a )
27		zcn 9217	. . . . . . . . . . . 12 |- ( b e. ZZ -> b e. CC )
28	27	3ad2ant2 1014	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> b e. CC )
29	28, 16, 18	divcanap4d 8713	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( b x. C ) / C ) = b )
30	26, 29	oveq12d 5871	. . . . . . . . 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 2213	. . . . . . . 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 5862	. . . . . . . . . 10 |- ( ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( a x. C ) gcd ( b x. C ) ) = ( A gcd B ) )
33	32	oveq1d 5868	. . . . . . . . 9 |- ( ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( ( a x. C ) gcd ( b x. C ) ) / C ) = ( ( A gcd B ) / C ) )
34		oveq1 5860	. . . . . . . . . 10 |- ( ( a x. C ) = A -> ( ( a x. C ) / C ) = ( A / C ) )
35		oveq1 5860	. . . . . . . . . 10 |- ( ( b x. C ) = B -> ( ( b x. C ) / C ) = ( B / C ) )
36	34, 35	oveqan12d 5872	. . . . . . . . 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 2185	. . . . . . . 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 154	. . . . . . 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 1198	. . . . . 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 115	. . . . 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 2594	. . . 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 1015	. . 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 149	. 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 123	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 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   E.wrex 2449   class class class wbr 3989  (class class class)co 5853   CCcc 7772   x. cmul 7779   / cdiv 8589   NNcn 8878   NN0cn0 9135   ZZcz 9212   || cdvds 11749   gcd cgcd 11897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898

This theorem is referenced by:  sqgcd  11984  divgcdodd  12097  divnumden  12150  hashgcdlem  12192  pythagtriplem19  12236