Theorem gcddiv 11948

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 9206	. . . . . . 7 |- ( C e. NN -> C e. ZZ )
2	1	3ad2ant3 1010	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> C e. ZZ )
3		simp1 987	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> A e. ZZ )
4		divides 11725	. . . . . 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 988	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> B e. ZZ )
7		divides 11725	. . . . . 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 465	. . . 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 2634	. . . 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 11895	. . . . . . . . . . . 12 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a gcd b ) e. NN0 )
13	12	nn0cnd 9165	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a gcd b ) e. CC )
14	13	3adant3 1007	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( a gcd b ) e. CC )
15		nncn 8861	. . . . . . . . . . 11 |- ( C e. NN -> C e. CC )
16	15	3ad2ant3 1010	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> C e. CC )
17		simp3 989	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> C e. NN )
18	17	nnap0d 8899	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> C # 0 )
19	14, 16, 18	divcanap4d 8688	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( ( a gcd b ) x. C ) / C ) = ( a gcd b ) )
20		nnnn0 9117	. . . . . . . . . . 11 |- ( C e. NN -> C e. NN0 )
21		mulgcdr 11947	. . . . . . . . . . 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 1263	. . . . . . . . . 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 5856	. . . . . . . . 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 9192	. . . . . . . . . . . 12 |- ( a e. ZZ -> a e. CC )
25	24	3ad2ant1 1008	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> a e. CC )
26	25, 16, 18	divcanap4d 8688	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( a x. C ) / C ) = a )
27		zcn 9192	. . . . . . . . . . . 12 |- ( b e. ZZ -> b e. CC )
28	27	3ad2ant2 1009	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> b e. CC )
29	28, 16, 18	divcanap4d 8688	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( b x. C ) / C ) = b )
30	26, 29	oveq12d 5859	. . . . . . . . 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 2208	. . . . . . . 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 5850	. . . . . . . . . 10 |- ( ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( a x. C ) gcd ( b x. C ) ) = ( A gcd B ) )
33	32	oveq1d 5856	. . . . . . . . 9 |- ( ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( ( a x. C ) gcd ( b x. C ) ) / C ) = ( ( A gcd B ) / C ) )
34		oveq1 5848	. . . . . . . . . 10 |- ( ( a x. C ) = A -> ( ( a x. C ) / C ) = ( A / C ) )
35		oveq1 5848	. . . . . . . . . 10 |- ( ( b x. C ) = B -> ( ( b x. C ) / C ) = ( B / C ) )
36	34, 35	oveqan12d 5860	. . . . . . . . 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 2180	. . . . . . . 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 1193	. . . . . 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 2589	. . . 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 1010	. . 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 968   = wceq 1343   e. wcel 2136   E.wrex 2444   class class class wbr 3981  (class class class)co 5841   CCcc 7747   x. cmul 7754   / cdiv 8564   NNcn 8853   NN0cn0 9110   ZZcz 9187   || cdvds 11723   gcd cgcd 11871

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868  ax-caucvg 7869

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-frec 6355  df-sup 6945  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-3 8913  df-4 8914  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-fz 9941  df-fzo 10074  df-fl 10201  df-mod 10254  df-seqfrec 10377  df-exp 10451  df-cj 10780  df-re 10781  df-im 10782  df-rsqrt 10936  df-abs 10937  df-dvds 11724  df-gcd 11872

This theorem is referenced by:  sqgcd  11958  divgcdodd  12071  divnumden  12124  hashgcdlem  12166  pythagtriplem19  12210