Theorem gcddiv 12580

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 9488	. . . . . . 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 12340	. . . . . 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 12340	. . . . . 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 12527	. . . . . . . . . . . 12 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a gcd b ) e. NN0 )
13	12	nn0cnd 9447	. . . . . . . . . . 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 9141	. . . . . . . . . . 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 9179	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> C # 0 )
19	14, 16, 18	divcanap4d 8966	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( ( a gcd b ) x. C ) / C ) = ( a gcd b ) )
20		nnnn0 9399	. . . . . . . . . . 11 |- ( C e. NN -> C e. NN0 )
21		mulgcdr 12579	. . . . . . . . . . 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 6028	. . . . . . . . 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 9474	. . . . . . . . . . . 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 8966	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( a x. C ) / C ) = a )
27		zcn 9474	. . . . . . . . . . . 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 8966	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( b x. C ) / C ) = b )
30	26, 29	oveq12d 6031	. . . . . . . . 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 6022	. . . . . . . . . 10 |- ( ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( a x. C ) gcd ( b x. C ) ) = ( A gcd B ) )
33	32	oveq1d 6028	. . . . . . . . 9 |- ( ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( ( a x. C ) gcd ( b x. C ) ) / C ) = ( ( A gcd B ) / C ) )
34		oveq1 6020	. . . . . . . . . 10 |- ( ( a x. C ) = A -> ( ( a x. C ) / C ) = ( A / C ) )
35		oveq1 6020	. . . . . . . . . 10 |- ( ( b x. C ) = B -> ( ( b x. C ) / C ) = ( B / C ) )
36	34, 35	oveqan12d 6032	. . . . . . . . 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 4086  (class class class)co 6013   CCcc 8020   x. cmul 8027   / cdiv 8842   NNcn 9133   NN0cn0 9392   ZZcz 9469   || cdvds 12338   gcd cgcd 12514

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-sup 7174  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-dvds 12339  df-gcd 12515

This theorem is referenced by:  sqgcd  12590  divgcdodd  12705  divnumden  12758  hashgcdlem  12800  pythagtriplem19  12845