Theorem gcddiv 11918

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 9191	. . . . . . 7 |- ( C e. NN -> C e. ZZ )
2	1	3ad2ant3 1005	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> C e. ZZ )
3		simp1 982	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> A e. ZZ )
4		divides 11696	. . . . . 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 983	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) -> B e. ZZ )
7		divides 11696	. . . . . 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 2626	. . . 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 11865	. . . . . . . . . . . 12 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a gcd b ) e. NN0 )
13	12	nn0cnd 9150	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a gcd b ) e. CC )
14	13	3adant3 1002	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( a gcd b ) e. CC )
15		nncn 8846	. . . . . . . . . . 11 |- ( C e. NN -> C e. CC )
16	15	3ad2ant3 1005	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> C e. CC )
17		simp3 984	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> C e. NN )
18	17	nnap0d 8884	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> C # 0 )
19	14, 16, 18	divcanap4d 8673	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( ( a gcd b ) x. C ) / C ) = ( a gcd b ) )
20		nnnn0 9102	. . . . . . . . . . 11 |- ( C e. NN -> C e. NN0 )
21		mulgcdr 11917	. . . . . . . . . . 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 1255	. . . . . . . . . 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 5841	. . . . . . . . 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 9177	. . . . . . . . . . . 12 |- ( a e. ZZ -> a e. CC )
25	24	3ad2ant1 1003	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> a e. CC )
26	25, 16, 18	divcanap4d 8673	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( a x. C ) / C ) = a )
27		zcn 9177	. . . . . . . . . . . 12 |- ( b e. ZZ -> b e. CC )
28	27	3ad2ant2 1004	. . . . . . . . . . 11 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> b e. CC )
29	28, 16, 18	divcanap4d 8673	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ /\ C e. NN ) -> ( ( b x. C ) / C ) = b )
30	26, 29	oveq12d 5844	. . . . . . . . 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 2200	. . . . . . . 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 5835	. . . . . . . . . 10 |- ( ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( a x. C ) gcd ( b x. C ) ) = ( A gcd B ) )
33	32	oveq1d 5841	. . . . . . . . 9 |- ( ( ( a x. C ) = A /\ ( b x. C ) = B ) -> ( ( ( a x. C ) gcd ( b x. C ) ) / C ) = ( ( A gcd B ) / C ) )
34		oveq1 5833	. . . . . . . . . 10 |- ( ( a x. C ) = A -> ( ( a x. C ) / C ) = ( A / C ) )
35		oveq1 5833	. . . . . . . . . 10 |- ( ( b x. C ) = B -> ( ( b x. C ) / C ) = ( B / C ) )
36	34, 35	oveqan12d 5845	. . . . . . . . 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 2172	. . . . . . . 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 1185	. . . . . 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 2581	. . . 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 1005	. . 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 963   = wceq 1335   e. wcel 2128   E.wrex 2436   class class class wbr 3967  (class class class)co 5826   CCcc 7732   x. cmul 7739   / cdiv 8549   NNcn 8838   NN0cn0 9095   ZZcz 9172   || cdvds 11694   gcd cgcd 11841

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-mulrcl 7833  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-precex 7844  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-apti 7849  ax-pre-ltadd 7850  ax-pre-mulgt0 7851  ax-pre-mulext 7852  ax-arch 7853  ax-caucvg 7854

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-ilim 4331  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-frec 6340  df-sup 6930  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-reap 8454  df-ap 8461  df-div 8550  df-inn 8839  df-2 8897  df-3 8898  df-4 8899  df-n0 9096  df-z 9173  df-uz 9445  df-q 9535  df-rp 9567  df-fz 9919  df-fzo 10051  df-fl 10178  df-mod 10231  df-seqfrec 10354  df-exp 10428  df-cj 10753  df-re 10754  df-im 10755  df-rsqrt 10909  df-abs 10910  df-dvds 11695  df-gcd 11842

This theorem is referenced by:  sqgcd  11928  divgcdodd  12033  divnumden  12086  hashgcdlem  12128  pythagtriplem19  12172