Theorem mulgcd 12153

Description: Distribute multiplication by a nonnegative integer over gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 30-May-2014.)

Assertion

Ref	Expression
mulgcd	|- ( ( K e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) )

Proof of Theorem mulgcd

Step	Hyp	Ref	Expression
1		elnn0 9242	. . 3 |- ( K e. NN0 <-> ( K e. NN \/ K = 0 ) )
2		simp1 999	. . . . . . . . 9 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. NN )
3	2	nnzd 9438	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
4		simp2 1000	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
5	3, 4	zmulcld 9445	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. ZZ )
6		simp3 1001	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
7	3, 6	zmulcld 9445	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
8	5, 7	gcdcld 12105	. . . . . 6 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) e. NN0 )
9	2	nnnn0d 9293	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. NN0 )
10		gcdcl 12103	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
11	10	3adant1 1017	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
12	9, 11	nn0mulcld 9298	. . . . . 6 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) e. NN0 )
13	8	nn0cnd 9295	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) e. CC )
14	2	nncnd 8996	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. CC )
15	2	nnap0d 9028	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K # 0 )
16	13, 14, 15	divcanap2d 8811	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( ( ( K x. M ) gcd ( K x. N ) ) / K ) ) = ( ( K x. M ) gcd ( K x. N ) ) )
17		gcddvds 12100	. . . . . . . . . . . . 13 |- ( ( ( K x. M ) e. ZZ /\ ( K x. N ) e. ZZ ) -> ( ( ( K x. M ) gcd ( K x. N ) ) || ( K x. M ) /\ ( ( K x. M ) gcd ( K x. N ) ) || ( K x. N ) ) )
18	5, 7, 17	syl2anc 411	. . . . . . . . . . . 12 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( K x. M ) gcd ( K x. N ) ) || ( K x. M ) /\ ( ( K x. M ) gcd ( K x. N ) ) || ( K x. N ) ) )
19	18	simpld 112	. . . . . . . . . . 11 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) || ( K x. M ) )
20	16, 19	eqbrtrd 4051	. . . . . . . . . 10 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( ( ( K x. M ) gcd ( K x. N ) ) / K ) ) || ( K x. M ) )
21		dvdsmul1 11956	. . . . . . . . . . . . . 14 |- ( ( K e. ZZ /\ M e. ZZ ) -> K || ( K x. M ) )
22	3, 4, 21	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K || ( K x. M ) )
23		dvdsmul1 11956	. . . . . . . . . . . . . 14 |- ( ( K e. ZZ /\ N e. ZZ ) -> K || ( K x. N ) )
24	3, 6, 23	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K || ( K x. N ) )
25		dvdsgcd 12149	. . . . . . . . . . . . . 14 |- ( ( K e. ZZ /\ ( K x. M ) e. ZZ /\ ( K x. N ) e. ZZ ) -> ( ( K || ( K x. M ) /\ K || ( K x. N ) ) -> K || ( ( K x. M ) gcd ( K x. N ) ) ) )
26	3, 5, 7, 25	syl3anc 1249	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || ( K x. M ) /\ K || ( K x. N ) ) -> K || ( ( K x. M ) gcd ( K x. N ) ) ) )
27	22, 24, 26	mp2and 433	. . . . . . . . . . . 12 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K || ( ( K x. M ) gcd ( K x. N ) ) )
28	2	nnne0d 9027	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K =/= 0 )
29	8	nn0zd 9437	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) e. ZZ )
30		dvdsval2 11933	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ K =/= 0 /\ ( ( K x. M ) gcd ( K x. N ) ) e. ZZ ) -> ( K || ( ( K x. M ) gcd ( K x. N ) ) <-> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) e. ZZ ) )
31	3, 28, 29, 30	syl3anc 1249	. . . . . . . . . . . 12 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K || ( ( K x. M ) gcd ( K x. N ) ) <-> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) e. ZZ ) )
32	27, 31	mpbid 147	. . . . . . . . . . 11 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) e. ZZ )
33		dvdscmulr 11963	. . . . . . . . . . 11 |- ( ( ( ( ( K x. M ) gcd ( K x. N ) ) / K ) e. ZZ /\ M e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( K x. ( ( ( K x. M ) gcd ( K x. N ) ) / K ) ) || ( K x. M ) <-> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || M ) )
34	32, 4, 3, 28, 33	syl112anc 1253	. . . . . . . . . 10 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. ( ( ( K x. M ) gcd ( K x. N ) ) / K ) ) || ( K x. M ) <-> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || M ) )
35	20, 34	mpbid 147	. . . . . . . . 9 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || M )
36	18	simprd 114	. . . . . . . . . . 11 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) || ( K x. N ) )
37	16, 36	eqbrtrd 4051	. . . . . . . . . 10 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( ( ( K x. M ) gcd ( K x. N ) ) / K ) ) || ( K x. N ) )
38		dvdscmulr 11963	. . . . . . . . . . 11 |- ( ( ( ( ( K x. M ) gcd ( K x. N ) ) / K ) e. ZZ /\ N e. ZZ /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( K x. ( ( ( K x. M ) gcd ( K x. N ) ) / K ) ) || ( K x. N ) <-> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || N ) )
39	32, 6, 3, 28, 38	syl112anc 1253	. . . . . . . . . 10 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. ( ( ( K x. M ) gcd ( K x. N ) ) / K ) ) || ( K x. N ) <-> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || N ) )
40	37, 39	mpbid 147	. . . . . . . . 9 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || N )
41		dvdsgcd 12149	. . . . . . . . . 10 |- ( ( ( ( ( K x. M ) gcd ( K x. N ) ) / K ) e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || M /\ ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || N ) -> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || ( M gcd N ) ) )
42	32, 4, 6, 41	syl3anc 1249	. . . . . . . . 9 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || M /\ ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || N ) -> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || ( M gcd N ) ) )
43	35, 40, 42	mp2and 433	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || ( M gcd N ) )
44	11	nn0zd 9437	. . . . . . . . 9 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. ZZ )
45		dvdscmul 11961	. . . . . . . . 9 |- ( ( ( ( ( K x. M ) gcd ( K x. N ) ) / K ) e. ZZ /\ ( M gcd N ) e. ZZ /\ K e. ZZ ) -> ( ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || ( M gcd N ) -> ( K x. ( ( ( K x. M ) gcd ( K x. N ) ) / K ) ) || ( K x. ( M gcd N ) ) ) )
46	32, 44, 3, 45	syl3anc 1249	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || ( M gcd N ) -> ( K x. ( ( ( K x. M ) gcd ( K x. N ) ) / K ) ) || ( K x. ( M gcd N ) ) ) )
47	43, 46	mpd 13	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( ( ( K x. M ) gcd ( K x. N ) ) / K ) ) || ( K x. ( M gcd N ) ) )
48	16, 47	eqbrtrrd 4053	. . . . . 6 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) || ( K x. ( M gcd N ) ) )
49		gcddvds 12100	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
50	49	3adant1 1017	. . . . . . . . 9 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
51	50	simpld 112	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) || M )
52		dvdscmul 11961	. . . . . . . . 9 |- ( ( ( M gcd N ) e. ZZ /\ M e. ZZ /\ K e. ZZ ) -> ( ( M gcd N ) || M -> ( K x. ( M gcd N ) ) || ( K x. M ) ) )
53	44, 4, 3, 52	syl3anc 1249	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M -> ( K x. ( M gcd N ) ) || ( K x. M ) ) )
54	51, 53	mpd 13	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) || ( K x. M ) )
55	50	simprd 114	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) || N )
56		dvdscmul 11961	. . . . . . . . 9 |- ( ( ( M gcd N ) e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( ( M gcd N ) || N -> ( K x. ( M gcd N ) ) || ( K x. N ) ) )
57	44, 6, 3, 56	syl3anc 1249	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || N -> ( K x. ( M gcd N ) ) || ( K x. N ) ) )
58	55, 57	mpd 13	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) || ( K x. N ) )
59	12	nn0zd 9437	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) e. ZZ )
60		dvdsgcd 12149	. . . . . . . 8 |- ( ( ( K x. ( M gcd N ) ) e. ZZ /\ ( K x. M ) e. ZZ /\ ( K x. N ) e. ZZ ) -> ( ( ( K x. ( M gcd N ) ) || ( K x. M ) /\ ( K x. ( M gcd N ) ) || ( K x. N ) ) -> ( K x. ( M gcd N ) ) || ( ( K x. M ) gcd ( K x. N ) ) ) )
61	59, 5, 7, 60	syl3anc 1249	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( K x. ( M gcd N ) ) || ( K x. M ) /\ ( K x. ( M gcd N ) ) || ( K x. N ) ) -> ( K x. ( M gcd N ) ) || ( ( K x. M ) gcd ( K x. N ) ) ) )
62	54, 58, 61	mp2and 433	. . . . . 6 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) || ( ( K x. M ) gcd ( K x. N ) ) )
63		dvdseq 11990	. . . . . 6 |- ( ( ( ( ( K x. M ) gcd ( K x. N ) ) e. NN0 /\ ( K x. ( M gcd N ) ) e. NN0 ) /\ ( ( ( K x. M ) gcd ( K x. N ) ) || ( K x. ( M gcd N ) ) /\ ( K x. ( M gcd N ) ) || ( ( K x. M ) gcd ( K x. N ) ) ) ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) )
64	8, 12, 48, 62, 63	syl22anc 1250	. . . . 5 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) )
65	64	3expib 1208	. . . 4 |- ( K e. NN -> ( ( M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) ) )
66		gcd0val 12097	. . . . . . 7 |- ( 0 gcd 0 ) = 0
67	10	3adant1 1017	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
68	67	nn0cnd 9295	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. CC )
69	68	mul02d 8411	. . . . . . 7 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. ( M gcd N ) ) = 0 )
70	66, 69	eqtr4id 2245	. . . . . 6 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 gcd 0 ) = ( 0 x. ( M gcd N ) ) )
71		simp1 999	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> K = 0 )
72	71	oveq1d 5933	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) = ( 0 x. M ) )
73		zcn 9322	. . . . . . . . . 10 |- ( M e. ZZ -> M e. CC )
74	73	3ad2ant2 1021	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
75	74	mul02d 8411	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. M ) = 0 )
76	72, 75	eqtrd 2226	. . . . . . 7 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) = 0 )
77	71	oveq1d 5933	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. N ) = ( 0 x. N ) )
78		zcn 9322	. . . . . . . . . 10 |- ( N e. ZZ -> N e. CC )
79	78	3ad2ant3 1022	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> N e. CC )
80	79	mul02d 8411	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. N ) = 0 )
81	77, 80	eqtrd 2226	. . . . . . 7 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. N ) = 0 )
82	76, 81	oveq12d 5936	. . . . . 6 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( 0 gcd 0 ) )
83	71	oveq1d 5933	. . . . . 6 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) = ( 0 x. ( M gcd N ) ) )
84	70, 82, 83	3eqtr4d 2236	. . . . 5 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) )
85	84	3expib 1208	. . . 4 |- ( K = 0 -> ( ( M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) ) )
86	65, 85	jaoi 717	. . 3 |- ( ( K e. NN \/ K = 0 ) -> ( ( M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) ) )
87	1, 86	sylbi 121	. 2 |- ( K e. NN0 -> ( ( M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) ) )
88	87	3impib 1203	1 |- ( ( K e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2164   =/= wne 2364   class class class wbr 4029  (class class class)co 5918   CCcc 7870   0cc0 7872   x. cmul 7877   / cdiv 8691   NNcn 8982   NN0cn0 9240   ZZcz 9317   || cdvds 11930   gcd cgcd 12079

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080

This theorem is referenced by:  absmulgcd  12154  mulgcdr  12155  mulgcddvds  12232  qredeu  12235  coprimeprodsq  12395  pythagtriplem4  12406  2sqlem8  15210