Theorem mulgcd 12712

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 9498	. . 3 |- ( K e. NN0 <-> ( K e. NN \/ K = 0 ) )
2		simp1 1024	. . . . . . . . 9 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. NN )
3	2	nnzd 9699	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
4		simp2 1025	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
5	3, 4	zmulcld 9706	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. ZZ )
6		simp3 1026	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
7	3, 6	zmulcld 9706	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
8	5, 7	gcdcld 12664	. . . . . 6 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) e. NN0 )
9	2	nnnn0d 9553	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. NN0 )
10		gcdcl 12662	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
11	10	3adant1 1042	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
12	9, 11	nn0mulcld 9558	. . . . . 6 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) e. NN0 )
13	8	nn0cnd 9555	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) e. CC )
14	2	nncnd 9251	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. CC )
15	2	nnap0d 9283	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K # 0 )
16	13, 14, 15	divcanap2d 9066	. . . . . . 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 12659	. . . . . . . . . . . . 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 4131	. . . . . . . . . 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 12499	. . . . . . . . . . . . . 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 12499	. . . . . . . . . . . . . 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 12708	. . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . 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 9282	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K =/= 0 )
29	8	nn0zd 9698	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) e. ZZ )
30		dvdsval2 12476	. . . . . . . . . . . . 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 1274	. . . . . . . . . . . 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 12506	. . . . . . . . . . 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 1278	. . . . . . . . . 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 4131	. . . . . . . . . 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 12506	. . . . . . . . . . 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 1278	. . . . . . . . . 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 12708	. . . . . . . . . 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 1274	. . . . . . . . 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 9698	. . . . . . . . 9 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. ZZ )
45		dvdscmul 12504	. . . . . . . . 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 1274	. . . . . . . 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 4133	. . . . . 6 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) || ( K x. ( M gcd N ) ) )
49		gcddvds 12659	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
50	49	3adant1 1042	. . . . . . . . 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 12504	. . . . . . . . 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 1274	. . . . . . . 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 12504	. . . . . . . . 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 1274	. . . . . . . 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 9698	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) e. ZZ )
60		dvdsgcd 12708	. . . . . . . 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 1274	. . . . . . 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 12534	. . . . . 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 1275	. . . . 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 1233	. . . 4 |- ( K e. NN -> ( ( M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) ) )
66		gcd0val 12656	. . . . . . 7 |- ( 0 gcd 0 ) = 0
67	10	3adant1 1042	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
68	67	nn0cnd 9555	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. CC )
69	68	mul02d 8665	. . . . . . 7 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. ( M gcd N ) ) = 0 )
70	66, 69	eqtr4id 2284	. . . . . 6 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 gcd 0 ) = ( 0 x. ( M gcd N ) ) )
71		simp1 1024	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> K = 0 )
72	71	oveq1d 6065	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) = ( 0 x. M ) )
73		zcn 9582	. . . . . . . . . 10 |- ( M e. ZZ -> M e. CC )
74	73	3ad2ant2 1046	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
75	74	mul02d 8665	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. M ) = 0 )
76	72, 75	eqtrd 2265	. . . . . . 7 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) = 0 )
77	71	oveq1d 6065	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. N ) = ( 0 x. N ) )
78		zcn 9582	. . . . . . . . . 10 |- ( N e. ZZ -> N e. CC )
79	78	3ad2ant3 1047	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> N e. CC )
80	79	mul02d 8665	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. N ) = 0 )
81	77, 80	eqtrd 2265	. . . . . . 7 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. N ) = 0 )
82	76, 81	oveq12d 6068	. . . . . 6 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( 0 gcd 0 ) )
83	71	oveq1d 6065	. . . . . 6 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) = ( 0 x. ( M gcd N ) ) )
84	70, 82, 83	3eqtr4d 2275	. . . . 5 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) )
85	84	3expib 1233	. . . 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 724	. . 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 1228	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 716   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   class class class wbr 4109  (class class class)co 6050   CCcc 8125   0cc0 8127   x. cmul 8132   / cdiv 8946   NNcn 9237   NN0cn0 9496   ZZcz 9577   || cdvds 12473   gcd cgcd 12649

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-sup 7275  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-dvds 12474  df-gcd 12650

This theorem is referenced by:  absmulgcd  12713  mulgcdr  12714  mulgcddvds  12791  qredeu  12794  coprimeprodsq  12955  pythagtriplem4  12966  2sqlem8  15996