Theorem mulgcd 15265

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 11294	. . 3 |- ( K e. NN0 <-> ( K e. NN \/ K = 0 ) )
2		simp1 1061	. . . . . . . . 9 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. NN )
3	2	nnzd 11481	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
4		simp2 1062	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
5	3, 4	zmulcld 11488	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. ZZ )
6		simp3 1063	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
7	3, 6	zmulcld 11488	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
8	5, 7	gcdcld 15230	. . . . . 6 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) e. NN0 )
9	2	nnnn0d 11351	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. NN0 )
10		gcdcl 15228	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
11	10	3adant1 1079	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
12	9, 11	nn0mulcld 11356	. . . . . 6 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) e. NN0 )
13	8	nn0cnd 11353	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) e. CC )
14	2	nncnd 11036	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. CC )
15	2	nnne0d 11065	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K =/= 0 )
16	13, 14, 15	divcan2d 10803	. . . . . . 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 15225	. . . . . . . . . . . . 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 693	. . . . . . . . . . . 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 475	. . . . . . . . . . 11 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) || ( K x. M ) )
20	16, 19	eqbrtrd 4675	. . . . . . . . . 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 15003	. . . . . . . . . . . . . 14 |- ( ( K e. ZZ /\ M e. ZZ ) -> K || ( K x. M ) )
22	3, 4, 21	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K || ( K x. M ) )
23		dvdsmul1 15003	. . . . . . . . . . . . . 14 |- ( ( K e. ZZ /\ N e. ZZ ) -> K || ( K x. N ) )
24	3, 6, 23	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K || ( K x. N ) )
25		dvdsgcd 15261	. . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . 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 715	. . . . . . . . . . . 12 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K || ( ( K x. M ) gcd ( K x. N ) ) )
28	8	nn0zd 11480	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) e. ZZ )
29		dvdsval2 14986	. . . . . . . . . . . . 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 ) )
30	3, 15, 28, 29	syl3anc 1326	. . . . . . . . . . . 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 ) )
31	27, 30	mpbid 222	. . . . . . . . . . 11 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) e. ZZ )
32		dvdscmulr 15010	. . . . . . . . . . 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 ) )
33	31, 4, 3, 15, 32	syl112anc 1330	. . . . . . . . . 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 ) )
34	20, 33	mpbid 222	. . . . . . . . 9 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || M )
35	18	simprd 479	. . . . . . . . . . 11 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) || ( K x. N ) )
36	16, 35	eqbrtrd 4675	. . . . . . . . . 10 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( ( ( K x. M ) gcd ( K x. N ) ) / K ) ) || ( K x. N ) )
37		dvdscmulr 15010	. . . . . . . . . . 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 ) )
38	31, 6, 3, 15, 37	syl112anc 1330	. . . . . . . . . 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 ) )
39	36, 38	mpbid 222	. . . . . . . . 9 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || N )
40		dvdsgcd 15261	. . . . . . . . . 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 ) ) )
41	31, 4, 6, 40	syl3anc 1326	. . . . . . . . 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 ) ) )
42	34, 39, 41	mp2and 715	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( K x. M ) gcd ( K x. N ) ) / K ) || ( M gcd N ) )
43	11	nn0zd 11480	. . . . . . . . 9 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. ZZ )
44		dvdscmul 15008	. . . . . . . . 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 ) ) ) )
45	31, 43, 3, 44	syl3anc 1326	. . . . . . . 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 ) ) ) )
46	42, 45	mpd 15	. . . . . . 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 ) ) )
47	16, 46	eqbrtrrd 4677	. . . . . 6 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) || ( K x. ( M gcd N ) ) )
48		gcddvds 15225	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
49	48	3adant1 1079	. . . . . . . . 9 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
50	49	simpld 475	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) || M )
51		dvdscmul 15008	. . . . . . . . 9 |- ( ( ( M gcd N ) e. ZZ /\ M e. ZZ /\ K e. ZZ ) -> ( ( M gcd N ) || M -> ( K x. ( M gcd N ) ) || ( K x. M ) ) )
52	43, 4, 3, 51	syl3anc 1326	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M -> ( K x. ( M gcd N ) ) || ( K x. M ) ) )
53	50, 52	mpd 15	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) || ( K x. M ) )
54	49	simprd 479	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) || N )
55		dvdscmul 15008	. . . . . . . . 9 |- ( ( ( M gcd N ) e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( ( M gcd N ) || N -> ( K x. ( M gcd N ) ) || ( K x. N ) ) )
56	43, 6, 3, 55	syl3anc 1326	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || N -> ( K x. ( M gcd N ) ) || ( K x. N ) ) )
57	54, 56	mpd 15	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) || ( K x. N ) )
58	12	nn0zd 11480	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) e. ZZ )
59		dvdsgcd 15261	. . . . . . . 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 ) ) ) )
60	58, 5, 7, 59	syl3anc 1326	. . . . . . 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 ) ) ) )
61	53, 57, 60	mp2and 715	. . . . . 6 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) || ( ( K x. M ) gcd ( K x. N ) ) )
62		dvdseq 15036	. . . . . 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 ) ) )
63	8, 12, 47, 61, 62	syl22anc 1327	. . . . 5 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) )
64	63	3expib 1268	. . . 4 |- ( K e. NN -> ( ( M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) ) )
65	10	3adant1 1079	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
66	65	nn0cnd 11353	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. CC )
67	66	mul02d 10234	. . . . . . 7 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. ( M gcd N ) ) = 0 )
68		gcd0val 15219	. . . . . . 7 |- ( 0 gcd 0 ) = 0
69	67, 68	syl6reqr 2675	. . . . . 6 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 gcd 0 ) = ( 0 x. ( M gcd N ) ) )
70		simp1 1061	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> K = 0 )
71	70	oveq1d 6665	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) = ( 0 x. M ) )
72		zcn 11382	. . . . . . . . . 10 |- ( M e. ZZ -> M e. CC )
73	72	3ad2ant2 1083	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
74	73	mul02d 10234	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. M ) = 0 )
75	71, 74	eqtrd 2656	. . . . . . 7 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) = 0 )
76	70	oveq1d 6665	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. N ) = ( 0 x. N ) )
77		zcn 11382	. . . . . . . . . 10 |- ( N e. ZZ -> N e. CC )
78	77	3ad2ant3 1084	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> N e. CC )
79	78	mul02d 10234	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. N ) = 0 )
80	76, 79	eqtrd 2656	. . . . . . 7 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. N ) = 0 )
81	75, 80	oveq12d 6668	. . . . . 6 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( 0 gcd 0 ) )
82	70	oveq1d 6665	. . . . . 6 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) = ( 0 x. ( M gcd N ) ) )
83	69, 81, 82	3eqtr4d 2666	. . . . 5 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) )
84	83	3expib 1268	. . . 4 |- ( K = 0 -> ( ( M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) ) )
85	64, 84	jaoi 394	. . 3 |- ( ( K e. NN \/ K = 0 ) -> ( ( M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) ) )
86	1, 85	sylbi 207	. 2 |- ( K e. NN0 -> ( ( M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) ) )
87	86	3impib 1262	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   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   || cdvds 14983   gcd cgcd 15216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217

This theorem is referenced by:  absmulgcd  15266  mulgcdr  15267  mulgcddvds  15369  qredeu  15372  coprimeprodsq  15513  pythagtriplem4  15524  odadd2  18252  2sqlem8  25151