Theorem mulgcd 12337

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 9297	. . 3 |- ( K e. NN0 <-> ( K e. NN \/ K = 0 ) )
2		simp1 1000	. . . . . . . . 9 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. NN )
3	2	nnzd 9494	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
4		simp2 1001	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
5	3, 4	zmulcld 9501	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. ZZ )
6		simp3 1002	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
7	3, 6	zmulcld 9501	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
8	5, 7	gcdcld 12289	. . . . . 6 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) e. NN0 )
9	2	nnnn0d 9348	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. NN0 )
10		gcdcl 12287	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
11	10	3adant1 1018	. . . . . . 7 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
12	9, 11	nn0mulcld 9353	. . . . . 6 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) e. NN0 )
13	8	nn0cnd 9350	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) e. CC )
14	2	nncnd 9050	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K e. CC )
15	2	nnap0d 9082	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K # 0 )
16	13, 14, 15	divcanap2d 8865	. . . . . . 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 12284	. . . . . . . . . . . . 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 4066	. . . . . . . . . 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 12124	. . . . . . . . . . . . . 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 12124	. . . . . . . . . . . . . 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 12333	. . . . . . . . . . . . . 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 1250	. . . . . . . . . . . . 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 9081	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> K =/= 0 )
29	8	nn0zd 9493	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) e. ZZ )
30		dvdsval2 12101	. . . . . . . . . . . . 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 1250	. . . . . . . . . . . 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 12131	. . . . . . . . . . 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 1254	. . . . . . . . . 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 4066	. . . . . . . . . 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 12131	. . . . . . . . . . 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 1254	. . . . . . . . . 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 12333	. . . . . . . . . 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 1250	. . . . . . . . 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 9493	. . . . . . . . 9 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. ZZ )
45		dvdscmul 12129	. . . . . . . . 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 1250	. . . . . . . 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 4068	. . . . . 6 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) || ( K x. ( M gcd N ) ) )
49		gcddvds 12284	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
50	49	3adant1 1018	. . . . . . . . 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 12129	. . . . . . . . 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 1250	. . . . . . . 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 12129	. . . . . . . . 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 1250	. . . . . . . 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 9493	. . . . . . . 8 |- ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) e. ZZ )
60		dvdsgcd 12333	. . . . . . . 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 1250	. . . . . . 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 12159	. . . . . 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 1251	. . . . 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 1209	. . . 4 |- ( K e. NN -> ( ( M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) ) )
66		gcd0val 12281	. . . . . . 7 |- ( 0 gcd 0 ) = 0
67	10	3adant1 1018	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
68	67	nn0cnd 9350	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. CC )
69	68	mul02d 8464	. . . . . . 7 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. ( M gcd N ) ) = 0 )
70	66, 69	eqtr4id 2257	. . . . . 6 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 gcd 0 ) = ( 0 x. ( M gcd N ) ) )
71		simp1 1000	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> K = 0 )
72	71	oveq1d 5959	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) = ( 0 x. M ) )
73		zcn 9377	. . . . . . . . . 10 |- ( M e. ZZ -> M e. CC )
74	73	3ad2ant2 1022	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
75	74	mul02d 8464	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. M ) = 0 )
76	72, 75	eqtrd 2238	. . . . . . 7 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) = 0 )
77	71	oveq1d 5959	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. N ) = ( 0 x. N ) )
78		zcn 9377	. . . . . . . . . 10 |- ( N e. ZZ -> N e. CC )
79	78	3ad2ant3 1023	. . . . . . . . 9 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> N e. CC )
80	79	mul02d 8464	. . . . . . . 8 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. N ) = 0 )
81	77, 80	eqtrd 2238	. . . . . . 7 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. N ) = 0 )
82	76, 81	oveq12d 5962	. . . . . 6 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( 0 gcd 0 ) )
83	71	oveq1d 5959	. . . . . 6 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( K x. ( M gcd N ) ) = ( 0 x. ( M gcd N ) ) )
84	70, 82, 83	3eqtr4d 2248	. . . . 5 |- ( ( K = 0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) )
85	84	3expib 1209	. . . 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 718	. . 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 1204	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 710   /\ w3a 981   = wceq 1373   e. wcel 2176   =/= wne 2376   class class class wbr 4044  (class class class)co 5944   CCcc 7923   0cc0 7925   x. cmul 7930   / cdiv 8745   NNcn 9036   NN0cn0 9295   ZZcz 9372   || cdvds 12098   gcd cgcd 12274

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-sup 7086  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-fz 10131  df-fzo 10265  df-fl 10413  df-mod 10468  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-dvds 12099  df-gcd 12275

This theorem is referenced by:  absmulgcd  12338  mulgcdr  12339  mulgcddvds  12416  qredeu  12419  coprimeprodsq  12580  pythagtriplem4  12591  2sqlem8  15600