Theorem dfgcd3 11943

Description: Alternate definition of the gcd operator. (Contributed by Jim Kingdon, 31-Dec-2021.)

Assertion

Ref	Expression
dfgcd3	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( iota_ d e. NN0 A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) )

Distinct variable groups:   M, d, z   N, d, z

Proof of Theorem dfgcd3

Dummy variables a b r w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gcd0val 11893	. . 3 |- ( 0 gcd 0 ) = 0
2		simprl 521	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> M = 0 )
3		simprr 522	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> N = 0 )
4	2, 3	oveq12d 5860	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( 0 gcd 0 ) )
5		0nn0 9129	. . . . 5 |- 0 e. NN0
6	5	a1i 9	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> 0 e. NN0 )
7		0dvds 11751	. . . . . . . . . . 11 |- ( M e. ZZ -> ( 0 || M <-> M = 0 ) )
8	7	ad2antrr 480	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( 0 || M <-> M = 0 ) )
9	2, 8	mpbird 166	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> 0 || M )
10		0dvds 11751	. . . . . . . . . . 11 |- ( N e. ZZ -> ( 0 || N <-> N = 0 ) )
11	10	ad2antlr 481	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( 0 || N <-> N = 0 ) )
12	3, 11	mpbird 166	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> 0 || N )
13	9, 12	jca 304	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( 0 || M /\ 0 || N ) )
14	13	ad2antrr 480	. . . . . . 7 |- ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) -> ( 0 || M /\ 0 || N ) )
15		0z 9202	. . . . . . . . 9 |- 0 e. ZZ
16		breq1 3985	. . . . . . . . . . 11 |- ( z = 0 -> ( z || d <-> 0 || d ) )
17		breq1 3985	. . . . . . . . . . . 12 |- ( z = 0 -> ( z || M <-> 0 || M ) )
18		breq1 3985	. . . . . . . . . . . 12 |- ( z = 0 -> ( z || N <-> 0 || N ) )
19	17, 18	anbi12d 465	. . . . . . . . . . 11 |- ( z = 0 -> ( ( z || M /\ z || N ) <-> ( 0 || M /\ 0 || N ) ) )
20	16, 19	bibi12d 234	. . . . . . . . . 10 |- ( z = 0 -> ( ( z || d <-> ( z || M /\ z || N ) ) <-> ( 0 || d <-> ( 0 || M /\ 0 || N ) ) ) )
21	20	rspcv 2826	. . . . . . . . 9 |- ( 0 e. ZZ -> ( A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) -> ( 0 || d <-> ( 0 || M /\ 0 || N ) ) ) )
22	15, 21	ax-mp 5	. . . . . . . 8 |- ( A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) -> ( 0 || d <-> ( 0 || M /\ 0 || N ) ) )
23	22	adantl 275	. . . . . . 7 |- ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) -> ( 0 || d <-> ( 0 || M /\ 0 || N ) ) )
24	14, 23	mpbird 166	. . . . . 6 |- ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) -> 0 || d )
25		simplr 520	. . . . . . . 8 |- ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) -> d e. NN0 )
26	25	nn0zd 9311	. . . . . . 7 |- ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) -> d e. ZZ )
27		0dvds 11751	. . . . . . 7 |- ( d e. ZZ -> ( 0 || d <-> d = 0 ) )
28	26, 27	syl 14	. . . . . 6 |- ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) -> ( 0 || d <-> d = 0 ) )
29	24, 28	mpbid 146	. . . . 5 |- ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) -> d = 0 )
30		dvds0 11746	. . . . . . . . 9 |- ( z e. ZZ -> z || 0 )
31	30	adantl 275	. . . . . . . 8 |- ( ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ d = 0 ) /\ z e. ZZ ) -> z || 0 )
32		breq2 3986	. . . . . . . . 9 |- ( d = 0 -> ( z || d <-> z || 0 ) )
33	32	ad2antlr 481	. . . . . . . 8 |- ( ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ d = 0 ) /\ z e. ZZ ) -> ( z || d <-> z || 0 ) )
34	31, 33	mpbird 166	. . . . . . 7 |- ( ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ d = 0 ) /\ z e. ZZ ) -> z || d )
35	2	ad3antrrr 484	. . . . . . . . 9 |- ( ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ d = 0 ) /\ z e. ZZ ) -> M = 0 )
36	31, 35	breqtrrd 4010	. . . . . . . 8 |- ( ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ d = 0 ) /\ z e. ZZ ) -> z || M )
37	3	ad3antrrr 484	. . . . . . . . 9 |- ( ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ d = 0 ) /\ z e. ZZ ) -> N = 0 )
38	31, 37	breqtrrd 4010	. . . . . . . 8 |- ( ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ d = 0 ) /\ z e. ZZ ) -> z || N )
39	36, 38	jca 304	. . . . . . 7 |- ( ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ d = 0 ) /\ z e. ZZ ) -> ( z || M /\ z || N ) )
40	34, 39	2thd 174	. . . . . 6 |- ( ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ d = 0 ) /\ z e. ZZ ) -> ( z || d <-> ( z || M /\ z || N ) ) )
41	40	ralrimiva 2539	. . . . 5 |- ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ d = 0 ) -> A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) )
42	29, 41	impbida 586	. . . 4 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) -> ( A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) <-> d = 0 ) )
43	6, 42	riota5 5823	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( iota_ d e. NN0 A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) = 0 )
44	1, 4, 43	3eqtr4a 2225	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( iota_ d e. NN0 A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) )
45		bezoutlembi 11938	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> E. r e. NN0 ( A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) /\ E. a e. ZZ E. b e. ZZ r = ( ( M x. a ) + ( N x. b ) ) ) )
46		simpl 108	. . . . . 6 |- ( ( A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) /\ E. a e. ZZ E. b e. ZZ r = ( ( M x. a ) + ( N x. b ) ) ) -> A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) )
47	46	reximi 2563	. . . . 5 |- ( E. r e. NN0 ( A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) /\ E. a e. ZZ E. b e. ZZ r = ( ( M x. a ) + ( N x. b ) ) ) -> E. r e. NN0 A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) )
48	45, 47	syl 14	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> E. r e. NN0 A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) )
49	48	adantr 274	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> E. r e. NN0 A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) )
50		simplll 523	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( r e. NN0 /\ A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) ) ) -> M e. ZZ )
51		simpllr 524	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( r e. NN0 /\ A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) ) ) -> N e. ZZ )
52		simprl 521	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( r e. NN0 /\ A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) ) ) -> r e. NN0 )
53		breq1 3985	. . . . . . . . 9 |- ( w = z -> ( w || r <-> z || r ) )
54		breq1 3985	. . . . . . . . . 10 |- ( w = z -> ( w || M <-> z || M ) )
55		breq1 3985	. . . . . . . . . 10 |- ( w = z -> ( w || N <-> z || N ) )
56	54, 55	anbi12d 465	. . . . . . . . 9 |- ( w = z -> ( ( w || M /\ w || N ) <-> ( z || M /\ z || N ) ) )
57	53, 56	bibi12d 234	. . . . . . . 8 |- ( w = z -> ( ( w || r <-> ( w || M /\ w || N ) ) <-> ( z || r <-> ( z || M /\ z || N ) ) ) )
58	57	cbvralv 2692	. . . . . . 7 |- ( A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) <-> A. z e. ZZ ( z || r <-> ( z || M /\ z || N ) ) )
59	58	biimpi 119	. . . . . 6 |- ( A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) -> A. z e. ZZ ( z || r <-> ( z || M /\ z || N ) ) )
60	59	ad2antll 483	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( r e. NN0 /\ A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) ) ) -> A. z e. ZZ ( z || r <-> ( z || M /\ z || N ) ) )
61		simplr 520	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( r e. NN0 /\ A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) ) ) -> -. ( M = 0 /\ N = 0 ) )
62	50, 51, 52, 60, 61	bezoutlemsup 11942	. . . 4 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( r e. NN0 /\ A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) ) ) -> r = sup ( { z e. ZZ | ( z || M /\ z || N ) } , RR , < ) )
63		breq1 3985	. . . . . . . . 9 |- ( w = z -> ( w || d <-> z || d ) )
64	63, 56	bibi12d 234	. . . . . . . 8 |- ( w = z -> ( ( w || d <-> ( w || M /\ w || N ) ) <-> ( z || d <-> ( z || M /\ z || N ) ) ) )
65	64	cbvralv 2692	. . . . . . 7 |- ( A. w e. ZZ ( w || d <-> ( w || M /\ w || N ) ) <-> A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) )
66	65	a1i 9	. . . . . 6 |- ( d e. NN0 -> ( A. w e. ZZ ( w || d <-> ( w || M /\ w || N ) ) <-> A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) )
67	66	riotabiia 5815	. . . . 5 |- ( iota_ d e. NN0 A. w e. ZZ ( w || d <-> ( w || M /\ w || N ) ) ) = ( iota_ d e. NN0 A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) )
68		simprr 522	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( r e. NN0 /\ A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) ) ) -> A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) )
69	50, 51, 52, 68	bezoutlemeu 11940	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( r e. NN0 /\ A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) ) ) -> E! d e. NN0 A. w e. ZZ ( w || d <-> ( w || M /\ w || N ) ) )
70		breq2 3986	. . . . . . . . . 10 |- ( d = r -> ( w || d <-> w || r ) )
71	70	bibi1d 232	. . . . . . . . 9 |- ( d = r -> ( ( w || d <-> ( w || M /\ w || N ) ) <-> ( w || r <-> ( w || M /\ w || N ) ) ) )
72	71	ralbidv 2466	. . . . . . . 8 |- ( d = r -> ( A. w e. ZZ ( w || d <-> ( w || M /\ w || N ) ) <-> A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) ) )
73	72	riota2 5820	. . . . . . 7 |- ( ( r e. NN0 /\ E! d e. NN0 A. w e. ZZ ( w || d <-> ( w || M /\ w || N ) ) ) -> ( A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) <-> ( iota_ d e. NN0 A. w e. ZZ ( w || d <-> ( w || M /\ w || N ) ) ) = r ) )
74	52, 69, 73	syl2anc 409	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( r e. NN0 /\ A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) ) ) -> ( A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) <-> ( iota_ d e. NN0 A. w e. ZZ ( w || d <-> ( w || M /\ w || N ) ) ) = r ) )
75	68, 74	mpbid 146	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( r e. NN0 /\ A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) ) ) -> ( iota_ d e. NN0 A. w e. ZZ ( w || d <-> ( w || M /\ w || N ) ) ) = r )
76	67, 75	eqtr3id 2213	. . . 4 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( r e. NN0 /\ A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) ) ) -> ( iota_ d e. NN0 A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) = r )
77		gcdn0val 11894	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = sup ( { z e. ZZ | ( z || M /\ z || N ) } , RR , < ) )
78	77	adantr 274	. . . 4 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( r e. NN0 /\ A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) ) ) -> ( M gcd N ) = sup ( { z e. ZZ | ( z || M /\ z || N ) } , RR , < ) )
79	62, 76, 78	3eqtr4rd 2209	. . 3 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( r e. NN0 /\ A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) ) ) -> ( M gcd N ) = ( iota_ d e. NN0 A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) )
80	49, 79	rexlimddv 2588	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( iota_ d e. NN0 A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) )
81		gcdmndc 11877	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
82		exmiddc 826	. . 3 |- (DECID ( M = 0 /\ N = 0 ) -> ( ( M = 0 /\ N = 0 ) \/ -. ( M = 0 /\ N = 0 ) ) )
83	81, 82	syl 14	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ N = 0 ) \/ -. ( M = 0 /\ N = 0 ) ) )
84	44, 80, 83	mpjaodan 788	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( iota_ d e. NN0 A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   E!wreu 2446   {crab 2448   class class class wbr 3982   iota_crio 5797  (class class class)co 5842   supcsup 6947   RRcr 7752   0cc0 7753   + caddc 7756   x. cmul 7758   < clt 7933   NN0cn0 9114   ZZcz 9191   || cdvds 11727   gcd cgcd 11875

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876

This theorem is referenced by:  bezout  11944