Theorem dfgcd3 11865

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 11816	. . 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 5832	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( 0 gcd 0 ) )
5		0nn0 9084	. . . . 5 |- 0 e. NN0
6	5	a1i 9	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> 0 e. NN0 )
7		0dvds 11680	. . . . . . . . . . 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 11680	. . . . . . . . . . 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 9157	. . . . . . . . 9 |- 0 e. ZZ
16		breq1 3964	. . . . . . . . . . 11 |- ( z = 0 -> ( z || d <-> 0 || d ) )
17		breq1 3964	. . . . . . . . . . . 12 |- ( z = 0 -> ( z || M <-> 0 || M ) )
18		breq1 3964	. . . . . . . . . . . 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 2809	. . . . . . . . 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 9263	. . . . . . 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 11680	. . . . . . 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 11675	. . . . . . . . 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 3965	. . . . . . . . 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 3988	. . . . . . . 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 3988	. . . . . . . 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 2527	. . . . 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 5795	. . 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 2213	. 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 11860	. . . . 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 2551	. . . . 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 3964	. . . . . . . . 9 |- ( w = z -> ( w || r <-> z || r ) )
54		breq1 3964	. . . . . . . . . 10 |- ( w = z -> ( w || M <-> z || M ) )
55		breq1 3964	. . . . . . . . . 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 2677	. . . . . . 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 11864	. . . 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 3964	. . . . . . . . 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 2677	. . . . . . 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 5787	. . . . 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 11862	. . . . . . 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 3965	. . . . . . . . . 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 2454	. . . . . . . 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 5792	. . . . . . 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	syl5eqr 2201	. . . 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 11817	. . . . 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 2198	. . 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 2576	. 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 11804	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
82		exmiddc 822	. . 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 820   = wceq 1332   e. wcel 2125   A.wral 2432   E.wrex 2433   E!wreu 2434   {crab 2436   class class class wbr 3961   iota_crio 5769  (class class class)co 5814   supcsup 6914   RRcr 7710   0cc0 7711   + caddc 7714   x. cmul 7716   < clt 7891   NN0cn0 9069   ZZcz 9146   || cdvds 11660   gcd cgcd 11802

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-mulrcl 7810  ax-addcom 7811  ax-mulcom 7812  ax-addass 7813  ax-mulass 7814  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-1rid 7818  ax-0id 7819  ax-rnegex 7820  ax-precex 7821  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-apti 7826  ax-pre-ltadd 7827  ax-pre-mulgt0 7828  ax-pre-mulext 7829  ax-arch 7830  ax-caucvg 7831

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rmo 2440  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-if 3502  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-id 4248  df-po 4251  df-iso 4252  df-iord 4321  df-on 4323  df-ilim 4324  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-1st 6078  df-2nd 6079  df-recs 6242  df-frec 6328  df-sup 6916  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-reap 8429  df-ap 8436  df-div 8525  df-inn 8813  df-2 8871  df-3 8872  df-4 8873  df-n0 9070  df-z 9147  df-uz 9419  df-q 9507  df-rp 9539  df-fz 9891  df-fzo 10020  df-fl 10147  df-mod 10200  df-seqfrec 10323  df-exp 10397  df-cj 10719  df-re 10720  df-im 10721  df-rsqrt 10875  df-abs 10876  df-dvds 11661  df-gcd 11803

This theorem is referenced by:  bezout  11866