Theorem dfgcd3 12765

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 12715	. . 3 |- ( 0 gcd 0 ) = 0
2		simprl 535	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> M = 0 )
3		simprr 537	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> N = 0 )
4	2, 3	oveq12d 6093	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( 0 gcd 0 ) )
5		0nn0 9557	. . . . 5 |- 0 e. NN0
6	5	a1i 9	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> 0 e. NN0 )
7		0dvds 12556	. . . . . . . . . . 11 |- ( M e. ZZ -> ( 0 || M <-> M = 0 ) )
8	7	ad2antrr 492	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( 0 || M <-> M = 0 ) )
9	2, 8	mpbird 167	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> 0 || M )
10		0dvds 12556	. . . . . . . . . . 11 |- ( N e. ZZ -> ( 0 || N <-> N = 0 ) )
11	10	ad2antlr 493	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( 0 || N <-> N = 0 ) )
12	3, 11	mpbird 167	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> 0 || N )
13	9, 12	jca 306	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( 0 || M /\ 0 || N ) )
14	13	ad2antrr 492	. . . . . . 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 9634	. . . . . . . . 9 |- 0 e. ZZ
16		breq1 4128	. . . . . . . . . . 11 |- ( z = 0 -> ( z || d <-> 0 || d ) )
17		breq1 4128	. . . . . . . . . . . 12 |- ( z = 0 -> ( z || M <-> 0 || M ) )
18		breq1 4128	. . . . . . . . . . . 12 |- ( z = 0 -> ( z || N <-> 0 || N ) )
19	17, 18	anbi12d 477	. . . . . . . . . . 11 |- ( z = 0 -> ( ( z || M /\ z || N ) <-> ( 0 || M /\ 0 || N ) ) )
20	16, 19	bibi12d 235	. . . . . . . . . 10 |- ( z = 0 -> ( ( z || d <-> ( z || M /\ z || N ) ) <-> ( 0 || d <-> ( 0 || M /\ 0 || N ) ) ) )
21	20	rspcv 2925	. . . . . . . . 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 277	. . . . . . 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 167	. . . . . 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 533	. . . . . . . 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 9745	. . . . . . 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 12556	. . . . . . 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 147	. . . . 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 12551	. . . . . . . . 9 |- ( z e. ZZ -> z || 0 )
31	30	adantl 277	. . . . . . . 8 |- ( ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ d = 0 ) /\ z e. ZZ ) -> z || 0 )
32		breq2 4129	. . . . . . . . 9 |- ( d = 0 -> ( z || d <-> z || 0 ) )
33	32	ad2antlr 493	. . . . . . . 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 167	. . . . . . 7 |- ( ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ d = 0 ) /\ z e. ZZ ) -> z || d )
35	2	ad3antrrr 496	. . . . . . . . 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 4153	. . . . . . . 8 |- ( ( ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) /\ d e. NN0 ) /\ d = 0 ) /\ z e. ZZ ) -> z || M )
37	3	ad3antrrr 496	. . . . . . . . 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 4153	. . . . . . . 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 306	. . . . . . 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 175	. . . . . 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 2623	. . . . 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 604	. . . 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 6056	. . 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 2297	. 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 12760	. . . . 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 109	. . . . . 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 2647	. . . . 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 276	. . 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 539	. . . . 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 540	. . . . 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 535	. . . . 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 4128	. . . . . . . . 9 |- ( w = z -> ( w || r <-> z || r ) )
54		breq1 4128	. . . . . . . . . 10 |- ( w = z -> ( w || M <-> z || M ) )
55		breq1 4128	. . . . . . . . . 10 |- ( w = z -> ( w || N <-> z || N ) )
56	54, 55	anbi12d 477	. . . . . . . . 9 |- ( w = z -> ( ( w || M /\ w || N ) <-> ( z || M /\ z || N ) ) )
57	53, 56	bibi12d 235	. . . . . . . 8 |- ( w = z -> ( ( w || r <-> ( w || M /\ w || N ) ) <-> ( z || r <-> ( z || M /\ z || N ) ) ) )
58	57	cbvralv 2786	. . . . . . 7 |- ( A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) <-> A. z e. ZZ ( z || r <-> ( z || M /\ z || N ) ) )
59	58	biimpi 120	. . . . . 6 |- ( A. w e. ZZ ( w || r <-> ( w || M /\ w || N ) ) -> A. z e. ZZ ( z || r <-> ( z || M /\ z || N ) ) )
60	59	ad2antll 495	. . . . 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 533	. . . . 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 12764	. . . 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 4128	. . . . . . . . 9 |- ( w = z -> ( w || d <-> z || d ) )
64	63, 56	bibi12d 235	. . . . . . . 8 |- ( w = z -> ( ( w || d <-> ( w || M /\ w || N ) ) <-> ( z || d <-> ( z || M /\ z || N ) ) ) )
65	64	cbvralv 2786	. . . . . . 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 6047	. . . . 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 537	. . . . . 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 12762	. . . . . . 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 4129	. . . . . . . . . 10 |- ( d = r -> ( w || d <-> w || r ) )
71	70	bibi1d 233	. . . . . . . . 9 |- ( d = r -> ( ( w || d <-> ( w || M /\ w || N ) ) <-> ( w || r <-> ( w || M /\ w || N ) ) ) )
72	71	ralbidv 2550	. . . . . . . 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 6052	. . . . . . 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 415	. . . . . 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 147	. . . . 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 2285	. . . 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 12716	. . . . 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 276	. . . 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 2282	. . 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 2673	. 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 12710	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
82		exmiddc 848	. . 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 810	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 104   <-> wb 105   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   E!wreu 2530   {crab 2532   class class class wbr 4125   iota_crio 6027  (class class class)co 6075   supcsup 7312   RRcr 8168   0cc0 8169   + caddc 8172   x. cmul 8174   < clt 8350   NN0cn0 9542   ZZcz 9623   || cdvds 12532   gcd cgcd 12708

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-sup 7314  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-dvds 12533  df-gcd 12709

This theorem is referenced by:  bezout  12766