Theorem bezoutlemeu 11488

Description: Lemma for Bézout's identity. There is exactly one nonnegative integer meeting the greatest common divisor condition. (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.)

Hypotheses

Ref	Expression
bezoutlemgcd.1	|- ( ph -> A e. ZZ )
bezoutlemgcd.2	|- ( ph -> B e. ZZ )
bezoutlemgcd.3	|- ( ph -> D e. NN0 )
bezoutlemgcd.4	|- ( ph -> A. z e. ZZ ( z || D <-> ( z || A /\ z || B ) ) )

Assertion

Ref	Expression
bezoutlemeu	|- ( ph -> E! d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )

Distinct variable groups:   z, D   A, d, z   B, d, z   ph, d

Allowed substitution hints:   ph( z)   D( d)

Proof of Theorem bezoutlemeu

Dummy variables e w s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bezoutlemgcd.1	. . 3 |- ( ph -> A e. ZZ )
2		bezoutlemgcd.2	. . 3 |- ( ph -> B e. ZZ )
3		bezoutlembi 11486	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> E. d e. NN0 ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. s e. ZZ E. t e. ZZ d = ( ( A x. s ) + ( B x. t ) ) ) )
4		simpl 108	. . . . 5 |- ( ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. s e. ZZ E. t e. ZZ d = ( ( A x. s ) + ( B x. t ) ) ) -> A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )
5	4	reximi 2488	. . . 4 |- ( E. d e. NN0 ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. s e. ZZ E. t e. ZZ d = ( ( A x. s ) + ( B x. t ) ) ) -> E. d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )
6	3, 5	syl 14	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> E. d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )
7	1, 2, 6	syl2anc 406	. 2 |- ( ph -> E. d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )
8	1	ad2antrr 475	. . . . . 6 |- ( ( ( ph /\ ( d e. NN0 /\ e e. NN0 ) ) /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ A. z e. ZZ ( z || e <-> ( z || A /\ z || B ) ) ) ) -> A e. ZZ )
9	2	ad2antrr 475	. . . . . 6 |- ( ( ( ph /\ ( d e. NN0 /\ e e. NN0 ) ) /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ A. z e. ZZ ( z || e <-> ( z || A /\ z || B ) ) ) ) -> B e. ZZ )
10		simplrl 505	. . . . . 6 |- ( ( ( ph /\ ( d e. NN0 /\ e e. NN0 ) ) /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ A. z e. ZZ ( z || e <-> ( z || A /\ z || B ) ) ) ) -> d e. NN0 )
11		simprl 501	. . . . . . 7 |- ( ( ( ph /\ ( d e. NN0 /\ e e. NN0 ) ) /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ A. z e. ZZ ( z || e <-> ( z || A /\ z || B ) ) ) ) -> A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )
12		breq1 3878	. . . . . . . . 9 |- ( z = w -> ( z || d <-> w || d ) )
13		breq1 3878	. . . . . . . . . 10 |- ( z = w -> ( z || A <-> w || A ) )
14		breq1 3878	. . . . . . . . . 10 |- ( z = w -> ( z || B <-> w || B ) )
15	13, 14	anbi12d 460	. . . . . . . . 9 |- ( z = w -> ( ( z || A /\ z || B ) <-> ( w || A /\ w || B ) ) )
16	12, 15	bibi12d 234	. . . . . . . 8 |- ( z = w -> ( ( z || d <-> ( z || A /\ z || B ) ) <-> ( w || d <-> ( w || A /\ w || B ) ) ) )
17	16	cbvralv 2612	. . . . . . 7 |- ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) <-> A. w e. ZZ ( w || d <-> ( w || A /\ w || B ) ) )
18	11, 17	sylib 121	. . . . . 6 |- ( ( ( ph /\ ( d e. NN0 /\ e e. NN0 ) ) /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ A. z e. ZZ ( z || e <-> ( z || A /\ z || B ) ) ) ) -> A. w e. ZZ ( w || d <-> ( w || A /\ w || B ) ) )
19		simplrr 506	. . . . . 6 |- ( ( ( ph /\ ( d e. NN0 /\ e e. NN0 ) ) /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ A. z e. ZZ ( z || e <-> ( z || A /\ z || B ) ) ) ) -> e e. NN0 )
20		simprr 502	. . . . . . 7 |- ( ( ( ph /\ ( d e. NN0 /\ e e. NN0 ) ) /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ A. z e. ZZ ( z || e <-> ( z || A /\ z || B ) ) ) ) -> A. z e. ZZ ( z || e <-> ( z || A /\ z || B ) ) )
21		breq1 3878	. . . . . . . . 9 |- ( z = w -> ( z || e <-> w || e ) )
22	21, 15	bibi12d 234	. . . . . . . 8 |- ( z = w -> ( ( z || e <-> ( z || A /\ z || B ) ) <-> ( w || e <-> ( w || A /\ w || B ) ) ) )
23	22	cbvralv 2612	. . . . . . 7 |- ( A. z e. ZZ ( z || e <-> ( z || A /\ z || B ) ) <-> A. w e. ZZ ( w || e <-> ( w || A /\ w || B ) ) )
24	20, 23	sylib 121	. . . . . 6 |- ( ( ( ph /\ ( d e. NN0 /\ e e. NN0 ) ) /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ A. z e. ZZ ( z || e <-> ( z || A /\ z || B ) ) ) ) -> A. w e. ZZ ( w || e <-> ( w || A /\ w || B ) ) )
25	8, 9, 10, 18, 19, 24	bezoutlemmo 11487	. . . . 5 |- ( ( ( ph /\ ( d e. NN0 /\ e e. NN0 ) ) /\ ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ A. z e. ZZ ( z || e <-> ( z || A /\ z || B ) ) ) ) -> d = e )
26	25	ex 114	. . . 4 |- ( ( ph /\ ( d e. NN0 /\ e e. NN0 ) ) -> ( ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ A. z e. ZZ ( z || e <-> ( z || A /\ z || B ) ) ) -> d = e ) )
27	26	ralrimivva 2473	. . 3 |- ( ph -> A. d e. NN0 A. e e. NN0 ( ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ A. z e. ZZ ( z || e <-> ( z || A /\ z || B ) ) ) -> d = e ) )
28		breq2 3879	. . . . . 6 |- ( d = e -> ( z || d <-> z || e ) )
29	28	bibi1d 232	. . . . 5 |- ( d = e -> ( ( z || d <-> ( z || A /\ z || B ) ) <-> ( z || e <-> ( z || A /\ z || B ) ) ) )
30	29	ralbidv 2396	. . . 4 |- ( d = e -> ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) <-> A. z e. ZZ ( z || e <-> ( z || A /\ z || B ) ) ) )
31	30	rmo4 2830	. . 3 |- ( E* d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) <-> A. d e. NN0 A. e e. NN0 ( ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ A. z e. ZZ ( z || e <-> ( z || A /\ z || B ) ) ) -> d = e ) )
32	27, 31	sylibr 133	. 2 |- ( ph -> E* d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )
33		reu5 2601	. 2 |- ( E! d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) <-> ( E. d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E* d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) ) )
34	7, 32, 33	sylanbrc 411	1 |- ( ph -> E! d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1299   e. wcel 1448   A.wral 2375   E.wrex 2376   E!wreu 2377   E*wrmo 2378   class class class wbr 3875  (class class class)co 5706   + caddc 7503   x. cmul 7505   NN0cn0 8829   ZZcz 8906   || cdvds 11288

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-fz 9632  df-fl 9884  df-mod 9937  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-dvds 11289

This theorem is referenced by:  dfgcd3  11491  bezout  11492