Theorem bezoutlemeu 11695

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 11693	. . . 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 2529	. . . 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 408	. 2 |- ( ph -> E. d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )
8	1	ad2antrr 479	. . . . . 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 479	. . . . . 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 524	. . . . . 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 520	. . . . . . 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 3932	. . . . . . . . 9 |- ( z = w -> ( z || d <-> w || d ) )
13		breq1 3932	. . . . . . . . . 10 |- ( z = w -> ( z || A <-> w || A ) )
14		breq1 3932	. . . . . . . . . 10 |- ( z = w -> ( z || B <-> w || B ) )
15	13, 14	anbi12d 464	. . . . . . . . 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 2654	. . . . . . 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 525	. . . . . 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 521	. . . . . . 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 3932	. . . . . . . . 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 2654	. . . . . . 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 11694	. . . . 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 2514	. . 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 3933	. . . . . 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 2437	. . . 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 2877	. . 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 2643	. 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 413	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 1331   e. wcel 1480   A.wral 2416   E.wrex 2417   E!wreu 2418   E*wrmo 2419   class class class wbr 3929  (class class class)co 5774   + caddc 7623   x. cmul 7625   NN0cn0 8977   ZZcz 9054   || cdvds 11493

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fl 10043  df-mod 10096  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-dvds 11494

This theorem is referenced by:  dfgcd3  11698  bezout  11699