Theorem bezoutlemeu 11940

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 11938	. . . 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 2563	. . . 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 409	. 2 |- ( ph -> E. d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )
8	1	ad2antrr 480	. . . . . 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 480	. . . . . 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 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 ) ) ) ) -> d e. NN0 )
11		simprl 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 || d <-> ( z || A /\ z || B ) ) )
12		breq1 3985	. . . . . . . . 9 |- ( z = w -> ( z || d <-> w || d ) )
13		breq1 3985	. . . . . . . . . 10 |- ( z = w -> ( z || A <-> w || A ) )
14		breq1 3985	. . . . . . . . . 10 |- ( z = w -> ( z || B <-> w || B ) )
15	13, 14	anbi12d 465	. . . . . . . . 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 2692	. . . . . . 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 526	. . . . . 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 522	. . . . . . 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 3985	. . . . . . . . 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 2692	. . . . . . 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 11939	. . . . 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 2548	. . 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 3986	. . . . . 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 2466	. . . 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 2919	. . 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 2678	. 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 414	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 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   E!wreu 2446   E*wrmo 2447   class class class wbr 3982  (class class class)co 5842   + caddc 7756   x. cmul 7758   NN0cn0 9114   ZZcz 9191   || cdvds 11727

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-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-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

This theorem is referenced by:  dfgcd3  11943  bezout  11944