Theorem bezoutlemeu 11991

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 11989	. . . 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 109	. . . . 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 2574	. . . 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 411	. 2 |- ( ph -> E. d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )
8	1	ad2antrr 488	. . . . . 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 488	. . . . . 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 535	. . . . . 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 529	. . . . . . 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 4003	. . . . . . . . 9 |- ( z = w -> ( z || d <-> w || d ) )
13		breq1 4003	. . . . . . . . . 10 |- ( z = w -> ( z || A <-> w || A ) )
14		breq1 4003	. . . . . . . . . 10 |- ( z = w -> ( z || B <-> w || B ) )
15	13, 14	anbi12d 473	. . . . . . . . 9 |- ( z = w -> ( ( z || A /\ z || B ) <-> ( w || A /\ w || B ) ) )
16	12, 15	bibi12d 235	. . . . . . . 8 |- ( z = w -> ( ( z || d <-> ( z || A /\ z || B ) ) <-> ( w || d <-> ( w || A /\ w || B ) ) ) )
17	16	cbvralv 2703	. . . . . . 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 122	. . . . . 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 536	. . . . . 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 531	. . . . . . 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 4003	. . . . . . . . 9 |- ( z = w -> ( z || e <-> w || e ) )
22	21, 15	bibi12d 235	. . . . . . . 8 |- ( z = w -> ( ( z || e <-> ( z || A /\ z || B ) ) <-> ( w || e <-> ( w || A /\ w || B ) ) ) )
23	22	cbvralv 2703	. . . . . . 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 122	. . . . . 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 11990	. . . . 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 115	. . . 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 2559	. . 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 4004	. . . . . 6 |- ( d = e -> ( z || d <-> z || e ) )
29	28	bibi1d 233	. . . . 5 |- ( d = e -> ( ( z || d <-> ( z || A /\ z || B ) ) <-> ( z || e <-> ( z || A /\ z || B ) ) ) )
30	29	ralbidv 2477	. . . 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 2930	. . 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 134	. 2 |- ( ph -> E* d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )
33		reu5 2689	. 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 417	1 |- ( ph -> E! d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   E!wreu 2457   E*wrmo 2458   class class class wbr 4000  (class class class)co 5869   + caddc 7805   x. cmul 7807   NN0cn0 9165   ZZcz 9242   || cdvds 11778

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779

This theorem is referenced by:  dfgcd3  11994  bezout  11995