Theorem bezoutlemeu 10774

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 10772	. . . 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 107	. . . . 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 2464	. . . 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 403	. 2 |- ( ph -> E. d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )
8	1	ad2antrr 472	. . . . . 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 472	. . . . . 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 502	. . . . . 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 498	. . . . . . 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 3814	. . . . . . . . 9 |- ( z = w -> ( z || d <-> w || d ) )
13		breq1 3814	. . . . . . . . . 10 |- ( z = w -> ( z || A <-> w || A ) )
14		breq1 3814	. . . . . . . . . 10 |- ( z = w -> ( z || B <-> w || B ) )
15	13, 14	anbi12d 457	. . . . . . . . 9 |- ( z = w -> ( ( z || A /\ z || B ) <-> ( w || A /\ w || B ) ) )
16	12, 15	bibi12d 233	. . . . . . . 8 |- ( z = w -> ( ( z || d <-> ( z || A /\ z || B ) ) <-> ( w || d <-> ( w || A /\ w || B ) ) ) )
17	16	cbvralv 2583	. . . . . . 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 120	. . . . . 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 503	. . . . . 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 499	. . . . . . 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 3814	. . . . . . . . 9 |- ( z = w -> ( z || e <-> w || e ) )
22	21, 15	bibi12d 233	. . . . . . . 8 |- ( z = w -> ( ( z || e <-> ( z || A /\ z || B ) ) <-> ( w || e <-> ( w || A /\ w || B ) ) ) )
23	22	cbvralv 2583	. . . . . . 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 120	. . . . . 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 10773	. . . . 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 113	. . . 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 2449	. . 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 3815	. . . . . 6 |- ( d = e -> ( z || d <-> z || e ) )
29	28	bibi1d 231	. . . . 5 |- ( d = e -> ( ( z || d <-> ( z || A /\ z || B ) ) <-> ( z || e <-> ( z || A /\ z || B ) ) ) )
30	29	ralbidv 2374	. . . 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 2796	. . 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 132	. 2 |- ( ph -> E* d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )
33		reu5 2572	. 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 408	1 |- ( ph -> E! d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   e. wcel 1434   A.wral 2353   E.wrex 2354   E!wreu 2355   E*wrmo 2356   class class class wbr 3811  (class class class)co 5589   + caddc 7254   x. cmul 7256   NN0cn0 8563   ZZcz 8644   || cdvds 10574

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365  ax-cnex 7337  ax-resscn 7338  ax-1cn 7339  ax-1re 7340  ax-icn 7341  ax-addcl 7342  ax-addrcl 7343  ax-mulcl 7344  ax-mulrcl 7345  ax-addcom 7346  ax-mulcom 7347  ax-addass 7348  ax-mulass 7349  ax-distr 7350  ax-i2m1 7351  ax-0lt1 7352  ax-1rid 7353  ax-0id 7354  ax-rnegex 7355  ax-precex 7356  ax-cnre 7357  ax-pre-ltirr 7358  ax-pre-ltwlin 7359  ax-pre-lttrn 7360  ax-pre-apti 7361  ax-pre-ltadd 7362  ax-pre-mulgt0 7363  ax-pre-mulext 7364  ax-arch 7365  ax-caucvg 7366

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-po 4086  df-iso 4087  df-iord 4156  df-on 4158  df-ilim 4159  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-riota 5545  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-frec 6086  df-pnf 7425  df-mnf 7426  df-xr 7427  df-ltxr 7428  df-le 7429  df-sub 7556  df-neg 7557  df-reap 7950  df-ap 7957  df-div 8036  df-inn 8315  df-2 8373  df-3 8374  df-4 8375  df-n0 8564  df-z 8645  df-uz 8913  df-q 8998  df-rp 9028  df-fz 9318  df-fl 9564  df-mod 9617  df-iseq 9739  df-iexp 9790  df-cj 10101  df-re 10102  df-im 10103  df-rsqrt 10256  df-abs 10257  df-dvds 10575

This theorem is referenced by:  dfgcd3  10777  bezout  10778