Theorem bezoutlemle 11992

Description: Lemma for Bézout's identity. The number satisfying the greatest common divisor condition is the largest number which divides both A and B. (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 ) ) )
bezoutlemgcd.5	|- ( ph -> -. ( A = 0 /\ B = 0 ) )

Assertion

Ref	Expression
bezoutlemle	|- ( ph -> A. z e. ZZ ( ( z || A /\ z || B ) -> z <_ D ) )

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

Proof of Theorem bezoutlemle

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( z || A /\ z || B ) )
2		breq1 4003	. . . . . . . 8 |- ( z = w -> ( z || D <-> w || D ) )
3		breq1 4003	. . . . . . . . 9 |- ( z = w -> ( z || A <-> w || A ) )
4		breq1 4003	. . . . . . . . 9 |- ( z = w -> ( z || B <-> w || B ) )
5	3, 4	anbi12d 473	. . . . . . . 8 |- ( z = w -> ( ( z || A /\ z || B ) <-> ( w || A /\ w || B ) ) )
6	2, 5	bibi12d 235	. . . . . . 7 |- ( z = w -> ( ( z || D <-> ( z || A /\ z || B ) ) <-> ( w || D <-> ( w || A /\ w || B ) ) ) )
7		equcom 1706	. . . . . . 7 |- ( z = w <-> w = z )
8		bicom 140	. . . . . . 7 |- ( ( ( z || D <-> ( z || A /\ z || B ) ) <-> ( w || D <-> ( w || A /\ w || B ) ) ) <-> ( ( w || D <-> ( w || A /\ w || B ) ) <-> ( z || D <-> ( z || A /\ z || B ) ) ) )
9	6, 7, 8	3imtr3i 200	. . . . . 6 |- ( w = z -> ( ( w || D <-> ( w || A /\ w || B ) ) <-> ( z || D <-> ( z || A /\ z || B ) ) ) )
10		bezoutlemgcd.4	. . . . . . . 8 |- ( ph -> A. z e. ZZ ( z || D <-> ( z || A /\ z || B ) ) )
11	6	cbvralv 2703	. . . . . . . 8 |- ( A. z e. ZZ ( z || D <-> ( z || A /\ z || B ) ) <-> A. w e. ZZ ( w || D <-> ( w || A /\ w || B ) ) )
12	10, 11	sylib 122	. . . . . . 7 |- ( ph -> A. w e. ZZ ( w || D <-> ( w || A /\ w || B ) ) )
13	12	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> A. w e. ZZ ( w || D <-> ( w || A /\ w || B ) ) )
14		simplr 528	. . . . . 6 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> z e. ZZ )
15	9, 13, 14	rspcdva 2846	. . . . 5 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( z || D <-> ( z || A /\ z || B ) ) )
16	1, 15	mpbird 167	. . . 4 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> z || D )
17		bezoutlemgcd.3	. . . . . . 7 |- ( ph -> D e. NN0 )
18	17	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> D e. NN0 )
19		bezoutlemgcd.5	. . . . . . . . 9 |- ( ph -> -. ( A = 0 /\ B = 0 ) )
20	19	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> -. ( A = 0 /\ B = 0 ) )
21		breq1 4003	. . . . . . . . . . . 12 |- ( z = 0 -> ( z || D <-> 0 || D ) )
22		breq1 4003	. . . . . . . . . . . . 13 |- ( z = 0 -> ( z || A <-> 0 || A ) )
23		breq1 4003	. . . . . . . . . . . . 13 |- ( z = 0 -> ( z || B <-> 0 || B ) )
24	22, 23	anbi12d 473	. . . . . . . . . . . 12 |- ( z = 0 -> ( ( z || A /\ z || B ) <-> ( 0 || A /\ 0 || B ) ) )
25	21, 24	bibi12d 235	. . . . . . . . . . 11 |- ( z = 0 -> ( ( z || D <-> ( z || A /\ z || B ) ) <-> ( 0 || D <-> ( 0 || A /\ 0 || B ) ) ) )
26		0zd 9254	. . . . . . . . . . 11 |- ( ph -> 0 e. ZZ )
27	25, 10, 26	rspcdva 2846	. . . . . . . . . 10 |- ( ph -> ( 0 || D <-> ( 0 || A /\ 0 || B ) ) )
28	27	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( 0 || D <-> ( 0 || A /\ 0 || B ) ) )
29	18	nn0zd 9362	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> D e. ZZ )
30		0dvds 11802	. . . . . . . . . 10 |- ( D e. ZZ -> ( 0 || D <-> D = 0 ) )
31	29, 30	syl 14	. . . . . . . . 9 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( 0 || D <-> D = 0 ) )
32		bezoutlemgcd.1	. . . . . . . . . . . 12 |- ( ph -> A e. ZZ )
33	32	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> A e. ZZ )
34		0dvds 11802	. . . . . . . . . . 11 |- ( A e. ZZ -> ( 0 || A <-> A = 0 ) )
35	33, 34	syl 14	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( 0 || A <-> A = 0 ) )
36		bezoutlemgcd.2	. . . . . . . . . . . 12 |- ( ph -> B e. ZZ )
37	36	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> B e. ZZ )
38		0dvds 11802	. . . . . . . . . . 11 |- ( B e. ZZ -> ( 0 || B <-> B = 0 ) )
39	37, 38	syl 14	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( 0 || B <-> B = 0 ) )
40	35, 39	anbi12d 473	. . . . . . . . 9 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( ( 0 || A /\ 0 || B ) <-> ( A = 0 /\ B = 0 ) ) )
41	28, 31, 40	3bitr3d 218	. . . . . . . 8 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( D = 0 <-> ( A = 0 /\ B = 0 ) ) )
42	20, 41	mtbird 673	. . . . . . 7 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> -. D = 0 )
43	42	neqned 2354	. . . . . 6 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> D =/= 0 )
44		elnnne0 9179	. . . . . 6 |- ( D e. NN <-> ( D e. NN0 /\ D =/= 0 ) )
45	18, 43, 44	sylanbrc 417	. . . . 5 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> D e. NN )
46		dvdsle 11833	. . . . 5 |- ( ( z e. ZZ /\ D e. NN ) -> ( z || D -> z <_ D ) )
47	14, 45, 46	syl2anc 411	. . . 4 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( z || D -> z <_ D ) )
48	16, 47	mpd 13	. . 3 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> z <_ D )
49	48	ex 115	. 2 |- ( ( ph /\ z e. ZZ ) -> ( ( z || A /\ z || B ) -> z <_ D ) )
50	49	ralrimiva 2550	1 |- ( ph -> A. z e. ZZ ( ( z || A /\ z || B ) -> z <_ D ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   =/= wne 2347   A.wral 2455   class class class wbr 4000   0cc0 7802   <_ cle 7983   NNcn 8908   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-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  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

This theorem depends on definitions:  df-bi 117  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-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-id 4290  df-po 4293  df-iso 4294  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-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  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-n0 9166  df-z 9243  df-q 9609  df-dvds 11779

This theorem is referenced by:  bezoutlemsup  11993