Theorem bezoutlemle 11963

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 109	. . . . 5 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( z || A /\ z || B ) )
2		breq1 3992	. . . . . . . 8 |- ( z = w -> ( z || D <-> w || D ) )
3		breq1 3992	. . . . . . . . 9 |- ( z = w -> ( z || A <-> w || A ) )
4		breq1 3992	. . . . . . . . 9 |- ( z = w -> ( z || B <-> w || B ) )
5	3, 4	anbi12d 470	. . . . . . . 8 |- ( z = w -> ( ( z || A /\ z || B ) <-> ( w || A /\ w || B ) ) )
6	2, 5	bibi12d 234	. . . . . . 7 |- ( z = w -> ( ( z || D <-> ( z || A /\ z || B ) ) <-> ( w || D <-> ( w || A /\ w || B ) ) ) )
7		equcom 1699	. . . . . . 7 |- ( z = w <-> w = z )
8		bicom 139	. . . . . . 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 199	. . . . . 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 2696	. . . . . . . 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 121	. . . . . . 7 |- ( ph -> A. w e. ZZ ( w || D <-> ( w || A /\ w || B ) ) )
13	12	ad2antrr 485	. . . . . 6 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> A. w e. ZZ ( w || D <-> ( w || A /\ w || B ) ) )
14		simplr 525	. . . . . 6 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> z e. ZZ )
15	9, 13, 14	rspcdva 2839	. . . . 5 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( z || D <-> ( z || A /\ z || B ) ) )
16	1, 15	mpbird 166	. . . 4 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> z || D )
17		bezoutlemgcd.3	. . . . . . 7 |- ( ph -> D e. NN0 )
18	17	ad2antrr 485	. . . . . 6 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> D e. NN0 )
19		bezoutlemgcd.5	. . . . . . . . 9 |- ( ph -> -. ( A = 0 /\ B = 0 ) )
20	19	ad2antrr 485	. . . . . . . 8 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> -. ( A = 0 /\ B = 0 ) )
21		breq1 3992	. . . . . . . . . . . 12 |- ( z = 0 -> ( z || D <-> 0 || D ) )
22		breq1 3992	. . . . . . . . . . . . 13 |- ( z = 0 -> ( z || A <-> 0 || A ) )
23		breq1 3992	. . . . . . . . . . . . 13 |- ( z = 0 -> ( z || B <-> 0 || B ) )
24	22, 23	anbi12d 470	. . . . . . . . . . . 12 |- ( z = 0 -> ( ( z || A /\ z || B ) <-> ( 0 || A /\ 0 || B ) ) )
25	21, 24	bibi12d 234	. . . . . . . . . . 11 |- ( z = 0 -> ( ( z || D <-> ( z || A /\ z || B ) ) <-> ( 0 || D <-> ( 0 || A /\ 0 || B ) ) ) )
26		0zd 9224	. . . . . . . . . . 11 |- ( ph -> 0 e. ZZ )
27	25, 10, 26	rspcdva 2839	. . . . . . . . . 10 |- ( ph -> ( 0 || D <-> ( 0 || A /\ 0 || B ) ) )
28	27	ad2antrr 485	. . . . . . . . 9 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( 0 || D <-> ( 0 || A /\ 0 || B ) ) )
29	18	nn0zd 9332	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> D e. ZZ )
30		0dvds 11773	. . . . . . . . . 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 485	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> A e. ZZ )
34		0dvds 11773	. . . . . . . . . . 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 485	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> B e. ZZ )
38		0dvds 11773	. . . . . . . . . . 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 470	. . . . . . . . 9 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( ( 0 || A /\ 0 || B ) <-> ( A = 0 /\ B = 0 ) ) )
41	28, 31, 40	3bitr3d 217	. . . . . . . 8 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> ( D = 0 <-> ( A = 0 /\ B = 0 ) ) )
42	20, 41	mtbird 668	. . . . . . 7 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> -. D = 0 )
43	42	neqned 2347	. . . . . 6 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> D =/= 0 )
44		elnnne0 9149	. . . . . 6 |- ( D e. NN <-> ( D e. NN0 /\ D =/= 0 ) )
45	18, 43, 44	sylanbrc 415	. . . . 5 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> D e. NN )
46		dvdsle 11804	. . . . 5 |- ( ( z e. ZZ /\ D e. NN ) -> ( z || D -> z <_ D ) )
47	14, 45, 46	syl2anc 409	. . . 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 114	. 2 |- ( ( ph /\ z e. ZZ ) -> ( ( z || A /\ z || B ) -> z <_ D ) )
50	49	ralrimiva 2543	1 |- ( ph -> A. z e. ZZ ( ( z || A /\ z || B ) -> z <_ D ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   =/= wne 2340   A.wral 2448   class class class wbr 3989   0cc0 7774   <_ cle 7955   NNcn 8878   NN0cn0 9135   ZZcz 9212   || cdvds 11749

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-q 9579  df-dvds 11750

This theorem is referenced by:  bezoutlemsup  11964