Theorem bezoutlemle 12023

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 4018	. . . . . . . 8 |- ( z = w -> ( z || D <-> w || D ) )
3		breq1 4018	. . . . . . . . 9 |- ( z = w -> ( z || A <-> w || A ) )
4		breq1 4018	. . . . . . . . 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 1716	. . . . . . 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 2715	. . . . . . . 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 2858	. . . . 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 4018	. . . . . . . . . . . 12 |- ( z = 0 -> ( z || D <-> 0 || D ) )
22		breq1 4018	. . . . . . . . . . . . 13 |- ( z = 0 -> ( z || A <-> 0 || A ) )
23		breq1 4018	. . . . . . . . . . . . 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 9279	. . . . . . . . . . 11 |- ( ph -> 0 e. ZZ )
27	25, 10, 26	rspcdva 2858	. . . . . . . . . 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 9387	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> D e. ZZ )
30		0dvds 11832	. . . . . . . . . 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 11832	. . . . . . . . . . 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 11832	. . . . . . . . . . 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 674	. . . . . . 7 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> -. D = 0 )
43	42	neqned 2364	. . . . . 6 |- ( ( ( ph /\ z e. ZZ ) /\ ( z || A /\ z || B ) ) -> D =/= 0 )
44		elnnne0 9204	. . . . . 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 11864	. . . . 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 2560	1 |- ( ph -> A. z e. ZZ ( ( z || A /\ z || B ) -> z <_ D ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1363   e. wcel 2158   =/= wne 2357   A.wral 2465   class class class wbr 4015   0cc0 7825   <_ cle 8007   NNcn 8933   NN0cn0 9190   ZZcz 9267   || cdvds 11808

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943

This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-n0 9191  df-z 9268  df-q 9634  df-dvds 11809

This theorem is referenced by:  bezoutlemsup  12024