Theorem bezoutlemsup 11942

Description: Lemma for Bézout's identity. The number satisfying the greatest common divisor condition is the supremum of divisors of 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
bezoutlemsup	|- ( ph -> D = sup ( { z e. ZZ | ( z || A /\ z || B ) } , RR , < ) )

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

Proof of Theorem bezoutlemsup

Dummy variables w f g u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bezoutlemgcd.3	. . . 4 |- ( ph -> D e. NN0 )
2	1	nn0red 9168	. . 3 |- ( ph -> D e. RR )
3		elrabi 2879	. . . . . . 7 |- ( w e. { z e. ZZ | ( z || A /\ z || B ) } -> w e. ZZ )
4	3	adantl 275	. . . . . 6 |- ( ( ph /\ w e. { z e. ZZ | ( z || A /\ z || B ) } ) -> w e. ZZ )
5	4	zred 9313	. . . . 5 |- ( ( ph /\ w e. { z e. ZZ | ( z || A /\ z || B ) } ) -> w e. RR )
6	2	adantr 274	. . . . 5 |- ( ( ph /\ w e. { z e. ZZ | ( z || A /\ z || B ) } ) -> D e. RR )
7		breq1 3985	. . . . . . . . . 10 |- ( z = w -> ( z || A <-> w || A ) )
8		breq1 3985	. . . . . . . . . 10 |- ( z = w -> ( z || B <-> w || B ) )
9	7, 8	anbi12d 465	. . . . . . . . 9 |- ( z = w -> ( ( z || A /\ z || B ) <-> ( w || A /\ w || B ) ) )
10	9	elrab 2882	. . . . . . . 8 |- ( w e. { z e. ZZ | ( z || A /\ z || B ) } <-> ( w e. ZZ /\ ( w || A /\ w || B ) ) )
11	10	simprbi 273	. . . . . . 7 |- ( w e. { z e. ZZ | ( z || A /\ z || B ) } -> ( w || A /\ w || B ) )
12	11	adantl 275	. . . . . 6 |- ( ( ph /\ w e. { z e. ZZ | ( z || A /\ z || B ) } ) -> ( w || A /\ w || B ) )
13		breq1 3985	. . . . . . . . 9 |- ( z = w -> ( z <_ D <-> w <_ D ) )
14	9, 13	imbi12d 233	. . . . . . . 8 |- ( z = w -> ( ( ( z || A /\ z || B ) -> z <_ D ) <-> ( ( w || A /\ w || B ) -> w <_ D ) ) )
15		bezoutlemgcd.1	. . . . . . . . . 10 |- ( ph -> A e. ZZ )
16		bezoutlemgcd.2	. . . . . . . . . 10 |- ( ph -> B e. ZZ )
17		bezoutlemgcd.4	. . . . . . . . . 10 |- ( ph -> A. z e. ZZ ( z || D <-> ( z || A /\ z || B ) ) )
18		bezoutlemgcd.5	. . . . . . . . . 10 |- ( ph -> -. ( A = 0 /\ B = 0 ) )
19	15, 16, 1, 17, 18	bezoutlemle 11941	. . . . . . . . 9 |- ( ph -> A. z e. ZZ ( ( z || A /\ z || B ) -> z <_ D ) )
20	19	adantr 274	. . . . . . . 8 |- ( ( ph /\ w e. ZZ ) -> A. z e. ZZ ( ( z || A /\ z || B ) -> z <_ D ) )
21		simpr 109	. . . . . . . 8 |- ( ( ph /\ w e. ZZ ) -> w e. ZZ )
22	14, 20, 21	rspcdva 2835	. . . . . . 7 |- ( ( ph /\ w e. ZZ ) -> ( ( w || A /\ w || B ) -> w <_ D ) )
23	3, 22	sylan2 284	. . . . . 6 |- ( ( ph /\ w e. { z e. ZZ | ( z || A /\ z || B ) } ) -> ( ( w || A /\ w || B ) -> w <_ D ) )
24	12, 23	mpd 13	. . . . 5 |- ( ( ph /\ w e. { z e. ZZ | ( z || A /\ z || B ) } ) -> w <_ D )
25	5, 6, 24	lensymd 8020	. . . 4 |- ( ( ph /\ w e. { z e. ZZ | ( z || A /\ z || B ) } ) -> -. D < w )
26	25	ralrimiva 2539	. . 3 |- ( ph -> A. w e. { z e. ZZ | ( z || A /\ z || B ) } -. D < w )
27	1	nn0zd 9311	. . . . . . . . . 10 |- ( ph -> D e. ZZ )
28		iddvds 11744	. . . . . . . . . 10 |- ( D e. ZZ -> D || D )
29	27, 28	syl 14	. . . . . . . . 9 |- ( ph -> D || D )
30		breq1 3985	. . . . . . . . . . 11 |- ( z = D -> ( z || D <-> D || D ) )
31		breq1 3985	. . . . . . . . . . . 12 |- ( z = D -> ( z || A <-> D || A ) )
32		breq1 3985	. . . . . . . . . . . 12 |- ( z = D -> ( z || B <-> D || B ) )
33	31, 32	anbi12d 465	. . . . . . . . . . 11 |- ( z = D -> ( ( z || A /\ z || B ) <-> ( D || A /\ D || B ) ) )
34	30, 33	bibi12d 234	. . . . . . . . . 10 |- ( z = D -> ( ( z || D <-> ( z || A /\ z || B ) ) <-> ( D || D <-> ( D || A /\ D || B ) ) ) )
35	34, 17, 27	rspcdva 2835	. . . . . . . . 9 |- ( ph -> ( D || D <-> ( D || A /\ D || B ) ) )
36	29, 35	mpbid 146	. . . . . . . 8 |- ( ph -> ( D || A /\ D || B ) )
37	36	ad2antrr 480	. . . . . . 7 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> ( D || A /\ D || B ) )
38	1	ad2antrr 480	. . . . . . . . 9 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> D e. NN0 )
39	38	nn0zd 9311	. . . . . . . 8 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> D e. ZZ )
40	33	elrab3 2883	. . . . . . . 8 |- ( D e. ZZ -> ( D e. { z e. ZZ | ( z || A /\ z || B ) } <-> ( D || A /\ D || B ) ) )
41	39, 40	syl 14	. . . . . . 7 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> ( D e. { z e. ZZ | ( z || A /\ z || B ) } <-> ( D || A /\ D || B ) ) )
42	37, 41	mpbird 166	. . . . . 6 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> D e. { z e. ZZ | ( z || A /\ z || B ) } )
43		breq2 3986	. . . . . . 7 |- ( u = D -> ( w < u <-> w < D ) )
44	43	rspcev 2830	. . . . . 6 |- ( ( D e. { z e. ZZ | ( z || A /\ z || B ) } /\ w < D ) -> E. u e. { z e. ZZ | ( z || A /\ z || B ) } w < u )
45	42, 44	sylancom 417	. . . . 5 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> E. u e. { z e. ZZ | ( z || A /\ z || B ) } w < u )
46	45	ex 114	. . . 4 |- ( ( ph /\ w e. RR ) -> ( w < D -> E. u e. { z e. ZZ | ( z || A /\ z || B ) } w < u ) )
47	46	ralrimiva 2539	. . 3 |- ( ph -> A. w e. RR ( w < D -> E. u e. { z e. ZZ | ( z || A /\ z || B ) } w < u ) )
48		lttri3 7978	. . . . 5 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
49	48	adantl 275	. . . 4 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
50	49	eqsupti 6961	. . 3 |- ( ph -> ( ( D e. RR /\ A. w e. { z e. ZZ | ( z || A /\ z || B ) } -. D < w /\ A. w e. RR ( w < D -> E. u e. { z e. ZZ | ( z || A /\ z || B ) } w < u ) ) -> sup ( { z e. ZZ | ( z || A /\ z || B ) } , RR , < ) = D ) )
51	2, 26, 47, 50	mp3and 1330	. 2 |- ( ph -> sup ( { z e. ZZ | ( z || A /\ z || B ) } , RR , < ) = D )
52	51	eqcomd 2171	1 |- ( ph -> D = sup ( { z e. ZZ | ( z || A /\ z || B ) } , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   {crab 2448   class class class wbr 3982   supcsup 6947   RRcr 7752   0cc0 7753   < clt 7933   <_ cle 7934   NN0cn0 9114   ZZcz 9191   || cdvds 11727

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-n0 9115  df-z 9192  df-q 9558  df-dvds 11728

This theorem is referenced by:  dfgcd3  11943