Theorem bezoutlemsup 11733

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 9055	. . 3 |- ( ph -> D e. RR )
3		elrabi 2841	. . . . . . 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 9197	. . . . 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 3940	. . . . . . . . . 10 |- ( z = w -> ( z || A <-> w || A ) )
8		breq1 3940	. . . . . . . . . 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 2844	. . . . . . . 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 3940	. . . . . . . . 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 11732	. . . . . . . . 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 2798	. . . . . . 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 7908	. . . 4 |- ( ( ph /\ w e. { z e. ZZ | ( z || A /\ z || B ) } ) -> -. D < w )
26	25	ralrimiva 2508	. . 3 |- ( ph -> A. w e. { z e. ZZ | ( z || A /\ z || B ) } -. D < w )
27	1	nn0zd 9195	. . . . . . . . . 10 |- ( ph -> D e. ZZ )
28		iddvds 11542	. . . . . . . . . 10 |- ( D e. ZZ -> D || D )
29	27, 28	syl 14	. . . . . . . . 9 |- ( ph -> D || D )
30		breq1 3940	. . . . . . . . . . 11 |- ( z = D -> ( z || D <-> D || D ) )
31		breq1 3940	. . . . . . . . . . . 12 |- ( z = D -> ( z || A <-> D || A ) )
32		breq1 3940	. . . . . . . . . . . 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 2798	. . . . . . . . 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 9195	. . . . . . . 8 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> D e. ZZ )
40	33	elrab3 2845	. . . . . . . 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 3941	. . . . . . 7 |- ( u = D -> ( w < u <-> w < D ) )
44	43	rspcev 2793	. . . . . 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 2508	. . 3 |- ( ph -> A. w e. RR ( w < D -> E. u e. { z e. ZZ | ( z || A /\ z || B ) } w < u ) )
48		lttri3 7868	. . . . 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 6891	. . 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 1319	. 2 |- ( ph -> sup ( { z e. ZZ | ( z || A /\ z || B ) } , RR , < ) = D )
52	51	eqcomd 2146	1 |- ( ph -> D = sup ( { z e. ZZ | ( z || A /\ z || B ) } , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421   class class class wbr 3937   supcsup 6877   RRcr 7643   0cc0 7644   < clt 7824   <_ cle 7825   NN0cn0 9001   ZZcz 9078   || cdvds 11529

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-po 4226  df-iso 4227  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-sup 6879  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-n0 9002  df-z 9079  df-q 9439  df-dvds 11530

This theorem is referenced by:  dfgcd3  11734