Theorem bezoutlemsup 12530

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 9423	. . 3 |- ( ph -> D e. RR )
3		elrabi 2956	. . . . . . 7 |- ( w e. { z e. ZZ | ( z || A /\ z || B ) } -> w e. ZZ )
4	3	adantl 277	. . . . . 6 |- ( ( ph /\ w e. { z e. ZZ | ( z || A /\ z || B ) } ) -> w e. ZZ )
5	4	zred 9569	. . . . 5 |- ( ( ph /\ w e. { z e. ZZ | ( z || A /\ z || B ) } ) -> w e. RR )
6	2	adantr 276	. . . . 5 |- ( ( ph /\ w e. { z e. ZZ | ( z || A /\ z || B ) } ) -> D e. RR )
7		breq1 4086	. . . . . . . . . 10 |- ( z = w -> ( z || A <-> w || A ) )
8		breq1 4086	. . . . . . . . . 10 |- ( z = w -> ( z || B <-> w || B ) )
9	7, 8	anbi12d 473	. . . . . . . . 9 |- ( z = w -> ( ( z || A /\ z || B ) <-> ( w || A /\ w || B ) ) )
10	9	elrab 2959	. . . . . . . 8 |- ( w e. { z e. ZZ | ( z || A /\ z || B ) } <-> ( w e. ZZ /\ ( w || A /\ w || B ) ) )
11	10	simprbi 275	. . . . . . 7 |- ( w e. { z e. ZZ | ( z || A /\ z || B ) } -> ( w || A /\ w || B ) )
12	11	adantl 277	. . . . . 6 |- ( ( ph /\ w e. { z e. ZZ | ( z || A /\ z || B ) } ) -> ( w || A /\ w || B ) )
13		breq1 4086	. . . . . . . . 9 |- ( z = w -> ( z <_ D <-> w <_ D ) )
14	9, 13	imbi12d 234	. . . . . . . 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 12529	. . . . . . . . 9 |- ( ph -> A. z e. ZZ ( ( z || A /\ z || B ) -> z <_ D ) )
20	19	adantr 276	. . . . . . . 8 |- ( ( ph /\ w e. ZZ ) -> A. z e. ZZ ( ( z || A /\ z || B ) -> z <_ D ) )
21		simpr 110	. . . . . . . 8 |- ( ( ph /\ w e. ZZ ) -> w e. ZZ )
22	14, 20, 21	rspcdva 2912	. . . . . . 7 |- ( ( ph /\ w e. ZZ ) -> ( ( w || A /\ w || B ) -> w <_ D ) )
23	3, 22	sylan2 286	. . . . . 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 8268	. . . 4 |- ( ( ph /\ w e. { z e. ZZ | ( z || A /\ z || B ) } ) -> -. D < w )
26	25	ralrimiva 2603	. . 3 |- ( ph -> A. w e. { z e. ZZ | ( z || A /\ z || B ) } -. D < w )
27	1	nn0zd 9567	. . . . . . . . . 10 |- ( ph -> D e. ZZ )
28		iddvds 12315	. . . . . . . . . 10 |- ( D e. ZZ -> D || D )
29	27, 28	syl 14	. . . . . . . . 9 |- ( ph -> D || D )
30		breq1 4086	. . . . . . . . . . 11 |- ( z = D -> ( z || D <-> D || D ) )
31		breq1 4086	. . . . . . . . . . . 12 |- ( z = D -> ( z || A <-> D || A ) )
32		breq1 4086	. . . . . . . . . . . 12 |- ( z = D -> ( z || B <-> D || B ) )
33	31, 32	anbi12d 473	. . . . . . . . . . 11 |- ( z = D -> ( ( z || A /\ z || B ) <-> ( D || A /\ D || B ) ) )
34	30, 33	bibi12d 235	. . . . . . . . . 10 |- ( z = D -> ( ( z || D <-> ( z || A /\ z || B ) ) <-> ( D || D <-> ( D || A /\ D || B ) ) ) )
35	34, 17, 27	rspcdva 2912	. . . . . . . . 9 |- ( ph -> ( D || D <-> ( D || A /\ D || B ) ) )
36	29, 35	mpbid 147	. . . . . . . 8 |- ( ph -> ( D || A /\ D || B ) )
37	36	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> ( D || A /\ D || B ) )
38	1	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> D e. NN0 )
39	38	nn0zd 9567	. . . . . . . 8 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> D e. ZZ )
40	33	elrab3 2960	. . . . . . . 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 167	. . . . . 6 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> D e. { z e. ZZ | ( z || A /\ z || B ) } )
43		breq2 4087	. . . . . . 7 |- ( u = D -> ( w < u <-> w < D ) )
44	43	rspcev 2907	. . . . . 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 420	. . . . 5 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> E. u e. { z e. ZZ | ( z || A /\ z || B ) } w < u )
46	45	ex 115	. . . 4 |- ( ( ph /\ w e. RR ) -> ( w < D -> E. u e. { z e. ZZ | ( z || A /\ z || B ) } w < u ) )
47	46	ralrimiva 2603	. . 3 |- ( ph -> A. w e. RR ( w < D -> E. u e. { z e. ZZ | ( z || A /\ z || B ) } w < u ) )
48		lttri3 8226	. . . . 5 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
49	48	adantl 277	. . . 4 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
50	49	eqsupti 7163	. . 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 1374	. 2 |- ( ph -> sup ( { z e. ZZ | ( z || A /\ z || B ) } , RR , < ) = D )
52	51	eqcomd 2235	1 |- ( ph -> D = sup ( { z e. ZZ | ( z || A /\ z || B ) } , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   {crab 2512   class class class wbr 4083   supcsup 7149   RRcr 7998   0cc0 7999   < clt 8181   <_ cle 8182   NN0cn0 9369   ZZcz 9446   || cdvds 12298

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-sup 7151  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-n0 9370  df-z 9447  df-q 9815  df-dvds 12299

This theorem is referenced by:  dfgcd3  12531