Theorem bezoutlemsup 12730

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 9571	. . 3 |- ( ph -> D e. RR )
3		elrabi 2973	. . . . . . 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 9718	. . . . 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 4117	. . . . . . . . . 10 |- ( z = w -> ( z || A <-> w || A ) )
8		breq1 4117	. . . . . . . . . 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 2976	. . . . . . . 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 4117	. . . . . . . . 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 12729	. . . . . . . . 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 2928	. . . . . . 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 8411	. . . 4 |- ( ( ph /\ w e. { z e. ZZ | ( z || A /\ z || B ) } ) -> -. D < w )
26	25	ralrimiva 2617	. . 3 |- ( ph -> A. w e. { z e. ZZ | ( z || A /\ z || B ) } -. D < w )
27	1	nn0zd 9716	. . . . . . . . . 10 |- ( ph -> D e. ZZ )
28		iddvds 12515	. . . . . . . . . 10 |- ( D e. ZZ -> D || D )
29	27, 28	syl 14	. . . . . . . . 9 |- ( ph -> D || D )
30		breq1 4117	. . . . . . . . . . 11 |- ( z = D -> ( z || D <-> D || D ) )
31		breq1 4117	. . . . . . . . . . . 12 |- ( z = D -> ( z || A <-> D || A ) )
32		breq1 4117	. . . . . . . . . . . 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 2928	. . . . . . . . 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 9716	. . . . . . . 8 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> D e. ZZ )
40	33	elrab3 2977	. . . . . . . 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 4118	. . . . . . 7 |- ( u = D -> ( w < u <-> w < D ) )
44	43	rspcev 2923	. . . . . 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 2617	. . 3 |- ( ph -> A. w e. RR ( w < D -> E. u e. { z e. ZZ | ( z || A /\ z || B ) } w < u ) )
48		lttri3 8369	. . . . 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 7300	. . 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 1377	. 2 |- ( ph -> sup ( { z e. ZZ | ( z || A /\ z || B ) } , RR , < ) = D )
52	51	eqcomd 2240	1 |- ( ph -> D = sup ( { z e. ZZ | ( z || A /\ z || B ) } , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   {crab 2526   class class class wbr 4114   supcsup 7286   RRcr 8142   0cc0 8143   < clt 8324   <_ cle 8325   NN0cn0 9513   ZZcz 9594   || cdvds 12498

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-n0 9514  df-z 9595  df-q 9970  df-dvds 12499

This theorem is referenced by:  dfgcd3  12731