Theorem bezoutlemsup 12786

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 9621	. . 3 |- ( ph -> D e. RR )
3		elrabi 2979	. . . . . . 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 9768	. . . . 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 4133	. . . . . . . . . 10 |- ( z = w -> ( z || A <-> w || A ) )
8		breq1 4133	. . . . . . . . . 10 |- ( z = w -> ( z || B <-> w || B ) )
9	7, 8	anbi12d 477	. . . . . . . . 9 |- ( z = w -> ( ( z || A /\ z || B ) <-> ( w || A /\ w || B ) ) )
10	9	elrab 2982	. . . . . . . 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 4133	. . . . . . . . 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 12785	. . . . . . . . 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 2934	. . . . . . 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 8448	. . . 4 |- ( ( ph /\ w e. { z e. ZZ | ( z || A /\ z || B ) } ) -> -. D < w )
26	25	ralrimiva 2623	. . 3 |- ( ph -> A. w e. { z e. ZZ | ( z || A /\ z || B ) } -. D < w )
27	1	nn0zd 9766	. . . . . . . . . 10 |- ( ph -> D e. ZZ )
28		iddvds 12571	. . . . . . . . . 10 |- ( D e. ZZ -> D || D )
29	27, 28	syl 14	. . . . . . . . 9 |- ( ph -> D || D )
30		breq1 4133	. . . . . . . . . . 11 |- ( z = D -> ( z || D <-> D || D ) )
31		breq1 4133	. . . . . . . . . . . 12 |- ( z = D -> ( z || A <-> D || A ) )
32		breq1 4133	. . . . . . . . . . . 12 |- ( z = D -> ( z || B <-> D || B ) )
33	31, 32	anbi12d 477	. . . . . . . . . . 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 2934	. . . . . . . . 9 |- ( ph -> ( D || D <-> ( D || A /\ D || B ) ) )
36	29, 35	mpbid 147	. . . . . . . 8 |- ( ph -> ( D || A /\ D || B ) )
37	36	ad2antrr 492	. . . . . . 7 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> ( D || A /\ D || B ) )
38	1	ad2antrr 492	. . . . . . . . 9 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> D e. NN0 )
39	38	nn0zd 9766	. . . . . . . 8 |- ( ( ( ph /\ w e. RR ) /\ w < D ) -> D e. ZZ )
40	33	elrab3 2983	. . . . . . . 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 4134	. . . . . . 7 |- ( u = D -> ( w < u <-> w < D ) )
44	43	rspcev 2929	. . . . . 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 424	. . . . 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 2623	. . 3 |- ( ph -> A. w e. RR ( w < D -> E. u e. { z e. ZZ | ( z || A /\ z || B ) } w < u ) )
48		lttri3 8405	. . . . 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 7336	. . 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 1381	. 2 |- ( ph -> sup ( { z e. ZZ | ( z || A /\ z || B ) } , RR , < ) = D )
52	51	eqcomd 2244	1 |- ( ph -> D = sup ( { z e. ZZ | ( z || A /\ z || B ) } , RR , < ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   class class class wbr 4130   supcsup 7322   RRcr 8178   0cc0 8179   < clt 8360   <_ cle 8361   NN0cn0 9563   ZZcz 9644   || cdvds 12554

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-n0 9564  df-z 9645  df-q 10020  df-dvds 12555

This theorem is used by:  dfgcd3  12787