Theorem gcdsupex 11441

Description: Existence of the supremum used in defining gcd. (Contributed by Jim Kingdon, 12-Dec-2021.)

Assertion

Ref	Expression
gcdsupex	|- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> E. x e. ZZ ( A. y e. { n e. ZZ | ( n || X /\ n || Y ) } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ( n || X /\ n || Y ) } y < z ) ) )

Distinct variable groups:   n, X, x, y, z   n, Y, x, y, z

Proof of Theorem gcdsupex

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1zzd 8933	. 2 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> 1 e. ZZ )
2		breq1 3878	. . 3 |- ( n = 1 -> ( n || X <-> 1 || X ) )
3		breq1 3878	. . 3 |- ( n = 1 -> ( n || Y <-> 1 || Y ) )
4	2, 3	anbi12d 460	. 2 |- ( n = 1 -> ( ( n || X /\ n || Y ) <-> ( 1 || X /\ 1 || Y ) ) )
5		1dvds 11302	. . . 4 |- ( X e. ZZ -> 1 || X )
6		1dvds 11302	. . . 4 |- ( Y e. ZZ -> 1 || Y )
7	5, 6	anim12i 334	. . 3 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( 1 || X /\ 1 || Y ) )
8	7	adantr 272	. 2 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( 1 || X /\ 1 || Y ) )
9		elnnuz 9212	. . . . . 6 |- ( n e. NN <-> n e. ( ZZ>= ` 1 ) )
10	9	biimpri 132	. . . . 5 |- ( n e. ( ZZ>= ` 1 ) -> n e. NN )
11		simpll 499	. . . . 5 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> X e. ZZ )
12		dvdsdc 11296	. . . . 5 |- ( ( n e. NN /\ X e. ZZ ) -> DECID n || X )
13	10, 11, 12	syl2an2 564	. . . 4 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID n || X )
14		simplr 500	. . . . 5 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> Y e. ZZ )
15		dvdsdc 11296	. . . . 5 |- ( ( n e. NN /\ Y e. ZZ ) -> DECID n || Y )
16	10, 14, 15	syl2an2 564	. . . 4 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID n || Y )
17		dcan 886	. . . 4 |- (DECID n || X -> (DECID n || Y -> DECID ( n || X /\ n || Y ) ) )
18	13, 16, 17	sylc 62	. . 3 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID ( n || X /\ n || Y ) )
19	18	adantlr 464	. 2 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID ( n || X /\ n || Y ) )
20		simplll 503	. . . 4 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ X =/= 0 ) -> X e. ZZ )
21		dvdsbnd 11440	. . . . . 6 |- ( ( X e. ZZ /\ X =/= 0 ) -> E. j e. NN A. n e. ( ZZ>= ` j ) -. n || X )
22		nnuz 9211	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
23	22	rexeqi 2589	. . . . . 6 |- ( E. j e. NN A. n e. ( ZZ>= ` j ) -. n || X <-> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || X )
24	21, 23	sylib 121	. . . . 5 |- ( ( X e. ZZ /\ X =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || X )
25		id 19	. . . . . . . 8 |- ( -. n || X -> -. n || X )
26	25	intnanrd 885	. . . . . . 7 |- ( -. n || X -> -. ( n || X /\ n || Y ) )
27	26	ralimi 2454	. . . . . 6 |- ( A. n e. ( ZZ>= ` j ) -. n || X -> A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
28	27	reximi 2488	. . . . 5 |- ( E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || X -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
29	24, 28	syl 14	. . . 4 |- ( ( X e. ZZ /\ X =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
30	20, 29	sylancom 414	. . 3 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ X =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
31		simpllr 504	. . . 4 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ Y =/= 0 ) -> Y e. ZZ )
32		dvdsbnd 11440	. . . . . 6 |- ( ( Y e. ZZ /\ Y =/= 0 ) -> E. j e. NN A. n e. ( ZZ>= ` j ) -. n || Y )
33	22	rexeqi 2589	. . . . . 6 |- ( E. j e. NN A. n e. ( ZZ>= ` j ) -. n || Y <-> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || Y )
34	32, 33	sylib 121	. . . . 5 |- ( ( Y e. ZZ /\ Y =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || Y )
35		id 19	. . . . . . . 8 |- ( -. n || Y -> -. n || Y )
36	35	intnand 884	. . . . . . 7 |- ( -. n || Y -> -. ( n || X /\ n || Y ) )
37	36	ralimi 2454	. . . . . 6 |- ( A. n e. ( ZZ>= ` j ) -. n || Y -> A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
38	37	reximi 2488	. . . . 5 |- ( E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || Y -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
39	34, 38	syl 14	. . . 4 |- ( ( Y e. ZZ /\ Y =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
40	31, 39	sylancom 414	. . 3 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ Y =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
41		simpr 109	. . . . 5 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> -. ( X = 0 /\ Y = 0 ) )
42		simpll 499	. . . . . . 7 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> X e. ZZ )
43		0z 8917	. . . . . . 7 |- 0 e. ZZ
44		zdceq 8978	. . . . . . 7 |- ( ( X e. ZZ /\ 0 e. ZZ ) -> DECID X = 0 )
45	42, 43, 44	sylancl 407	. . . . . 6 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> DECID X = 0 )
46		ianordc 843	. . . . . 6 |- (DECID X = 0 -> ( -. ( X = 0 /\ Y = 0 ) <-> ( -. X = 0 \/ -. Y = 0 ) ) )
47	45, 46	syl 14	. . . . 5 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( -. ( X = 0 /\ Y = 0 ) <-> ( -. X = 0 \/ -. Y = 0 ) ) )
48	41, 47	mpbid 146	. . . 4 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( -. X = 0 \/ -. Y = 0 ) )
49		df-ne 2268	. . . . 5 |- ( X =/= 0 <-> -. X = 0 )
50		df-ne 2268	. . . . 5 |- ( Y =/= 0 <-> -. Y = 0 )
51	49, 50	orbi12i 722	. . . 4 |- ( ( X =/= 0 \/ Y =/= 0 ) <-> ( -. X = 0 \/ -. Y = 0 ) )
52	48, 51	sylibr 133	. . 3 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( X =/= 0 \/ Y =/= 0 ) )
53	30, 40, 52	mpjaodan 753	. 2 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
54	1, 4, 8, 19, 53	zsupcllemex 11434	1 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> E. x e. ZZ ( A. y e. { n e. ZZ | ( n || X /\ n || Y ) } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ( n || X /\ n || Y ) } y < z ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 670  DECID wdc 786   = wceq 1299   e. wcel 1448   =/= wne 2267   A.wral 2375   E.wrex 2376   {crab 2379   class class class wbr 3875   ` cfv 5059   RRcr 7499   0cc0 7500   1c1 7501   < clt 7672   NNcn 8578   ZZcz 8906   ZZ>=cuz 9176   || cdvds 11288

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-fz 9632  df-fzo 9761  df-fl 9884  df-mod 9937  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-dvds 11289

This theorem is referenced by:  gcddvds  11447  dvdslegcd  11448