Theorem gcdsupcl 11647

Description: Closure of the supremum used in defining gcd. A lemma for gcdval 11648 and gcdn0cl 11651. (Contributed by Jim Kingdon, 11-Dec-2021.)

Assertion

Ref	Expression
gcdsupcl	|- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> sup ( { n e. ZZ | ( n || X /\ n || Y ) } , RR , < ) e. NN )

Distinct variable groups:   n, X   n, Y

Proof of Theorem gcdsupcl

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1zzd 9081	. . 3 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> 1 e. ZZ )
2		breq1 3932	. . . 4 |- ( n = 1 -> ( n || X <-> 1 || X ) )
3		breq1 3932	. . . 4 |- ( n = 1 -> ( n || Y <-> 1 || Y ) )
4	2, 3	anbi12d 464	. . 3 |- ( n = 1 -> ( ( n || X /\ n || Y ) <-> ( 1 || X /\ 1 || Y ) ) )
5		1dvds 11507	. . . . 5 |- ( X e. ZZ -> 1 || X )
6		1dvds 11507	. . . . 5 |- ( Y e. ZZ -> 1 || Y )
7	5, 6	anim12i 336	. . . 4 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( 1 || X /\ 1 || Y ) )
8	7	adantr 274	. . 3 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( 1 || X /\ 1 || Y ) )
9		elnnuz 9362	. . . . . . 7 |- ( n e. NN <-> n e. ( ZZ>= ` 1 ) )
10	9	biimpri 132	. . . . . 6 |- ( n e. ( ZZ>= ` 1 ) -> n e. NN )
11		simpll 518	. . . . . 6 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> X e. ZZ )
12		dvdsdc 11501	. . . . . 6 |- ( ( n e. NN /\ X e. ZZ ) -> DECID n || X )
13	10, 11, 12	syl2an2 583	. . . . 5 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID n || X )
14		simplr 519	. . . . . 6 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> Y e. ZZ )
15		dvdsdc 11501	. . . . . 6 |- ( ( n e. NN /\ Y e. ZZ ) -> DECID n || Y )
16	10, 14, 15	syl2an2 583	. . . . 5 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID n || Y )
17		dcan 918	. . . . 5 |- (DECID n || X -> (DECID n || Y -> DECID ( n || X /\ n || Y ) ) )
18	13, 16, 17	sylc 62	. . . 4 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID ( n || X /\ n || Y ) )
19	18	adantlr 468	. . 3 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID ( n || X /\ n || Y ) )
20		simplll 522	. . . . 5 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ X =/= 0 ) -> X e. ZZ )
21		dvdsbnd 11645	. . . . . . 7 |- ( ( X e. ZZ /\ X =/= 0 ) -> E. j e. NN A. n e. ( ZZ>= ` j ) -. n || X )
22		nnuz 9361	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
23	22	rexeqi 2631	. . . . . . 7 |- ( 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	. . . . . 6 |- ( ( X e. ZZ /\ X =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || X )
25		id 19	. . . . . . . . 9 |- ( -. n || X -> -. n || X )
26	25	intnanrd 917	. . . . . . . 8 |- ( -. n || X -> -. ( n || X /\ n || Y ) )
27	26	ralimi 2495	. . . . . . 7 |- ( A. n e. ( ZZ>= ` j ) -. n || X -> A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
28	27	reximi 2529	. . . . . 6 |- ( 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	. . . . 5 |- ( ( X e. ZZ /\ X =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
30	20, 29	sylancom 416	. . . 4 |- ( ( ( ( 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 523	. . . . 5 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ Y =/= 0 ) -> Y e. ZZ )
32		dvdsbnd 11645	. . . . . . 7 |- ( ( Y e. ZZ /\ Y =/= 0 ) -> E. j e. NN A. n e. ( ZZ>= ` j ) -. n || Y )
33	22	rexeqi 2631	. . . . . . 7 |- ( 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	. . . . . 6 |- ( ( Y e. ZZ /\ Y =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || Y )
35		id 19	. . . . . . . . 9 |- ( -. n || Y -> -. n || Y )
36	35	intnand 916	. . . . . . . 8 |- ( -. n || Y -> -. ( n || X /\ n || Y ) )
37	36	ralimi 2495	. . . . . . 7 |- ( A. n e. ( ZZ>= ` j ) -. n || Y -> A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
38	37	reximi 2529	. . . . . 6 |- ( 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	. . . . 5 |- ( ( Y e. ZZ /\ Y =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
40	31, 39	sylancom 416	. . . 4 |- ( ( ( ( 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	. . . . . 6 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> -. ( X = 0 /\ Y = 0 ) )
42		simpll 518	. . . . . . . 8 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> X e. ZZ )
43		0z 9065	. . . . . . . 8 |- 0 e. ZZ
44		zdceq 9126	. . . . . . . 8 |- ( ( X e. ZZ /\ 0 e. ZZ ) -> DECID X = 0 )
45	42, 43, 44	sylancl 409	. . . . . . 7 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> DECID X = 0 )
46		ianordc 884	. . . . . . 7 |- (DECID X = 0 -> ( -. ( X = 0 /\ Y = 0 ) <-> ( -. X = 0 \/ -. Y = 0 ) ) )
47	45, 46	syl 14	. . . . . 6 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( -. ( X = 0 /\ Y = 0 ) <-> ( -. X = 0 \/ -. Y = 0 ) ) )
48	41, 47	mpbid 146	. . . . 5 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( -. X = 0 \/ -. Y = 0 ) )
49		df-ne 2309	. . . . . 6 |- ( X =/= 0 <-> -. X = 0 )
50		df-ne 2309	. . . . . 6 |- ( Y =/= 0 <-> -. Y = 0 )
51	49, 50	orbi12i 753	. . . . 5 |- ( ( X =/= 0 \/ Y =/= 0 ) <-> ( -. X = 0 \/ -. Y = 0 ) )
52	48, 51	sylibr 133	. . . 4 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( X =/= 0 \/ Y =/= 0 ) )
53	30, 40, 52	mpjaodan 787	. . 3 |- ( ( ( 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	zsupcl 11640	. 2 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> sup ( { n e. ZZ | ( n || X /\ n || Y ) } , RR , < ) e. ( ZZ>= ` 1 ) )
55	54, 22	eleqtrrdi 2233	1 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> sup ( { n e. ZZ | ( n || X /\ n || Y ) } , RR , < ) e. NN )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697  DECID wdc 819   = wceq 1331   e. wcel 1480   =/= wne 2308   A.wral 2416   E.wrex 2417   {crab 2420   class class class wbr 3929   ` cfv 5123   supcsup 6869   RRcr 7619   0cc0 7620   1c1 7621   < clt 7800   NNcn 8720   ZZcz 9054   ZZ>=cuz 9326   || cdvds 11493

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-fl 10043  df-mod 10096  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-dvds 11494

This theorem is referenced by:  gcdval  11648  gcdn0cl  11651